\section{Introduction}
Spin Glasses (SGs) are models whose mean field (MF) version \cite{SK} undergoes a phase transition, crossing
a critical line in the temperature-field $(T-h)$ plane.
The solution of the MF problem sees the introduction of replicas of the original system as a mathematical trick to perform computations. The resulting Hamiltonian is symmetric under
replica exchanges. However, quite surprising, one finds that in the low-temperature spin-glass phase the replica symmetry is broken.
While the MF behavior of the model is completely under control \cite{FRSB}, also from a rigorous viewpoint \cite{PAN}, we still do not have a confirmed theory for the finite dimensional version.
In particular, there is not agreement both on the upper and lower critical dimension, looking at theoretical, numerical and experimental data
\cite{Moore2011,ReplyMoore2011,SG_MK,Yaida,LR1,LR2,LR3,Janus,experimental}.

The project to perform a renormalization group (RG) analysis is an old one.
The spin glass transition in zero-field was already studied within the RG by Harris et al. \cite{Harris}, and in field by Bray and Roberts \cite{BR}, limiting at the
sector associated with the critical eigenvalue, the so called \textit{replicon}.
Their one-loop analysis was then repeated adding the other sectors, longitudinal and anomalous ones, in refs. \cite{Temesvari1,Temesvari2,Temesvari3,Temesvari4}.
In a recent work also the two loop computation in a field has been performed \cite{Yaida}, suggesting the possibility of a non-perturbative fixed point.

These works are expansions around the Fully Connected (FC) mean field model.
They start studying  the symmetric phase,  approaching  the transition from the high-temperature side, where replica symmetry holds.
Thus the replica symmetric Lagrangian is written, that in its most complete version has three bare masses and eight cubic couplings involving the
replica fields, which correspond to all the possible invariants under the replica symmetry.
At this point one can perform a renormalization \'a la Wilson, integrating the degrees of freedom over an infinitesimal momentum shell, extracting the leading, one-loop, order approximation
in $\epsilon=6-d$.
Although the scheme is clear, the computation is highly technical also for the algebraic viewpoint.

Recently a new loop expansion around the mean field Bethe solution was proposed in ref. \cite{Mlayer}. The new method can be applied to each model that is well defined on a Bethe lattice.
In this paper, we apply for the first time this new expansion to the SG in a field.
We restrict ourself to the limit of high connectivity $z\rightarrow\infty$ to perform computations analytically.
We compute the 1th order correction, and we show that in the $T>0$ region this new expansion is completely equivalent to the field theoretical one,
recovering the results of Bray and Roberts \cite{BR}.
However it has the advantage that the starting point is the original Hamiltonian of the model, with no need to define an associated field theory, nor to
know the initial values of the couplings, and the computations have a clear and simple
physical meaning: while in standard field theory Feynmann diagrams have no special meaning, here the important diagrams have a geometrical interpretation.
Moreover, the expansion is around Bethe lattice that has finite connectivity, an important characteristic shared with finite dimensional systems.
Even if in this work we obtain the same results as standard RG around the fully-connected model,
we will discuss the differences that could arise in the two methods in particular situations.
This work is first of all a verification of the correctness of the method proposed in ref. \cite{Mlayer} and it is a first step towards the generalization
of this new expansion to more complicated cases: finite small connectivity and zero temperature.

\section{Expanding around the Bethe lattice solution}

In this section we just recap the results of ref. \cite{Mlayer}, while in following sections we will apply for the first time these results to the SG model in a field.
Starting from a $D$ dimensional system, the $M$-layer construction of ref. \cite{Mlayer} consists of taking $M$ copies of the original model and rewire them.
In the $M\rightarrow\infty$ limit, the rewiring procedure leads to the Bethe lattice. One can then expand the observables in powers of $\frac{1}{M}$.
In particular, we will focus our attention on the correlation functions, connected over the disorder, let's name them $G$.
The $\frac{1}{M}$ expansion results in a topological expansion in the number of loops.
At order $\frac{1}{M}$, the correlation function between the origin and a point $d$ in the $D$ dimensional model results to be:
$G(d)=\sum_{L=1}^{\infty}{\cal B}(d,L) g^B(L)$, where ${\cal B}(d,L)$ is the number of non-backtracking walks that go from the origin to the point $d$ in the original $D$ dimensional model
and $g^B(L)$ is the correlation between points at distance $L$ on a Bethe lattice.

\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{DiagramBethe}
\caption{\label{Fig:loop} Spatial loop $\mathcal{L}$ that gives the first correction to the bare correlation functions in the expansion around the Bethe solution.}
\end{figure}
At order $\frac{1}{M^2}$, the correlation function on the original system receive a leading contribution that is
the product of the so-called \textit{line-connected} observable $g^{lc}(\cal L)$ computed on a Bethe lattice,
in which it has been manually injected a loop ${\cal L}$ of the type in Fig. \ref{Fig:loop},
multiplied by the number of such a structure ${\cal L}$ present on the original model.
The line-connected observable is just the observable computed on a Bethe lattice with the loop minus the observable computed on the two paths $L_0,L_1+1,L_3$ and $L_0,L_2+1,L_3$
considered as independent.

The quantities $g^B(L)$ and $g^{lc}(\cal L)$ are model dependent.
In the following we will compute them for the Spin Glass in a field.
To make the computation analytically feasible, we will compute things on a Bethe lattice in the high connectivity limit, at temperature $T>0$.
However things can be computed in finite connectivity and even at $T=0$ using numerically the Belief Propagation equations. This will be the subject of a subsequent paper.

We just want to recall that one could perform the same $M$-layer construction around the fully connected model instead of the Bethe lattice. In the former approach, the leading
divergences at each order are exactly given by the corresponding terms in the loop expansion of the continuum  field theory.
In ref. \cite{Mlayer} it is claimed that, if the critical behaviour of the Fully connected model and of the Bethe lattice model is the same, than also the two expansions will lead to the
same results. We will see that this is exactly the case for the SG with field at $T>0$.

\section{Model and definitions}

To be concrete, we are interested in the SG model in field, that has the following Hamiltonian:
$$
H(\{{\sigma\}})=-\sum_{i=1,N} \sigma_i h_i^R -\sum_{ij\in E} \sigma_i \sigma_j J_{ij}\,,
$$
where $E$ is the set of the edges of the lattice. On this model we should compute the quantities $g^B(L)$ and $g^{lc}(\cal L)$ introduced in the previous section.

We will consider the model on a Bethe lattice (for definiteness on a random $z$-regular lattice) in the large connectivity limit, i.e.\ $z$ large:
we will keep the leading terms and we will neglect the $1/z$ corrections. Only at the end we will perform the limit $z\to\infty$.
In this limit, computations are easier and the model has the same properties of the Sherrington Kirkpatrick one \cite{SK}.
The procedure of first computing the results for finite $z$ in the thermodynamic limit ($N\to\infty$) and later send $z$ to infinity makes the physical approach much clearer.

The couplings are i.i.d.\ random variables extracted from a distribution with the following properties: $\overline{J_{ij}}=0$, $\overline{J_{ij}^2}=1/z$.
Higher order moments are irrelevant in the $z\to\infty$ limit. We have indicated by $h_i^R$ the field on the site $i$.
It can be either a local random field extracted from a given distribution or a spatially uniform field. The physics is
equivalent in the two cases.
For simplicity here we consider the case where the fields $h_i^R$  are Gaussian variables with zero average and finite variance $v_h$
\footnote{The attentive reader could notice that the variance $v_h$ never appears in the following. This is not because things do not depend on $v_h$, but because the dependence is
hidden in the definitions of the magnetizations: $m^2\equiv \overline{\langle \sigma\rangle^2}$ and higher moments will implicitly depend on $v_h$.}.
With standard notation, we indicate with $\langle\cdot\rangle$ the thermal average and with $\overline{\:\cdot\:}$ the average over the disorder (random couplings and fields).

In the thermodynamic limit we will compute different kinds of correlation functions first between point at distance $L$ on a standard Bethe lattice;
this will lead to the \textit{bare} propagator
and we will compute its exact expression with two different methods in the high-temperature region: the replica method and the cavity method.
Then we will compute the first correction to this result due to the presence of one spatial loop.
The limit $z\rightarrow\infty$ allows us to compute all the quantities analytically. For finite connectivity, one could
compute everything numerically using Belief-Propagation as usually done on Bethe lattices.

\section{The replica computation of line-correlations}\label{Sec:line_rep}

In the high connectivity limit (that corresponds to small couplings limit) we can expand the replicated partition function as:
\begin{align}
\nonumber
\overline{Z^n}=&\overline{\sum_{\{{\sigma\}}}e^{\beta\sum_a\sum_i\sigma_i^ah_i^R}e^{\beta\sum_a\sum_{ij}\sigma_i^a\sigma_j^aJ_{ij}}}=\\
\simeq&\overline{\sum_{\{{\sigma\}}}e^{\beta\sum_a\sum_i\sigma_i^ah_i^R}\prod_{ij}\((1+\beta\sum_a\sigma_i^a\sigma_j^aJ_{ij}+\frac{\beta^2}{2}\sum_{a,b}\sigma_i^a\sigma_i^b\sigma_j^a\sigma_j^bJ^2_{ij}\))}
\label{eq:Zn_largez}
\end{align}

The neglected terms give sub-leading contributions for large $z$.

As usual in SG computations, $a,b,c,\ldots\in[1,n]$ indicate the replica index, where the replicas are independent copies of the system with the same disorder realization.

A Bethe random regular graphs in the $N\to\infty$ limit becomes locally loop-less. The distance on the graph between two generic points (i.e. the length of the shortest path between them)
is of order $\log(N)$. We are interested in the computation of the correlation functions of spins that are on points at a distance $L$ between them,
in the limit where $N$ goes to infinity at fixed $L$.
In this limit with probability one there is an unique path (of finite length) connecting them so the computation can be done on a single line.

In general we will be interested in correlation functions that are connected
with respect to the disorder. At this end it is convenient to compute
\begin{equation}
 G_{a,b;c,d}(L)\equiv\overline{\langle\sigma_0^a\sigma_0^b\sigma_L^c\sigma_L^d\rangle}-\overline{\langle\sigma_0\rangle^2}\cdot \overline{\langle\sigma_L\rangle^2}\,,
\end{equation}
from which we can extract connected and disconnected (with respect to thermal average) correlation functions.
In the following, for simplicity of notation, we will indicate with $\overline{ (\cdot) }^c$ the correlations connected with respect to the disorder:
$\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c\equiv
\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}-\overline{\langle\sigma_0\rangle^2}\cdot\overline{\langle\sigma_d\rangle^2}$.
We define the matrix $T\in \mathbb{R}^{n(n-1)\times n(n-1)}$ such as $T_{ab,cd}(i)\equiv\frac{\beta^2}{2}\overline{\langle\sigma_i^a\sigma_i^b\sigma_i^c\sigma_i^d\rangle}$.
We define $T$ only for $a\neq b$, $c\neq d$, following what is usually done for the matrix $Q_{ab}(i)=\langle\sigma_i^a\sigma_i^b\rangle$ in replica calculations
\footnote{Please be careful to not confuse $T$ with a tensor. We could use two superindices $i,j\in[1,n(n-1)]$ instead of the couples $ab$, $cd$. However we choose this notation
because it will be useful to define the different types of correlation functions in the following.}.
Remembering that $\overline{J_{ij}}=0$, $\overline{J_{ij}^2}=\frac{1}{z}$,
from eq. (\ref{eq:Zn_largez}) we find that:
\begin{equation}
G_{a,b;c,d}(L)=\frac{2}{\beta^2z^L}\left[\prod_{i=0}^LT(i)\right]_{ab,cd}.
\label{eq:corr_d}
\end{equation}
Please notice that all the $i\in[0,L]$ are present in eq. (\ref{eq:corr_d}). In fact on a Bethe lattice, two spins are linked just by a path. If the link between two spins is cut,
the spins become disconnected. This means that if a coupling is 0, all the correlation functions between the two spins linked by that coupling are zero. This automatically implies that
a correlation function should be proportional to the product of all the couplings on the path between the two spins.

From eq. (\ref{eq:corr_d}), we need to compute powers of the matrix $T$.
It is easy to show that:
\small
\begin{align}
\label{eq:T1}
T_{ab,cd}(i)=&\frac{\beta^2}{2}\overline{\langle\sigma_i^a\sigma_i^b\sigma_i^c\sigma_i^d\rangle}=\\
\nonumber
=&\frac{\beta^2}{2}\cdot
\begin{cases}
 1    &\text{ if }a=c\text{ , }b=d\text{ or if }a=d\text{ , }b=c \\
 m_2 &\text{ if }a=c\text{ or }b=d\text{ or }a=d\text{ or } b=c \\
 m_4 &\text{ if } a\neq b\neq c\neq d
 \end{cases}
 \end{align}
\normalsize
with $m_2=\overline{\langle\sigma_i\rangle^2}$ and $m_4=\overline{\langle\sigma_i\rangle^4}$.
Let us just mention that for eq. (\ref{eq:corr_d}) to hold, $T$ should be defined as the so-called ``cavity'' average:
the average over the rest of the system with the exception of the neighbouring spins on the considered line. However, in the large $z$ limit, cavity averages are equal to standard averages (see
also the Supplementary Material).

Eq. (\ref{eq:T1}) can be written in the form:
\begin{align}
 \nonumber
 T_{ab,cd}=&\frac{\beta^2}{2}\cdot\left[m_4+(m_2-m_4)(\delta_{ad}+\delta_{bc}+\delta_{bd}+\delta_{ac})+\right.\\
 &\left.+(1-2m_2+m_4)(\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{cb})\right].
\label{Eq:T}
\end{align}

In order to compute correlation functions, we need to compute  powers of $T$; For this reason, we proceed to the diagonalization of $T$.
The whole calculation is explained in the Supplementary Material. Here we just sketch the main steps and state the final result.
First of all we look for eigenvalues and eigenvectors of $T_{ab,cd}$, of the form:
$\sum_{cd}T_{ab,cd}\psi_{cd}=\lambda\psi_{ab}$.
Because of the symmetry of the matrix $T$ under permutations of the replica indices, we know that there are three symmetry classes of eigenvectors
(in an analogous way to what one does when looking to the stability of the Sherrington Kirkpatrick solution for the FC model \cite{dAT}) and three associated eigenvalues:

In the limit $n\rightarrow 0$ the first two eigenvalues (longitudinal and anomalous) are $\lambda_{L/A}=\beta^2(3m_4-4m_2+1),$
while the third one (replicon) is $\lambda_R=\beta^2(1-2m_2+m_4).$

We just want to point out that the eigenvalues are not the same ones as the usual ``replicon, anomalous, longitudinal'' eigenvalues
that comes out from the diagonalization of the Hessian in ref. \cite{dAT}, but we called them in the same way because they identify the same sub-spaces with the same replica symmetries.
In particular at the spin-glass transition the usual replicon goes to zero, while $\lambda_R$ as defined in this paper goes to $\lambda_R=1$ leading to the
divergence of the spin-glass susceptibility (see the following Section).

At this point, we construct the projectors on the sub-spaces of the eigenvectors and write $T$ as a combination of the projectors. In this representation, it is
easy to compute powers of $T$. A special care should be taken in performing the limit $n\rightarrow0$, because of the degeneration of $\lambda_L$ and $\lambda_A$.
The final result is:
\small
\begin{align*}
T^L(n=0)=&\frac{\beta^2}{2}L\lambda_{L/A}^{L-1}\((3m_4-2m_2\))R+\\
&+\lambda_{L/A}^L\((-\frac{R}{2}-Q\))+\lambda_R^L\((\frac{R}{2}+Q+P\)).
\end{align*}
\normalsize
where we defined the matrices $R_{ab,cd}=1$, $Q_{ab,cd}=\frac{1}{4}\left[\delta_{ac}+\delta_{ad}+\delta_{bc}+\delta_{bd}\right]$,
$P_{ab,cd}=\frac{1}{2}\left[\delta_{ac}\cdot\delta_{bd}+\delta_{bc}\cdot\delta_{ad}\right]$.

For a spin-glass model, for each realization of the the system different correlation functions can be defined.
Because of the symmetry of the coupling distribution, the average over the realizations of all the ``linear'' correlations will be zero, and the relevant ones
will be the squared correlations. We will define three main correlations averaged over the thermal noise:
\begin{itemize}
 \item The total correlation: $\langle\sigma_0 \sigma_L\rangle$
 \item The disconnected correlation: $\langle\sigma_0\rangle\langle\sigma_L\rangle$
 \item The connected correlation: $\langle\sigma_0\sigma_L\rangle_c=\langle\sigma_0\sigma_L\rangle-\langle\sigma_0\rangle\langle\sigma_L\rangle$
\end{itemize}
Obviously only two of these correlations are linearly independent.

With these two-spins correlations, we can build different squared correlations:
\begin{itemize}
 \item  The total-total correlation at distance $L$:

\begin{align}
\nonumber
\overline{\langle\sigma_0\sigma_{L}\rangle^2}^c=&\frac{2}{\beta^2z^{L}}\lim_{n\rightarrow 0}\left[\frac{1}{n(n-1)}\sum_{a\neq b}\((T^{L+1}\))_{ab,ab}\right]=\\
=&\frac{2}{\beta^2z^{L}}\((\frac{3}{2}\lambda_R^{L+1}-\lambda_{L/A}^{L+1}+(L+1)\lambda_{L/A}^{L}\frac{\beta^2}{2}(3m_4-2m_2)\)).
\label{Eq:corrTT_rep}
\end{align}
\item  The disconnected-disconnected correlation at distance $L$:
\begin{align}
 \nonumber
\overline{\langle\sigma_0\rangle^2\langle\sigma_{L}\rangle^2}^c=&\frac{2}{\beta^2z^{L}}\lim_{n\rightarrow 0}\left[\frac{1}{n(n-1)(n-2)(n-3)}\sum_{a\neq b \neq c\neq d}\((T^{L+1}\))_{ab,cd}\right]=\\
=&\frac{2}{\beta^2z^{L}}\((\frac{\lambda_R^{L+1}}{2}-\frac{\lambda_{L/A}^{L+1}}{2}+(L+1)\lambda_{L/A}^{L}\frac{\beta^2}{2}(3m_4-2m_2)\)).
\label{Eq:corrDD_rep}
\end{align}
\item The total-disconnected correlation at distance $L$:
\begin{align}
 \nonumber
\overline{\langle\sigma_0\sigma_{L}\rangle\langle\sigma_0\rangle\langle\sigma_{L}\rangle}^c=&\frac{2}{\beta^2z^{L}}\lim_{n\rightarrow 0}\left[\frac{1}{n(n-1)(n-2)}\sum_{a\neq b\neq c}\((T^{L+1}\))_{ab,ac}\right]=\\
=&\frac{2}{\beta^2z^{L}}\((\frac{3 \lambda_R^{L+1}}{4}-\frac{3\lambda_{L/A}^{L+1}}{4}+(L+1)\lambda_{L/A}^{L}\frac{\beta^2}{2}(3m_4-2m_2)\)).
\label{Eq:corrTD_rep}
\end{align}
\item The connected-connected correlation at distance $L$, whose expression can be obtained as a combination of the previous ones:

\begin{align}
 \nonumber
\overline{\langle\sigma_0\sigma_{L}\rangle_c^2}=&\overline{\((\langle\sigma_0\sigma_{L}\rangle-\langle\sigma_0\rangle\langle\sigma_{L}\rangle\))^2}=\overline{\langle\sigma_0\sigma_{L}\rangle^2}+\\
&\nonumber -2\overline{\langle\sigma_0\sigma_{L}\rangle\langle\sigma_0\rangle\langle\sigma_{L}\rangle}+\overline{\langle\sigma_0\rangle^2\langle\sigma_{L}\rangle^2}=\\
=&\frac{1}{\beta^2z^{L}}\lambda_R^{L+1}.
\label{Eq:corrCC_rep}
\end{align}

\end{itemize}
Also in this case only three of these correlations are linearly independent.
\footnote{The expression for the connected correlation in the Bethe lattice is obtained in this paper from a large $z$ expansion.
However we numerically checked that also for small $z$ eq. (\ref{Eq:corrCC_rep}) is valid substituting $z$ with $z-1$ (that is equivalent in the large $z$ limit). }

Others correlations can be readily obtained from the previous one by integration by part.
In the Supplementary Material, we show how to obtain the connected and disconnected bare correlation functions in a cavity approach leading to the same results.

\section{The dominant contribution in the correlation functions} \label{Sec:susc}

To build the susceptibility associated with a given correlation function $C(0,L)$ in a Bethe lattice (where $C$ can be one among the correlation functions considered in the previous section),
one should sum over all the spins that are at distance $L$ from the spin 0, and then over all the distances $L$:
\begin{equation}
\chi_C^B\propto\sum_{L=1}^{\infty}\mathcal{N}_L C(0,L),
\label{eq:chi}
\end{equation}
where $\mathcal{N}_L=z(z-1)^{L-1}$ is the number of spins at distance $L$ from a given spin in a Bethe lattice with connectivity $z$.
The SG transition line is commonly associated to the divergence of the SG susceptibility, that is the susceptibility associated to the connected correlation function.
Substituting eq. (\ref{Eq:corrCC_rep}) and the expression for ${\cal N}_L$ in eq. (\ref{eq:chi}), we discover that the critical line is identified by $\lambda_R=1$
\footnote{Indeed one can check that in the limit of $z\rightarrow\infty$ $\lambda_R$, as defined in this paper, is deeply related to the replicon eigenvalue $\lambda$
of eq. (15) from ref. \cite{dAT}. Using eq. (11) of ref. \cite{dAT}, it can be demonstrated that the relation $\big(\frac{KT}{J}\big)^2 \lambda=-\lambda_R+1$ holds.
The usual SG line associated with $\lambda=0$ translates in $\lambda_R=1$}.
Looking at the eigenvalues, we can numerically check that $\lambda_{L/A}<\lambda_R$.
All the bare correlation functions, as shown in the two precedent sections, have a term proportional to $\lambda_R^L$, that is thus the dominant one.
Recovering the result of standard theory, the critical behavior of all the correlation functions is the same because all depend on the only critical eigenvalue \cite{DedominicisGiardina}.
The susceptibility associated to the different correlations, computed at the critical point, is divergent at the critical line.
On the Bethe lattice, this divergence is a consequence of the exponential decay of the correlations multiplied by the exponential numbers of neighbours at a given distance.
Until now we have computed $\chi_C^B$ on the Bethe lattice.
If now we want to use the Bethe approximation to compute the susceptibility in a $D$ dimensional lattice,
following ref. \cite{Mlayer} (recall Sec. ``Expanding around the Bethe lattice solution''), we should replace $\mathcal{N}_L$ in eq. (\ref{eq:chi}) with the total
number ${\cal B}(d,L)$ of non-backtracking paths of length $L$ between two points at distance $d$ on the original lattice, and add a sum over $d$.
For large $d$ and $L$
\begin{equation}
{\cal B}(d,L) \propto (2D-1)^L |L|^{-D/2}\exp(-|d|^2/(2L))
\end{equation}
implying that at this order the divergence of the susceptibility in a finite dimensional lattice
takes place in correspondence with the divergence in a Bethe lattice with connectivity $z=2D$.

\section{One spatial loop in the RS phase}

In a Bethe lattice in the thermodynamic limit, the density of loops of finite length vanishes, while spatial loops of finite length are common in finite dimensional lattices. In ref.~\cite{Mlayer} a new expansion around the Bethe lattice is performed.
As a result, the first correction to the bare propagator computed in the precedent sections comes from one spatial loop.
In this section, we will compute this contribution confirming that it is totally equivalent to the first correction computed in the usual field theoretical loop expansion, as stated in ref. \cite{Mlayer}.
Following the prescriptions of ref. \cite{Mlayer}, we construct a spatial loop structure $\mathcal{L}$, shown in Fig. \ref{Fig:loop},
formed by two paths of length $L_1+1$ and $L_2+1$ between the points $x$ and $y$ (the length of the internal paths are defined of length $L_1+1$ and $L_2+1$
because in this way results are more compact), plus two external legs of length $L_0$ and $L_3$
to the external spins $\sigma_0$ and $\sigma_d$.  The rest of the lattice is a Bethe lattice without loops in the thermodynamic limit.

We will compute the correction to the ``bare'' correlation functions computed in the previous section, that are those connected over the disorder.
Analogously to the definition of $T$, we define the vertex
\small
\begin{align*}
V_{ab,cd,ef}(x)=&\overline{\langle\sigma_x^a\sigma_x^b\sigma_x^c\sigma_x^d\sigma_x^e\sigma_x^f\rangle}=\\
=&\begin{cases}
1 &\mbox{if  three pairs of indices are equal}\\
m_2 &\mbox{if  two pairs of indices are equal}\\
m_4 &\mbox{if  one pair of indices is equal}\\
m_6 &\mbox{if  the indices are all different}\\
\end{cases}
\end{align*}
\normalsize
As in the case of $T$, $a\neq b$, $c\neq d$, $e\neq f$.
We write the partition function in the presence of one loop $\mathcal{L}$ and we expand it for small $J$s analogously to eq. (\ref{eq:Zn_largez}).
From it, following the same reasoning of Sec. "The replica computation of line-correlations", we obtain the form of the correlation function when a structure $\mathcal{L}$ is present:
\begin{align}
\nonumber
\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{\mathcal{L}}=&\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{L_0,L_1+1,L_3}+\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{L_0,L_2+1,L_3}+\\
&+\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}
\label{eq:oneloopcorr}
\end{align}
where the first and second terms turn out to be exactly the ``bare'' correlations computed as the two paths $L_0+(L_1+1)+L_3$ and $L_0+(L_2+1)+L_3$ were independent.
They have respectively $L_0+(L_1+1)+L_3$ and $L_0+(L_2+1)+L_3$ couplings, that are the minimal number of couplings to have a non zero correlation.
The last term has $L_0+(L_1+1)+(L_2+1)+L_3$ couplings and turns out to be
\begin{align}
 \nonumber
 \overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}=&\((\frac{\beta^2}{2}\))^2\frac{1}{z^{L_0+(L_1+1)+(L_2+1)+L_3}}\times\\
&\times\sum_{q,r,s,t}\((T^{L_0}\))_{ab,qr}\left[\sum_{e,f,g,h,l,m,o,p}
V_{qr,ef,gh}\((T^{L_1}\))_{ef,lm}\((T^{L_2}\))_{gh,op}V_{st,lm,op}\right]\((T^{L_3}\))_{st,cd}.
\label{eq:loop_corr}
\end{align}
Thus in our case, from eq. (\ref{eq:oneloopcorr}) we see that $\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}$ is exactly the line-connected correlation function,
that gives the one-loop correction in the $\frac{1}{M}$ expansion around the Bethe solution.
The one loop contribution takes a very intuitive form.

To compute the explicit form for $\overline{\langle\sigma_0^a\sigma_0^b\sigma_d^c\sigma_d^d\rangle}^c_{lc}$,
we performed sums and products in eq. (\ref{eq:loop_corr}) using Mathematica
\footnote{The same results can be concluded from Eq.(62)
of Ref. \cite{Temesvari1} after a suitable correspondence between quantities like the
propagators and 3-point vertex.}.
At this point we can compute the one-loop contribution to the different correlation functions
as in Sec. ``The replica computation of line-correlations``. The whole expressions are reported in the Supplementary Material.
As for the bare term, the dominant terms are those with the highest power of $\lambda_R$, that are:

\begin{align}
 \overline{\((\langle\sigma_0\sigma_d\rangle_c\))^2}_{lc}\simeq(\overline{\langle\sigma_0\rangle^2\langle\sigma_d\rangle^2}^c)_{lc}\simeq32 \lambda_R^{L_0+L_1+L_2+L_3} [&1+44 m_2^2+101 m_4^2+m_4 (22-90 m_6)+\\
&-2 m_2 (7+67 m_4-30 m_6)-10 m_6+20 m_6^2]
\label{eq:DominantTermLoop}
\end{align}

As explained in Sec. ``The dominant contribution in the correlation functions'', following ref. \cite{Mlayer},
we can now compute the correction to the susceptibility summing over all the length $L_0, L_1, L_2, L_3$, once we have multiplied
by the number of non-backtracking walks.

\section{Relation with previous RG studies}

In ref.~\cite{BR}, the authors perform standard RG calculations in $6-\epsilon$ dimensions for SG with a field.
They write the field-theoretic Hamiltonian in the vicinity of the critical line projected on the replicon eigenspace as
a function of the order parameter $q_{\alpha\beta}$ as:

\begin{align*}
H=&\frac{1}{4}r\sum q_{\alpha\beta}^2+\frac{1}{4}\sum (\nabla q_{\alpha\beta})^2+\\
&-\frac{1}{6} w_1\sum q_{\alpha\beta}q_{\beta\gamma}q_{\gamma\alpha}-\frac{1}{6} w_2\sum q^3_{\alpha\beta}
\end{align*}
with $r$ the reduced temperature, and $w_1$, $w_2$ being coupling constants.
The correlation functions in the momentum space $k$ (projected on the replicon eigenspace) are proportional to $(r+k^2)^{-1}$.
Integrating over an infinitesimal shell $e^{-dl}<k<1$ in the momentum space, they obtain the recursion relation for $r$:

\begin{equation}
\frac{\text{d}r}{\text{d}l}=(2-\eta)r-\frac{K_d}{(1+r)^2}(4 w_1^2-16 w_1w_2+11w_2^2),
\label{Eq:BR}
\end{equation}
with $\eta=\frac{1}{3}K_d(4 w_1^2-16 w_1w_2+11w_2^2)$ and $K_d$ the usual geometrical factor, together with analogous recursion relations for $w_1$ and $w_2$.
The first term in Eq. (\ref{Eq:BR}) is the contribution connected to a renormalization of the critical temperature,
and that we can compute in our approach computing the correction given by a spatial tadpole structure.
The second term is the one coming from the non-trivial loop.

Following ref. \cite{w1w2Parisi},\cite{w1w2Sompolinsky}, in a Bethe lattice in finite and infinite connectivity, as well as in the fully-connected model,
$w_1\propto1-3m_2+3m_4-m_6$ and $w_2\propto2m_2-4m_4+2m_6$.
Inserting these expressions in the loop term of eq. (\ref{Eq:BR}), we obtain:
\small
\begin{align*}
4 w_1-16 w_1w_2+11w_2^2\propto &4 - 56 m_2 + 176 m_2^2 + 88 m_4 +\\
&- 536 m_2 m_4 + 404 m_4^2 - 40 m_6 +\\
&+240 m_2 m_6 - 360 m_4 m_6 + 80 m_6^2
\end{align*}
that is exactly proportional to the coefficient of the dominant term
for the spatial-loop correction to the connected correlation function eq.~(\ref{eq:DominantTermLoop})\footnote{Please note that the replicon contribution, that is the one multiplied by the coefficient $b_4$, that corresponds to the
one found by Bray and Roberts, is the dominant one for the correlation functions. In our
approach, we obtain also the sub-dominant corrections coming from the other sectors, as in ref. \cite{Temesvari1,Temesvari2}.}.
Our approach permits to find the same results in a  clearer, simpler and more physically intuitive way.

\section{Conclusions and perspectives}

In ref.~\cite{Mlayer} a new expansion is introduced around the Bethe lattice. This expansion can be applied to all the models that can be defined on a Bethe lattice.
It is supposed to give the same results as standard perturbation theory for fully connected models when the physics of fully connected and finite connectivity MF models is the same.
We test the new expansion of ref. \cite{Mlayer} for the first time in the case of the spin glass model at finite temperature and with an external field in the limit of high connectivity.
We analytically find the same results of standard RG \cite{BR}, confirming the validity of the new $M$-layer topological expansion.

The new method has, however, the advantage that the equations are physically very intuitive:
the bare correlations are just the correlation on a Bethe lattice (i.e.\ on a line), and
the first-order correction to the correlation function is just
the value of the correlation function computed on a spatial loop (finite loops are absent in the Bethe lattice and present in finite dimensional systems) once
the contributions of the two lines forming the loop, considered as independent graphs, are subtracted.
Even if for the case of the SG in a field in the limit $z\rightarrow \infty$ and finite temperature
the results of the new expansion add nothing to what already known about the SGs in finite dimensions, there are cases in which things should be different.

It was already underlined how the Bethe lattice is more similar to finite dimensional systems \cite{CammarotaBiroliTarziaTarjus} with respect to the FC version
for different disordered spin models. In the particular case of SG in a field, the critical line on the FC model tends to infinite field when the temperature goes to zero,
while in the Bethe lattice it ends at a finite field $h_c$ at $T=0$ \cite{w1w2Parisi}.
If the critical point for SGs in field in finite dimension is a zero temperature one, as supposed by some authors \cite{ParisiTemesvariT0dFinite}\cite{SG_MK}, it is crucial
to perform an expansion around the Bethe solution, instead of around FC model, since the latter model is always in the SG phase at $T=0$.
This paper is a first step in this direction. The application of the topological expansion to the SG model in a field at $T=0$ and finite connectivity is at the moment under study.
It could be done according to lines similar to what done in the case of the random field Ising model \cite{RFIM_Mlayer}.
\vspace{.5cm}

We thank Tommaso Rizzo for very useful discussions. We acknowledge funding from the European Research Council (ERC) under
the European Unions Horizon 2020 research and innovation programme
(grant agreement No [694925]).


\section{Introduction: a brief overview on cold molecules}
\label{sec:introduction_a_brief_overview_on_cold_molecules}

The research field on translationally cold ($\approx 1$K and lower) and ultracold ($\approx 1$mK and lower) molecules is continuously expanding in many directions, involving an increasing number of groups throughout the world. The availability of gaseous samples of cold neutral molecules opens entirely new avenues for fascinating researches, offering the possibility to control all degrees of freedom of a quantum system. Molecular ions can also be trapped and cooled down inside atomic ion traps behaving like ionic crystals, where each ion is kept at a specific position with a residual motion equivalent to a temperature smaller than 1~K \cite{gerlich2008a}. Cold neutral molecules brought new perspectives in high-resolution molecular spectroscopy \cite{stwalley1999,jones2006,meerakker2005,gilijamse2007}. The expected accuracy of the envisioned measurements with ultracold molecules makes them appear as a promising class of quantum systems for precision measurements related to fundamental issues: the existence of the permanent electric dipole moment of the electron \cite{demille2000,hudson2002,kozlov2002,kawall2004} related to CP-parity violation \cite{hinds1980,cho1991,ravaine2005}, and the time-independence of the electron-to-nuclear and nuclear-to-nuclear mass ratios \cite{veldhoven2004,karr2005,schiller2005,chin2006,demille2008,zelevinsky2008}, or of the fine-structure constant \cite{hudson2006a}. Various proposals have been suggested for achieving quantum information devices \cite{demille2002,rabl2006,kotochigova2006,rabl2007} based on cold polar molecules (i.e. molecules exhibiting a permanent electric dipole moment). Elementary chemical reactions at very low temperatures could be manipulated by external electric or magnetic fields, therefore offering an extra flexibility for their control \cite{krems2005}. The large anisotropic interaction between cold polar molecules is also expected to give rise to quantum magnetism \cite{barnett2006}, and to novel quantum phases \cite{baranov2002,goral2002}. The achievement of quantum degeneracy with cold molecular gases \cite{jaksch2002,donley2002,herbig2003,jochim2003,greiner2003,zwierlein2003,regal2003,bourdel2004} together with the mastering of optical lattices \cite{bloch2005} built a fantastic bridge between condensed matter and dilute matter physics. Indeed, the smooth crossover between the Bose-Einstein condensation (BEC) of fermionic atomic pairs and the Bardeen-Cooper-Schrieffer (BCS) delocalized pairing of fermions related to superconductivity and superfluidity, has been observed experimentally \cite{bourdel2004,chin2004,bartenstein2004,zwierlein2004}.

Cold and ultracold neutral molecules can be formed along two paths:
	\begin{itemize}
	\item Existing molecules in their vibrational ground state can be slowed down and cooled by various means, such as interactions with external electric and magnetic fields, or by collision with surrounding particles (see for instance the review articles in refs.\cite{bethlem2003,krems2005,hutson2006}. This approach concerns a broad variety of small molecules (OH, NH, NO, SO$_2$, ND$_3$, CO, YbF, C$_7$H$_5$N, H$_2$CO, LiH, CaH,...), but is currently limited to the production of molecules with a translational temperature around 10~mK.
	\item Pairs of trapped ultracold atoms can be associated using laser fields (photoassociation, or PA) \cite{jones2006}, or time-varying magnetic fields (magneto- or Feshbach- association) \cite{kohler2006}. Translational temperatures as low as a few tens of microKelvins can be reached, and even lower when quantum degeneracy is achieved. In contrast to the previous case, these results are up to now limited to alkali diatomic molecules which are most often formed in highly-excited vibrational levels, i.e. with high internal energy.
	\end{itemize}

In the latter case, a breakthrough occurred in 2008, when the possibility to create ultracold bialkali molecules in the lowest vibrational level of their ground state or their lowest metastable triplet state, has been demonstrated. Caesium dimers have been created in their ground state $v=0$ level after a PA step followed by spontaneous emission and further vibrational pumping, using a sequence of shaped laser pulses \cite{viteau2008}. Dipolar molecules, namely LiCs, have been observed in their absolute ground state rovibrational level $v=0, J=0$ after PA and spontaneous emission \cite{deiglmayr2008a}. Samples close to the degeneracy regime of ultracold molecules in the $v=0$ level of their ground state have been observed for Cs$_2$ \cite{danzl2008,mark2009}, and KRb \cite{ni2008}, and in the $v=0$ level of their lowest triplet state for Rb$_2$ \cite{lang2008}, using the STIRAP technique (Stimulated Rapid Adiabatic Passage) to transfer the population from initial high-lying vibrational levels.

\section{Motivation of the present work}
\label{sec:motivation_of_the_present_work}

The growing availability of ultracold samples of alkali diatomics brings the possibility to observe and study their interactions with neighboring ultracold atoms and molecules. In two independent - and almost simultaneous- experiments, inelastic rate constants for atom-molecule and molecule-molecule collisions have been extracted, using optically trapped Cs$_2$ molecules created by PA and spontaneous emission \cite{staanum2006,zahzam2006}. The rate constants have been found independent from the initially populated rovibrational level. A resonant feature observed in the loss rate after collisions between trapped ultracold Cs$_2$ molecules created by Feshbach association has been interpreted as Cs$_{2}$-Cs$_{2}$ bound states \cite{chin2005}. Inelastic atom-molecule collisions in a trapped sample of RbCs molecules have been studied for both RbCs-Cs and RbCs-Rb cases~\cite{hudson2008}. No systematic dependence on the internal state of the molecule has been probed within the experimental precision.

Theoretical knowledge of the electronic structure of alkali triatomic systems is strongly needed for theoretical dynamical studies, which would support these experiments. There has been a significant research activity concerning the theoretical study of the structure and dynamical properties of homonuclear alkali-metal trimers. Non-additive effects in spin-polarized alkali-metal trimers were studied by Sold\'{a}n \textit{et.al.} \cite{soldan2003} and the three-dimensional potential energy surfaces for the lowest quartet states were constructed for Li$_{3}$~\cite{colavecchia2003a,cvitas2007}, Na$_{3}$~\cite{higgins2000,simoni2009}, K$_{3}$ \cite{quemener2005} and Rb$_{3}$ \cite{soldan2008u}. These were then used to study the corresponding spin-polarized reactive atom-dimer collisions at very low temperatures \cite{cvitas2007,quemener2005,soldan2002,quemener2004,cvitas2005a,cvitas2005,quemener2007,li2008a,li2008b,li2008c}. General trends for the lowest quartet state of homonuclear alkali trimers - correlated to three spin-polarized alkali atoms - have been described by Hutson and Sold\'an \cite{hutson2007}. One of the main characteristic of these systems is the importance of non-additive three-body forces, which are predicted to be large especially close to the equilibrium geometries \cite{soldan2003}. Much less has been done in the case of heteronuclear alkali-metal trimers. One of us has actually started with a systematic study the lowest quartet states of Li$_{2}$A mixed systems (with A=Na, K, Rb, Cs) \cite{soldan2008}. He concluded that the single-reference coupled-cluster approach, which has been successfully used for homonuclear alkali-metal trimers \cite{quemener2005,soldan2008}, would be for various reasons very difficult to employ for calculations of the potential energy surfaces of heteronuclear alkali-metal trimers, and that an alternative approach to the problem should be sought.

In the present paper, we present such an alternative approach. We extend our previous works on effective two-electron diatomic molecules like alkali dimers \cite{aymar2005,aymar2006,deiglmayr2008} and alkali hydrides \cite{aymar2009} to alkali trimers. We propose a preliminary study of the potential energy surfaces of the heaviest alkali trimer, Cs$_3$, modeled as an effective three-electron molecule. Our approach is based on large-core effective core potentials (ECP) with core polarization potentials (CPP). We focused our work on two aspects: (i) the estimation of repulsion effects between the three Cs$^+$ ionic cores at short distances within the present ECP+CPP approach; (ii) the comparison of the results with the results obtained by the MOLPRO package for {\it ab-initio} calculations.

\section{Method of calculation}
\label{sec:method_of_calculation}

	\subsection{\emph{Ab initio} methods and basis sets}
	\label{sub:ab_initio_methods_and_basis_sets}

The potential energies were calculated making use of the CIPSI package (Configuration Interaction by Perturbation of a multiconfigurational wave function Selected Iteratively)~\cite{huron1973}. As in our previous studies on alkali dimers, the alkali atom A is described by an $\ell$-dependent Effective Core Potential (ECP) for the ionic core A$^+$ \cite{durand1974,durand1975} including effective core polarization potential (CPP) terms \cite{muller1984,foucrault1992}, and by a large set of uncontracted Gaussian functions for the valence electron. The atom is modeled as an effective one-electron system, permitting us to perform Full Configuration Interaction (FCI) calculations for any alkali dimer or trimer, considered as an effective two-electron or three-electron system, respectively.

As the distance between different cores becomes smaller than the equilibrium distance of the system, it is well known that their electrostatic repulsion is not accounted for by this effective large-core potential. At the same time, the addition of the CPP term leads to an non-physical attractive behavior at short distances. For diatomic molecules, an empirical repulsion term is usually added to the calculation (see e.g. Refs.~\cite{spiegelmann1989,pavolini1989,magnier1993,magnier1996}). If this short-range repulsive interaction in the triatomic molecule can be treated as the sum of pairwise interaction terms, computation of the potential surfaces would be greatly simplified. In order to check if such an approximation could be justified, we calculated the core-core repulsion at the Hartree-Fock level making use of the MOLPRO 2006.1 Quantum Chemistry Package~\cite{MOLPRO_brief}. For lithium and sodium, the all-electron correlation-consistent polarized core-valence quadruple-$\zeta$ cc-pCVQZ basis sets~\cite{iron2003} were used. For potassium, rubidium, and caesium, the small-core scalar relativistic effective core potentials ECP10MDF, ECP28MDF, and ECP46MDF, respectively, together with the corresponding uncontracted valence basis sets~\cite{lim2005a} were employed.

For the caesium dimer and trimer, the counterpoise-corrected dimer interaction energies were optimized using an algorithm implemented in MOLPRO. In this algorithm the interaction energies were calculated at the coupled-clusters level making use of a single-reference restricted open-shell variant \cite{knowles1993} of the coupled cluster method \cite{cizek1966} with single, double and non-iterative triple excitations [RCCSD(T)]. The small-core scalar relativistic effective core potentials ECP46MDF was employed together with its uncontracted valence basis set, which was augmented by one set of diffuse function in an even-tempered manner (aug-ECP46MDF). All electrons from the ``outer-core'' $5s5p$ orbitals were included in the RCSSD(T) calculations. The full counterpoise correction of Boys and Bernardi \cite{boys1970} was applied to all the interaction energies in order to compensate for the basis set superposition errors.

	\subsection{Core Polarization Potential for a triatomic molecule}
	\label{sub:core_polarization_potential}

The core polarization potential (CPP) is given by~\cite{muller1984}:

		\begin{equation}
		  \label{eq:VCPP}
		  V_{CPP}=-\frac{1}{2} \sum_{c} \alpha_{c}\,\mathbf{f}_{c}\cdot\mathbf{f}_{c}
		\end{equation}

The electric field $\mathbf{f}_{c}$ is the one produced at point $\mathbf{r}_{c}$ by all the other electrons and the ionic cores with static dipole polarizability $\alpha_{c}$:

		\begin{equation}
		  \label{eq:fc}
		  \mathbf{f}_{c}=\sum_{i}\frac{\mathbf{r}_{ci}}{r_{ci}^{3}}h(r_{ci},\rho_{c})\
			-\sum_{c^{\prime}\neq c}\frac{\mathbf{R}_{cc^{\prime}}}{R_{cc^{\prime}}^{3}}Z_{c^{\prime}}
		\end{equation}

where $\mathbf{r}_{ci}$ is the position vector from core $c$ to electron $i$, $\mathbf{R}_{cc^{\prime}}$ the one between cores $c$ and $c^{\prime}$. The cutoff function $h(r,\rho)$  ensures the convergence of the integral over electronic coordinates. The $V_{CPP}$ term contains a purely geometrical	part $v_{nn}$ depending only on the relative positions of the nuclei:

		\begin{equation}
		  \label{eq:vnn}
		  v_{nn}=-\frac{1}{2}\sum_{c}\alpha_{c}\sum_{c^{\prime},c^{\prime\prime}\neq c}\
			\frac{\mathbf{R}_{cc^{\prime}}\cdot \mathbf{R}_{cc^{\prime\prime}}}{R_{cc^{\prime}}^{3}\
			R_{cc^{\prime\prime}}^{3}}Z_{c^{\prime}}Z_{c^{\prime\prime}}
		\end{equation}

For a diatomic molecule, $v_{nn}$ is given by the familiar formula: $v_{nn}=-\frac{Z_{2}^{2}\alpha_{1}+Z_{1}^{2}\alpha_{2}}{2 R^{4}}$. For a triatomic system, we have:

		\begin{equation}
			\label{eq:CPP_tri}
			\begin{split}
				&v_{nn}=-\frac{Z_{2}^{2}\alpha_{1}+Z_{1}^{2}\alpha_{2}}{2 R_{12}^{4}}-\
				\frac{Z_{3}^{2}\alpha_{1}+Z_{1}^{2}\alpha_{3}}{2 R_{13}^{4}}\
				-\frac{Z_{3}^{2}\alpha_{2}+Z_{2}^{2}\alpha_{3}}{2 R_{23}^{4}}\\
				&-\frac{Z_{2}Z_{3}\alpha_{1}\,\mathbf{R}_{12}\cdot\mathbf{R}_{13}}{R_{12}^{3}R_{13}^{3}}\
				-\frac{Z_{1}Z_{3}\alpha_{2}\,\mathbf{R}_{12}\cdot\mathbf{R}_{23}}{R_{12}^{3}R_{23}^{3}}\
				-\frac{Z_{1}Z_{2}\alpha_{3}\,\mathbf{R}_{13}\cdot\mathbf{R}_{23}}{R_{13}^{3}R_{23}^{3}}
			\end{split}
		\end{equation}

In those formulas, $Z_{i}$ and $\alpha_{i}$ are, respectively, the net charge and static polarizability of ion core $i$. This term is independent of the \emph{ab initio} calculation itself and is added \emph{a posteriori}.

\section{Volume effect for triatomic systems}
\label{sec:volume_effect_for_triatomic_systems}

The part of the interaction between two nuclei other than the point charge repulsion has been dubbed “volume effect” by Jeung in his 1997 paper on alkali diatomics~\cite{jeung1997}. Here, we are interested in this volume effect for alkali triatomic systems. More precisely, we investigate whether it can be reasonably approximated as a sum of two-body terms.

The two-body repulsion energies $V_{cc}$ are calculated for all the alkali homonuclear diatomic molecules. The three-body repulsion energies $V_{ccc}$ are calculated for particular geometries, namely for D$_{\infty h}$ and D$_{3h}$ symmetries, to provide a representative investigation. Calculations ar done at the Hartree-Fock level of theory starting with the orbitals of the free atoms. Those orbitals are kept frozen as the internuclear distance is shortened. In this respect, our calculations for the two-body term are similar to those of the “A” column from the tables of ref.~\cite{jeung1997}. This comparison is shown in Figure~\ref{fig:compJeung} in logarithmic scale. Apart from a slight discrepancy in the case of Na$_{2}$, there is a good agreement between both calculations, and it is seen that the two-body core repulsion term could be well fitted by an exponential form, as already stated in ref.\cite{pavolini1989}.

 		\begin{figure}[htbp]
 			\begin{center}
 				\includegraphics[width=15cm]{CompJeung.eps}
 			\end{center}
 			\caption{Two-body core repulsion energies for alkali dimers. Blue : our work. Red dashed : Ref.~\cite{jeung1997}. Note the logarithmic scale used for the potential energy axis.}
 			\label{fig:compJeung}
 		\end{figure}

Three-body core repulsion energies have been calculated for all homonuclear alkali trimers at linear symmetric and equilateral geometries according to the schemes of Figure~\ref{fig:geom}. In Figure~\ref{fig:addLin}, we compare the three-body core repulsion term for linear geometries $V_{ccc}^{lin}(R)$ (the internuclear distance $R$ being defined in Figure~\ref{fig:geom}), with the sum of two-body term $2 V_{cc}(R)+V_{cc}(2R)$. In the same manner, we compare in Figure~\ref{fig:addEqui} the three-body repulsion term for equilateral geometries $V_{ccc}^{eq}(R)$ to $3 V^{(2)}(R)$. In both cases, it is seen that the three-body core repulsion term is well described by a pairwise additive lemma, up to the region where it becomes negligible. Therefore, just like for alkali dimers, the $V_{ccc}$ term can be added {\it a posteriori} to the potential surface calculations, whatever the chosen grid is for them.

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=10cm]{geometries.eps}
		\end{center}
		\caption{Linear (D$_{\infty h}$) and equilateral (D$_{3h}$) geometries at which the three-body core repulsion term have been calculated. The Jacobi coordinates $\{r',R',\theta\}$ used in the calculation of the potential energy surfaces of Cs$_{3}$ are also presented.}
		\label{fig:geom}
	\end{figure}

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=15cm]{CoreCoreCore_linear.eps}
		\end{center}
		\caption{Additivity of the three-body core repulsion term for linear geometries.
		Blue : Three-body term $V_{ccc}^{lin}(R)$. Red dashed :  sum of the relevant two-body terms $2 V_{cc}(R)+V_{cc}(2R)$ (see text).}
		\label{fig:addLin}
	\end{figure}

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=15cm]{CoreCoreCore_equilateral.eps}
		\end{center}
		\caption{Additivity of the three-body core repulsion term for equilateral geometries.
		Blue : Three-body term $V_{ccc}^{eq}(R)$. Red dashed : sum of the relevant two-body terms $3 V^{(2)}(R)$ (see text).}
		\label{fig:addEqui}
	\end{figure}

\newpage
\section{Preliminary application to caesium trimer}
\label{sec:preliminary_application_to_caesium_trimer}

The inclusion of the three-body core repulsion term is considered in preliminary calculations concerning the $^{4}A_{2}^{\prime}$ state of Cs$_{3}$ which correlates to three spin-polarized caesium atoms. The used basis set is kept relatively small to maintain reasonable size for the FCI, reaching $152\,532$ Slater determinants in C$_{s}$ symmetry in which the calculation is performed. The basis set $(5s3p4d/4s3p3d)$ is detailed in Table~\ref{tab:basisSet}. We use $\ell$-dependent CPP's~\cite{foucrault1992} which allow us to adjust the calculated atomic energies to the experimental ones for the $6s\,^{2}S$, $6p\,^{2}P$ and $5d\,\,^{2}D$ atomic states of caesium. This is achieved by tuning the cutoff radii $\rho_{\ell}$ of Eq.~\ref{eq:fc} to the values displayed in Table~\ref{tab:cutoffRadii}.

	\begin{table}
		\begin{center}
			\begin{tabular}{ccc}
				\hline
				Angular momentum &  Exponent &  Contraction coefficients\\
				\hline
				s &  0.347926  &   0.411589\\
				s &  0.239900  &  -0.682422\\
				s &  0.050502  &   1.\\
				s &  0.036900  &   1.\\
				s &  0.00515   &   1.\\
				p &  0.1837    &   1.\\
				p &  0.0655    &   1.\\
				p &  0.0162    &   1.\\
				d &  0.2106894 &   0.18965\\
				d &  0.065471  &   0.22724\\
				d &  0.021948  &   1.\\
				d &  0.011200  &   1.\\
				\hline
			\end{tabular}
		\end{center}
		\caption{Gaussian basis set used on each caesium atom in the ECP+CPP-FCI approach.}
		\label{tab:basisSet}
	\end{table}

	\begin{table}
		\begin{center}
			\begin{tabular}{cc}
				\hline
				$\ell$ & $\rho_{\ell}$\\
				\hline
				s & 2.6248\\
				p & 1.87\\
				d & 2.8111\\
				\hline
			\end{tabular}
		\end{center}
		\caption{$\ell$-dependent cutoff radii $\rho_{\ell}$ (in units of $a_{0}$) used in the core polarization potential. The polarization for Cs$^{+}$ ionic core of 16.33 $a_{0}^{3}$ is taken from Ref~\cite{coker1976}.}
		\label{tab:cutoffRadii}
	\end{table}

	\subsection{The triplet state of Cs$_{2}$}
	\label{sub:the_triplet_state_of_cs2}

With these basis set and cut-off radii, we first calculate the lowest $^{3}\Sigma_{u}^{+}$ potential energy curve for Cs$_{2}$ dissociating into Cs($6s$)+Cs($6s$) in the framework of our ECP+CC-FCI method. We found an equilibrium internuclear distance R$_{e}=11.82\,a_{0}$ and a well depth of D$_{e}=380.6\,\text{cm}^{-1}$. These numbers can be compared to our optimized results obtained at the RCCSD(T)/aug-ECP46MDF level R$_{e}=12.19\,a_{0}$ and D$_{e}=256.4\,\text{cm}^{-1}$ and the results from Ref.~\cite{soldan2003,hutson2007}, who obtained R$_{e}=12.44\,a_{0}$ and D$_{e}=246.8\,\text{cm}^{-1}$ at the RCCSD(T)/ECP46MWB level of theory. The MOLPRO calculation yields a potential well which is deeper by is a more than $140\,\text{cm}^{-1}$ than the one of the ECP+CC-FCI calculation. Apparently, the small-core ECP in combination with a rich valence basis set in the RCCSD(T) calculations provides more realistic description of the interaction energy than the large-core ECP in combination with a smaller valence basis set used in the FCI calculations (see also the discussion in Section \ref{sec:conclusion}). The potential energy well of the triplet state results from the competition between the exchange energy at short distances and the attractive dispersion forces at long distances and is very sensitive to the quality of the basis. For this $^{3}\Sigma_{u}^{+}$ curve, the inclusion of the two-body core repulsion term does not change the characteristics of the well but ensures a correct exponential behavior at short internuclear distances (Figure~\ref{fig:cs2Triplet}).

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=13cm]{Different_Quartet_Curves.eps}
		\end{center}
		\caption{Calculated potential energy curves for the lowest $^{3}\Sigma_{u}^{+}$ state of Cs$_{2}$. Black plain curve : ECP+CPP-FCI calculation with the basis detailed in Table~\ref{tab:basisSet}. Red dashed curve : obtained at the RCCSD(T)/aug-ECP46MDF level. Green chain dotted curve : ECP+CPP full CI result obtained with basis set “A” from Ref.~\cite{aymar2005} (see the discussion in Section~\ref{sec:conclusion}). Blue dotted curve : Multiparameter Morse Long Range fitting to spectroscopic data measured by Xie \emph{et al}~\cite{xie2009}.}
		\label{fig:cs2Triplet}
	\end{figure}

	\subsection{Quartet state of Cs$_{3}$}
	\label{sub:quartet_state}

We then calculate the potential energy surface of the lowest quartet state of Cs$_{3}$. Calculations are carried out in Jacobi coordinates $\{r',R',\theta\}$ (depicted in Figure~\ref{fig:geom}) with the angle $\theta$ fixed at $\pi/2$ in order to explore C$_{2v}$ nuclear geometries, as a representative geometry. We have computed 984 points on a grid where $4.4\,a_{0}<r'<40\,a_{0}$ and $4\,a_{0}<R'<40\,a_{0}$. The minimum of the surface is found at a D$_{3h}$ geometry with a bond length of $R'_{e}=10.72\,a_{0}$ and a well depth of $D_{e}=1518\,\text{cm}^{-1}$ with the inclusion of the pairwise three-body core repulsion term, which ensures a realistic short-range repulsive wall. These numbers can be compared to our optimization results obtained at the RCCSD(T)/aug-ECP46MDF level $R'_{e}=11.18\,a_{0}$ and D$_{e}=1208\,\text{cm}^{-1}$ and to the results from Ref.~\cite{soldan2003,hutson2007}, who obtained $R'_{e}=11.33\,a_{0}$ and D$_{e}=1139\,\text{cm}^{-1}$ at the RCCSD(T)/ECP46MWB level of theory. It is worth noting that as the minimum of the quartet potential well is located at shorter distances than the minimum of the Cs$_2$ triplet state, the $V_{ccc}$ term indeed contributes to the depth of the well, and not only in the region of the repulsive wall. However, the magnitude of $V_{ccc}$ is still small, as the equilibrium internuclear distance is $R'_{e}=10.7\,a_{0}$ and the well depth is $D_{e}=1522\,\text{cm}^{-1}$ if we neglect it. As expected from the diatomic calculations above, we obtain a difference of about $300\,\text{cm}^{-1}$ on the well depth of the quartet state of Cs$_{3}$ compared to the MOLPRO calculations. This confirms the limited quality of our basis set at the current level of our computations.

	\subsection{Doublet states of Cs$_{3}$}
	\label{sub:doublet_states}

The three lowest potential surfaces for Cs$_{3}$ are presented in Figure~\ref{fig:cs3Surfaces}. The $^{2}B_{2}$ and $^{2}A_{1}$ states are the two components of a $^{2}E^{\prime}$ state at D$_{3h}$ symmetry subjected to Jahn-Teller effect. The left (resp. right) column shows the surfaces without (resp. with) the addition of the core repulsion term. Note the strongly unphysical attractive behavior in the region $r'<5\,a_{0}$, which is removed when $V_{ccc}$ is added. The global ground state $^{2}B_{2}$ is characterized by a well depth of $5437.1\,\text{cm}^{-1}$ at the geometry $\{r'=10.58\, a_{0},R'=7.36\, a_{0}\}$ with the inclusion of the core repulsion term. This changes to a well depth of $5489\,\text{cm}^{-1}$ at  $\{r'=10.57\, a_{0},R'=7.3\, a_{0}\}$ without the core repulsion term. Once again, the effect of $V_{ccc}$ is larger than in the dimer, as the equlibrium distance is shorter. The other component $^{2}A_{1}$ is characterized by a well depth of $5258.2\,\text{cm}^{-1}$ at $\{r'=8.75\, a_{0},R'=9.28\, a_{0}\}$ with the core repulsion term ($5312\,\text{cm}^{-1}$ at $\{r'=8.66\, a_{0},R'=9.26\, a_{0}\}$ without it).

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=13cm]{Cs3_Jacobi_surfaces.eps}
		\end{center}
		\caption{Lowest three calculated surfaces for Cs$_{3}$ in Jacobi coordinates with fixed $\theta=\pi/2$. Left (resp. right) column : without (resp. with) the inclusion of the cores repulsion term $V_{ccc}$. The red dashed curve locates the equilateral D$_{3h}$ geometry $R'=\frac{\sqrt{3}}{2}r'$. Contours are distant of $500\,\text{cm}^{-1}$ and the dashed contour is the energy of the dissociation $\text{Cs}(^{2}S)+\text{Cs}(^{2}S)+\text{Cs}(^{2}S)$.}
		\label{fig:cs3Surfaces}
	\end{figure}

We present in Figures~~\ref{fig:cs3CurvesQuartet} and \ref{fig:cs3CurvesDoublet} the cut through the D$_{3h}$ geometry of the previous surfaces. We recall that in the D$_{3h}$ symmetry point group, the two doublet states $^{2}B_{2}$ and $^{2}A_{1}$ discussed before correlates to the two components of a doubly degenerate $^{2}E'$ state. We clearly see in these figures the abrupt attractive behavior which occurs at short distances and that the addition of the core repulsion term gives us back a realistic repulsive wall of potential.

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=13cm]{Cs3_Jacobi_curves_quartet.eps}
		\end{center}
		\caption{Cut through the D$_{3h}$ geometry for the lowest quartet states of Cs$_{3}$. Plain black (resp. red dashed) curve : with (resp. without) the core repulsion term $V_{ccc}^{eq}(R)$. Green chain dotted curve : quartet state on the RCCSD(T)/aug-ECP46MDF level of theory (see the discussion in section \ref{sec:conclusion}.}
		\label{fig:cs3CurvesQuartet}
	\end{figure}

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=13cm]{Cs3_Jacobi_curves_doublet.eps}
		\end{center}
		\caption{Cut through the D$_{3h}$ geometry for the lowest doublet states of Cs$_{3}$. Plain black (resp. red dashed) curve : with (resp. without) the core repulsion term $V_{ccc}^{eq}(R)$. Both components of the $^{2}E'$ state are represented.}
		\label{fig:cs3CurvesDoublet}
	\end{figure}

\section{Future prospects}
\label{sec:conclusion}

Preliminary potential energy surfaces for doublet and quartet states of the Cs$_{3}$ molecule have been computed in the framework of a ECP+CPP-FCI quantum chemistry approach. The core repulsion interaction has been taken into account as an additional empirical correction as it is routinely done for diatomic molecules calculations. We showed that this term is well described by a sum of two-body terms. As mentioned before in ref.~\cite{pavolini1989}, it can be fitted by an exponential formula. Therefore, this term needs not to be calculated at each point of the three dimensional grid spanned by the degrees of freedom of a triatomic molecule on which the \emph{ab initio} calculations are performed. It can easily be added \emph{a posteriori} to cancel any non-physical behavior which might occur at short internuclear distances.

To estimate the quality of this basis set, we have compared our calculations to high level ones which uses a different method than ours, with the MOLPRO package. Such a comparison between different methods is crucial to assess their accuracy and their consistency, as no spectroscopic data are available for the Cs$_3$ system, as well as for most of the triatomic alkali systems. The calculations presented here are exploratory in nature: the small basis set used (Table~\ref{tab:basisSet}) makes the calculation time short enough to generate over a thousand \emph{ab initio} points. Figure~\ref{fig:cs2Triplet} shows the limitation of this basis set where the well depth of the triplet state of Cs$_{2}$ is overestimated by more than 100~cm$^{-1}$. For the quartet state of Cs$_{3}$, our calculated well depth is consequently 300~cm$^{-1}$ lower than the RCCSD(T) calculations. If we employ the basis set previously used for the caesium dimer~\cite{aymar2005} in the ECP+CPP-FCI method, this yields a very good agreement with the RCCSD(T) calculations, as the difference is now found at about $20\,\text{cm}^{-1}$ between them (Figure~\ref{fig:cs2Triplet}). The agreement is even more spectacular with the potential curve extracted from the spectroscopy of the lowest triplet state of Cs$_{2}$ \cite{xie2009}.

We are currently calculating a comprehensive set of \emph{ab initio} points with the ECP+CPP-FCI with this extended basis for the Cs$_{3}$ molecule which will be the subject of a future paper. Such FCI calculations now involve more than 500,000 Slater determinants. In Figure~\ref{fig:cs3Dinfh}, we display the first calculated points for the D$_{\infty h}$ symmetry of the lowest Cs$_{3}$ quartet state, compared to the MOLPRO calculation as described in Section~\ref{sec:preliminary_application_to_caesium_trimer}. The figure suggests that the results are in good agreement with each other as the position of the minimum looks similar. The point calculated around $12.3\,a_{0}$ with the ECP+CCP full CI method is deeper than the minimum of the MOLPRO curve by about $40\,\text{cm}^{-1}$, while the difference between corresponding triplet potential curves in Figure~\ref{fig:cs2Triplet} is seen to be about $20\,\text{cm}^{-1}$  This suggests that pairwise additivity of the forces is a reasonably good approximation in this case.

	\begin{figure}[htbp]
		\begin{center}
			\includegraphics[width=15cm]{Cs3_Quartet_Dinfh.eps}
		\end{center}
		\caption{Quartet $^{4}\Sigma_{u}^{+}$ state of Cs$_{3}$ in D$_{\infty h}$ symmetry. Green chain dotted curve : calculation at the RCCSD(T)/aug-ECP46MDF level of theory. Plus signs : preliminary calculations with the ECP+CPP-FCI approach using basis set “A” from Ref.~\cite{aymar2005}.}
		\label{fig:cs3Dinfh}
	\end{figure}

\section*{Acknowledgments}
This work is performed in the framework of the network "Quantum Dipolar Molecular Gases" (QuDipMol) of the EUROCORES/EUROQUAM program of the European Science Foundation. J.D. acknowledges partial support of the French-German University (http://www.dfh-ufa.org), and R.G. support from {\it Institut Francilien de Recherches sur les atomes froids - IFRAF} (http://www.ifraf.org).

\newpage
\section*{References}
\label{sec:references}


\section{Introduction}
In the evolution of a star, when the radius of a star becomes less than a
certain limit, called the Schwarzschild radius $R_{Sch}=2GM/c^2$,
where M is the mass of the object; $G$, Newton's universal
gravitational constant, and $c$, velocity of light; neither
light nor material particles can escape from the star. Thus, one finds
from above, the Schwarzschild radius corresponds to the situation when the
escape velocity is equal to the velocity of light, a particle (or a
photon) coming from a large distance when passes near by  the black
hole\cite{raine,wheel},
it is not only attracted towards it but also its orbit diverges from a
straight line.It may so happen that if  the particle goes too close to a
black hole it is likely to be trapped and hence, it cannot escape to
infinity. As a special case, if the particle travels straight to the
center of the black hole, it falls inside it and is lost forever.
An interesting
problem that is associated with the formation of a black hole is the final
collapse of a massive star. This happens when the nuclear fuel inside the
central core of the star gets exhausted. At this stage, it is the dominance
of the gravitational attraction among the particles within a star over the
internal outward pressure which makes the star to collapse. A black hole,
 though may be formed from baryons, leptons etc; the exterior observer
cannot
have access to the details of the inside of the black hole. The observer
probes the black hole mass M, electromagnetic charge Q and angular
momentum J. This is refered to as baldness of the black hole or as Wheeler
describes, a black hole has no hair\cite{wheel}. But actually it has only
three hairs,
M,\ Q,\ J. We in this paper derive the Schwarzschild radius of a system of
gravitating particles quantum mechanically and show that when quantum
exchange correction is taken into account, there is a thin
correction to this radius, which we designate as the skin of the black
hole.
It has been shown previously\cite{duff} that using the quantum field
theoretic method, by
including a single-closed-loop in the self energy, a quantum
correction to
the classical Schwarzschild solution of the order of $\sim G^2$ can be
found. This
comes from the gravity sector. The correction that we find, comes from the
exchange part of the matter-energy sector of the black hole. Our correction is
also of the
order of $\sim G^2$.

We here consider a black hole with mass only, a Schwarzchild\cite{raine}
type black
hole. As
described  above, since the black hole forms out of particles due to
immense gravitational attraction, we consider the system of particles as a
many particle system that ultimately forms a black hole.
We have succeeded in developing a quantum mechanical
approach\cite{sm1,sm2} for a system of self-gravitating particles using
Newtonian gravity. We calculate the
energy content of the system including exchange interaction. In order to
calculate the total energy of a star, we choose
a trial single-particle density to account for the distribution of
particles within star. The form of our single-particle density is such
that it has a singularity at the origin. Applying it to the case of a
neutron star, we not only arrive at a compact expression for the radius of
the neutron star, but also obtain an expression for the binding energy of
the star which varies with the particle number\cite{levy} as $N^{7/3}$,
where N is
the particle number. Such a dependence with $N$ is in agreement with those
of the earlier workers.\cite{levy} The aim of the present work is to give
a
derivation of the socalled Schwarzschild radius even without using GTR and
relativistic quantum mechanics. By accounting for the exchange effects due
to the interparticle correlations to the total ground state energy of the
system, we find a quantum correction to the Schwarzschild radius.

 \section{Mathematical formulation}

In order to describe a system of N self-gravitating particles in absence
of any source for radiation, we use a Hamiltonian of the form:
\begin{equation}
H=\sum_{i=1}^N\frac{-\hbar^2\nabla_i^2}{2m}+\frac{1}{2}\sum_{i=1}^N
\sum_{j=1,i\ne j}^Nv(|\vec X_i-\vec X_j|),
\end{equation}
where $v(|\vec X_i-\vec X_j|)=-g^2/|\vec X_i-\vec X_j|$, is the
interparticle interaction between  pair of gravitating particles
and $g^2=Gm^2$,
$m$ being the mass of particle and G being the universal gravitational
constant. In the present case we confine ourselves to the system of
neutrons only. Since the wavefunction of a neutron star is not known, we
proceed\cite{sm1,sm2} to evaluate the total kinetic energy of the system
using a model
density distribution function for particles (neutrons) within the neutron
star. The particles being fermions, we use the Thomas-Fermi formula for
calculating the total kinetic energy of the system. For an infinite
many-fermion system, the average particle density and the fermi momentum
of
a particle are related to each other as:
\begin{equation}
n=\frac{k_F^3}{3\pi^2}.
\end{equation}
For a finite system like  the star, since the density distribution is a
function of radius vector $\vec r$, the fermi energy of a particle is
supposed to be dependent on $r$. For an infinite many-fermion system, the
total kinetic energy of the system\cite{fetter} is given as
\begin{eqnarray}
<KE>_{inf}=\frac{3}{5}n\epsilon_F=\frac{3}{5}n(3\pi^2n)^{2/3}\frac{\hbar^2}{2m}
\nonumber\\
=\frac{3\hbar^2}{10m}(3\pi^2)^{2/3}n^{5/3}\label{ke},
\end{eqnarray}
In analogy with
Eq.(\ref{ke}),
for a finite system\cite{land}, we write,
\begin{equation}
<KE>=\frac{3\hbar^2}{10m}(3\pi^2)^{2/3}\int d\vec r [\rho(\vec
r)]^{5/3}\label{kee},
\end{equation}
The total energy E of the system is given as
\begin{equation}
E=<H>=<KE>+<PE>,
\end{equation}
where
\begin{equation}
<PE>=-\frac{g^2}{2}\int d\vec r d\vec r'\frac{\rho(\vec r)\rho(\vec r')}
{|\vec r-\vec r'|}\label{pe}
\end{equation}
The expression given in Eq.(\ref{pe}) is written in the Hartree
approximation. In order to find E, we choose a trial single-particle
density of the form
\begin{equation}
\rho(\vec r)=A\frac{exp[-(\frac{r}{\lambda})^{1/2}]}
{(\frac{r}{\lambda})^{3/2}}\label{den}
\end{equation}
where A is the normalization constant, which is determined using the
relation
\begin{equation}
\int\rho(\vec r)d\vec r=N .
\end{equation}
As one can see from Eq.(\ref{den}), $\rho(\vec r)$ is singular at $r=0$.
In general one could choose a single particle density of the form
\begin{equation}
\rho(\vec r)=A\frac{exp[-(\frac{r}{\lambda})^{\nu}]}
{(\frac{r}{\lambda})^{3\nu}},\label{deng}
\end{equation}
where $\nu=1,2,3,4...or\ \frac{1}{2},\frac{1}{3},\frac{1}{4},...$. Integer
values of $\nu$ are not permissible because they make the normalization
constant infinite. Out of the fractional values, $\nu=\frac{1}{2}$ is
found to be most appropriate, because, as we shall see later, it gives
the expected upper limit for the critical mass of a neutron
star\cite{sm1}, beyond which black hole formation takes place. Any other
value of $\nu$ would give rise to a different value for the critical mass.
Also because if $\nu$ goes to zero (like $1/n$, $n\rightarrow\infty$),
$\rho(r)$ would tend to the case of a constant density as found in an
infinite many-fermion system. Having accepted the value $\nu=\frac{1}{2}$,
the parameter $\lambda$ associated with $\rho(r)$ is determined after
minimizing $E(\lambda)=<H>$ with respect to $\lambda$. This is how, we are
able to find the total energy of the system corresponding to its lowest
energy state.
After evaluating the integral shown in Eq.(\ref{kee}) and Eq.(\ref{pe}),
we
obtain
\begin{equation}
E(\lambda)=\frac{\hbar^2}{m}\frac{12}{25\pi}(\frac{3\pi
N}{16})^{5/3}\frac{1}{\lambda^2}-\frac{g^2N^2}{16}\frac{1}{\lambda}\label{bige}
\end{equation}
Differentiating this with respect to $\lambda$ and then equating it with
zero, we obtain the value of $\lambda$  at which the minimum occurs. This
is found as:
\begin{equation}
\lambda_0=\frac{72}{25}\frac{\hbar^2}{mg^2}(\frac{3\pi}{16})^{2/3}
\frac{1}{N^{1/3}}\label{lam}
\end{equation}
Here we are only concerned with the total kinetic energy of the system. At
$\lambda=\lambda_0$, we have,
\begin{equation}
<KE>=0.015441\frac{mg^4}{\hbar^2}N^{7/3}\label{kes}
\end{equation}
The total energy E of the system is found to be just negative of this.
\section{Derivation of Schwarzschild Radius}

We
now try to calculate the average velocity of a particle within the neutron
star using Eq.(\ref{kes}). Let us denote it by $<\vec v^2>$. If M denotes
the total mass of the neutron star, one writes
\begin{equation}
<KE>=\frac{1}{2}M<\vec v^2>\label{ket},
\end{equation}
where $M=Nm$. By comparing Eq.(\ref{ket}) with Eq.(\ref{kes}), one
obtains,
\begin{equation}
<\vec v^2>=0.030882\frac{g^4}{\hbar^2}N^{4/3}.
\end{equation}
From the expression for the total kinetic energy of an infinite
many-fermion system\cite{fetter}, one finds that the average velocity of a
particle within the system is $\sim 0.77v_f$, $v_f$ being the fermi
velocity\cite{fetter} of the particle within the system, which is maximum
velocity of
that particle. From this, one clearly sees that the maximum velocity of a
particle belonging to an infinite many-fermion system is greater than the
average particle velocity. In view of this fact, we could write the
maximum velocity of a particle within a neutron star as
\begin{equation}
<\vec v^2>_{max}=\alpha <v^2>=0.030882\alpha\frac{g^4}{\hbar^2}N^{4/3}
\end{equation}
where $\alpha$ is a constant whose value is to be greater than unity and
it is to be calculated later. $v_{max}$ can be identified as the escape
velocity of a particle within a neutron star.
According to special theory of relativity, $\vec v^2_{max}$ is to be less
than $c^2$, c being the velocity of light. That is,
\begin{equation}
0.030882\alpha\frac{g^4}{\hbar^2}N^{4/3}\le c^2,
\end{equation}
From this it follows that
\begin{equation}
N\le\frac{13.574409}{\alpha^{3/4}}(\frac{\hbar
c}{g^2})^{3/2}=N_c\ (say),\label{nu}
\end{equation}
having $g^2=Gm^2$. Substituting Eq.(\ref{nu}) in Eq.(\ref{lam}), one
finds
that,
\begin{equation}
\lambda_0\ge\lambda_c=\frac{Gm}{c^2}\alpha^{1/4}[0.8483718(\frac{\hbar
c}{g^2})^{3/2}].
\end{equation}
If we define the radius of a neutron star as $R_0=2\lambda_0$, we have
the expression for the  critical radius as,
\begin{equation}
R_c=2\lambda_c=2\frac{Gm}{c^2}\alpha^{1/4}[0.8483718(\frac{\hbar
c}{g^2})^{3/2}]=\frac{2GM}{c^2}\label{rad}
\end{equation}

Our identification about the radius $R$ of the star with $2\lambda_0$ is
based on the use of socalled quantum mechanical tunneling\cite{karp}
effect. Classically, it is well known that a particle has a turning
point
where the potential energy becomes equal to the total energy. Since the
kinetic energy and therefore the velocity are equal to zero at such a
point, the classical particle is expected to be turned around or reflected
by the potential barrier. From the present theory it is seen that the
turning point occurs at a distance $R=2\lambda_0$. This is the reason why
we identify $2\lambda_0$ with the radius of a star. For $R>2\lambda_0$,
a particle, belonging to the system, may have an access to the region beyond
$R>2\lambda_0$, because of quantum mechanical tunneling, but is
forbidden by classical theory.

$R_c$ as given in Eq.(\ref{rad}) is being identified as
the so called Schwarzschild radius which we have derived here by treating
the system as a quantum many-body system. When $R_0\le R_c$, the
corresponding
neutron star becomes a black hole. From Eq.(\ref{nu}), we therefore find
that the lowest mass of the neutron star beyond which black hole formation
takes place is given as
\begin{equation}
M_c=mN_c=\frac{13.574409}{\alpha^{3/4}}m(\frac{\hbar c}{g^2})^{3/2}
\end{equation}
In order to determine $\alpha$, we now try to evaluate the limiting mass
of a neutron star following the general expression for the radius of a
star. Beyond this mass, the black hole formation is likely to take place.
For that, we consider the situation when
\begin{equation}
(R_0=2\lambda_0)= (R_{sch}=\frac{2GM}{c^2}),
\end{equation}
where $M=Nm$. From this, we arrive at
\begin{equation}
N\ge(1.696758)(\frac{\hbar c}{g^2})^{3/2}=N_c.\label{nc}
\end{equation}
Since the expression in the right hand side of Eq.(\ref{nc}) should be
equal to the one given in right hand side of Eq.(\ref{nu}), we must have
$\alpha=16$. Under this situation, we have
\begin{equation}
v^2_{max}=0.494112\frac{g^4}{\hbar^2}N_c^{4/3},
\end{equation}
where $N_c=1.696758(\frac{\hbar c}{g^2})^{3/2}$, which, when evaluated,
becomes $3.7390777\times 10^{57}$. For a neutron star in which the number
of neutrons exceeds $N_c$, it has the tendency of forming a black hole. In
that case, its mass must exceed $M=M_c=mN_c=3.12213\ M_{\odot}$,
$M_{\odot}$
being the solar mass.

\section{Quantum Correction}

So far we have been discussing about the quantum mechanical derivation of
the
Schwarzschild radius $R_{Sch}$. The very form of $R_{Sch}$ shows that it
is a classical result, leaving aside the fact the number of particles N
within a neutron star\cite{wes} is to be less than $N_c$ where
$N_c=1.70(\frac{\hbar
c}{Gm_n^2})^{3/2}$, which involves the Planck's constant $\hbar$.
Now, inorder to account for the
quantum corrections to $R_{Sch}$, we go beyond the Hartree
contribution to the  total energy of the system. That is the exchange
correction or Hartree-Fock(HF) term\cite{beth} over the Hartree result
(direct contribution). Since
the
HF-correction term is non-local we make use of the local density
approximation\cite{beth} to write it as,
\begin{equation}
<PE>_{ex}=\frac{3}{2\pi}(3\pi^2)^{1/3}g^2\int d\vec r[\rho(\vec r)]^{4/3}.
\end{equation}
This when evaluated gives
\begin{equation}
<PE>_{ex}=\frac{27}{4}(\frac{1}{16\pi})^{4/3}(3\pi^2)^{1/3}g^2
\frac{N^{4/3}}{\lambda}.
\end{equation}
With the inclusion of this extra term, the  expression for $E(\lambda)$,
Eq.(\ref{bige}) is minimized with respect to $\lambda$ and we  arrive at
\begin{equation}
\lambda_0'=\frac{72}{25}(\frac{3\pi}{16})^{2/3}\frac{\hbar^2}{mg^2}
\frac{1}{N^{1/3}}[1+\frac{1.8010}{N^{2/3}}]\label{lamor}
\end{equation}
Following the  argument discussed earlier, we identify the radius of the
neutron star by $R'_0=2\lambda'_0$. As before writing
${v'}^2_{max}=16<{v'}^2>$
and using the condition that ${v'}^2_{max}\le c^2$, we  obtain
\begin{equation}
N\le N'_c=1.696758(\frac{\hbar c}{g^2})^{3/2}[1+0.4747761(\frac{g^2}{\hbar
c})]
\end{equation}
Corresponding to $N'_c$ , the new  expression for the critical radius
$R'_c$ becomes
\begin{equation}
R'_c=2\lambda'_c=2\frac{GM_c}{c^2}[1+0.7912723(\frac{g^2}{\hbar
c})]\label{rp}
\end{equation}
where $M_c=mN_c=1.696758(\frac{\hbar c}{g^2})^{3/2}$. The above
expression, Eq. (\ref{rp}) is obtained by keeping  terms upto order
$(\frac{g^2}{\hbar c})$ only in Eq. (\ref{lamor}).
For $N>N'_c$, the neutron star is likely to go over the black hole stage.
From Eq.(\ref{rp}), we find that the second term within the square
bracket, forms
the quantum correction to the Schwarzschild radius. As expected, it
involves the gravitational fine structure constant $(\frac{g^2}{\hbar
c})$. Since it is of the order $10^{-39}$,  obviously it makes  an
extremely small correction to $R_{Sch}$.
It has been shown earlier\cite{duff} that using the quantum field
theoretic method and by
including a single-closed-loop in the self energy, a quantum correction to
the classical Schwarzschild solution of the order of $\sim G^2$ can be
found. This
comes from the gravity sector. The correction that we get is also of the
order $\sim G^2$ but it comes from the
exchange part of the matter-energy sector of the black hole.

\section{Conclusion}
We in this paper derive the Schwarzschild radius of a black hole from a
condensed matter point of view by using a single particle density
distribution for the many-body self-gravitating system which ultimately
forms a black hole. By incorporating the quantum mechanical exchange
interaction, we also find a thin correction to the Schwarzschild radius
which we designate as the skin of the black hole.

We thank F. Hehl for critically reading the manuscript and bringing to our
notice the Ref.3.