\section{Introduction}
Throughout, we denote by $\sN, \, \sZ, \, \sQ$ and $\sR$ the
natural, integer, rational and real numbers, respectively, and we
fix a number $d \in \sN$. For arbitrary $n \in \sN$, we compare the
set of all integer points in the homothetic copy $n \Pi$ of a
lattice polytope $\Pi \subseteq \sR^d$ (that means $\Pi$ is the
convex hull of a finite number of points in $\sZ^d$) with the set of
sums $x_1 + \cdots + x_n$ with $x_1, \, \ldots, \, x_n \in \Pi \cap
\sZ^d$. It is easy to see that the latter set is always contained in
the first -- but in general, they are different. We give conditions
on the polytope under which these two sets coincide and we discuss
two notions of boundary for subsets of $\sZ^d$ and, more generally,
of a finitely generated (not necessarily commutative) discrete
group.

The motivation for this paper stems from the study of projection
methods for the approximate solution of operator equations. Let $A$
be a bounded linear operator acting on a Banach space. To solve the
operator equation $Au = f$ numerically, one chooses a sequence
$(Q_n)$ of projections (usually assumed to be of finite rank and to
converge strongly to the identity operator) and replaces the
equation $Au=f$ by the sequence of the linear systems $Q_n A Q_n u_n
= Q_n f$. What one expects is that, under suitable conditions, the
solutions $u_n$ of these systems converge to the solution $u$ of the
original equation. If the Banach space on which $A$ lives consists
of functions on a countable set $Y$ (think of a sequence space
$l^2(Y)$, for example) then it is convenient to choose an increasing
sequence $(Y_n)$ of finite subsets of $Y$ and to specify $Q_n$ as
the operator $P_{Y_n}$ which restricts a function on $Y$ to $Y_n$.

In this paper, we will be concerned with the case when $Y$ is a
finitely generated discrete group $\Gamma$. In case $\Gamma$ is the
additive group $\Z^d$, a typical (rather geometric) approach
\cite{RaRoSi:Book,Roch:FSM,Li:Habil,Li:FSMsubs} to design a projection method
is to fix a compact set (for example a lattice polytope) $\Pi
\subseteq \R^d$ and to consider the operator $P_{\Pi_n}$ of
restriction to (likewise, the operator of multiplication by the
characteristic function of) the set
\begin{equation} \label{eq:Pn}
\Pi_n := (n \Pi) \cap \Z^d
\end{equation}
for $n \in \sN$. If we assume that the origin is in the interior of
$\Pi$ then it follows that
\begin{equation} \label{eq:incmethod}
\Pi_n \subseteq \Pi_{n+1} \quad \textrm{for all} \; n \in \N \qquad
\textrm{and} \qquad \bigcup_{n \in \N} \Pi_n = \sZ^d,
\end{equation}
whence the sequence $(\Pi_n)_{n\in\N}$ gives rise to an increasing
sequence of finite-dimensional projection operators on $l^2(\sZ^d)$
that strongly converges to the identity operator as $n \to \infty$.

In the case of a general finitely generated group $\Gamma$, this
geometric approach is clearly infeasible. A natural idea here is to
fix a finite set $\Omega \subseteq \Gamma$ of generators of $\Gamma$
(that is, we assume that $\Omega$ generates $\Gamma$ as a
semi-group) and to consider the set
\begin{equation} \label{eq:Pn'}
\Omega_n :=  \{x_1 x_2 \cdots x_n : x_1, \, x_2, \, \dots, \, x_n \in \Omega \}
\end{equation}
of all words of length $n \ge 1$ over the alphabet $\Omega$ in place
of \eqref{eq:Pn}. If $\Omega$ is symmetric (in the sense that
$x^{-1} \in \Omega$ if $x \in \Omega$), contains the identity
element $e$ of $\Gamma$ (in analogy to the above approach in $\Z^d$)
and if $\Gamma$ is equipped with the word metric over $\Omega$ then
$\Omega_n$ is the disk of radius $n$ in $\Gamma$ around the identity
$e$. Moreover, one gets that also $(\Omega_n)_{n \in \N}$ yields an
increasing sequence of finite-dimensional projections with strong
limit identity, i.e., \eqref{eq:incmethod} holds with $\Pi_m$
replaced by $\Omega_m$ and $\Z^d$ by $\Gamma$.

A natural question before proposing the latter approach for general
finitely generated groups $\Gamma$ is whether or not the geometric
approach \eqref{eq:Pn} and the algebraic approach \eqref{eq:Pn'}
coincide if we have $\Gamma = \Z^d$ and use $\Omega := \Pi \cap
\Z^d$ as finite set of generators in \eqref{eq:Pn'} with a symmetric
lattice polytope $\Pi \subseteq \R^d$ containing the origin in its
interior. This question is discussed in Section \ref{sec:Pn-vs-Pn'}.
We will give conditions on the polytope $\Pi$ under which
\eqref{eq:Pn} and \eqref{eq:Pn'} coincide -- but in general they do
not.

In Section \ref{sec:bdry} we address a further question that arises
in the study of projection methods. In \cite{Roch:FSMgroup} it has
been pointed out that the ``boundaries'' $\partial_\Omega \Omega_n$
(if appropriately defined) of the sets $\Omega_n$ (or $\Pi_n$) hold
the key to the answer to whether or not the projection method
\[
P_{\Omega_n} A P_{\Omega_n} u_n = P_{\Omega_n} f, \qquad n \in \sN,
\]
yields stable approximations $u_n$ to the solution $u$ of $Au=f$. In
Section \ref{sec:bdry} we give special emphasis to the question
whether the ``algebraic boundaries'' $\partial_\Omega \Omega_n$
coincide with the corresponding ``numerical boundaries'' $\Omega_n
\setminus \Omega_{n-1}$.
\section{Lattice polytopes, enlargements and integer points} \label{sec:Pn-vs-Pn'}
Now fix $d\in\N$, let $e_1,...,e_d$ be the standard unit vectors of
$\R^d$, and denote the unit simplex $\conv \{0,e_1,...,e_d\}$ by
$\sigma_d$. (For standard notions on convex polytopes we recommend
\cite{Henk97,Gruenbaum,Ziegler}; for lattice polytopes see
\cite{BeckRobins,GritzWills,Gruber,Schrijver,White}.)

Given a non-empty subset $S$ of $\R^d$ and a positive integer $n$,
we write
\[
nS := \{ns:s\in S\}\quad\textrm{and}\quad n*S := \{s_1+\dots
+s_n:s_1,...,s_n\in S\} = S + \dots + S
\]
for the ratio-$n$ homothetic copy and $n$-fold Minkowski sum of $S$,
respectively. For convenience, we also set $0 S := \{0\}$ and $0 * S
:= \{0\}$. It is easy to see that $nS=n*S$ holds for all $n \in \N$
if $S$ is convex. Indeed, the inclusion $nS \subseteq n*S$ is always
true, and, by convexity of $S$, $s_1+\dots +s_n$ can be written as
$ns$ with $s=(s_1+\dots +s_n)/n\in S$ for all $n \in \N$ and
$s_1,...,s_n \in S$. (Note that equality of $nS$ and $n*S$ for all
$n\in\N$ does not imply convexity of $S$, as $S=\Q$ shows.)

For a convex set $\Pi\subseteq\R^d$ containing at least two integer
points, it is clear that $(n\Pi)\cap\Z^d\ne n(\Pi\cap\Z^d)$ as soon
as $n>1$. But (as motivated in the introduction) a much more
interesting question is whether or not
\begin{equation} \label{eq:2enlargements}
(n\Pi)\cap\Z^d\ =\ n*(\Pi\cap\Z^d)
\end{equation}
is true for all $n\in\N$. We will study this question for certain
polytopes $\Pi$.

Let $v_1,...,v_k$ be points of $\Z^d$ with their affine hull equal
to $\R^d$ and put $\Pi=\conv\{v_1,...,v_k\}$. $\Pi$ is a so-called
{\sl lattice polytope} as all its vertices are in $\Z^d$. We will
suppose that there is no proper subset $I$ of $\{1,...,k\}$ with
$\Pi=\conv\{v_i:i\in I\}$, so that $v_1,...,v_k$ are the vertices of
$\Pi$. If $\Pi\cap\Z^d$ only consists of the vertices of $\Pi$ then
$\Pi$ is called an {\sl elementary polytope}
\cite{Kantor98,Schrijver} (or a {\sl lattice-(point-)free polytope}
\cite{Henk98,Kantor99}). The following lemma is fairly standard:

\begin{lemma} \label{lem:fullyoccupied}
Let $A\in\Z^{d\times d}$ be a matrix and $a_1,...,a_d\in\Z^d$ its columns.

{\bf a)} The following conditions are equivalent:

\begin{tabular}{rl}
  (i)&$A(\Z^d)=\Z^d$.\\
 (ii)&$A^{-1}$ is an integer matrix.\\
(iii)&$\det A=\pm 1$.\\
 (iv)&The parallelotope $A([0,1]^d)$ spanned by $a_1,...,a_d$ has volume $1$.\\
  (v)&The parallelotope $A([0,1]^d)$ is elementary.
\end{tabular}

{\bf b)} The condition

\begin{tabular}{rl}
 (vi)&The simplex $A(\sigma_d)=\conv\{0,a_1,...,a_d\}$ is elementary.
\end{tabular}

is necessary for $(i)$--$(v)$; it is moreover sufficient iff $d\in\{1,2\}$.
\end{lemma}

If $(i)$--$(v)$ hold then $A$ is called an {\sl integer unimodular
matrix} \cite{Gruber,Schrijver} and the simplex $A(\sigma_d)$ has
volume $1/d!$ and is sometimes called a {\sl primitive} (or {\sl unimodular})
{\sl simplex} (e.g. \cite{Kantor98}). So primitive simplices are elementary,
and the converse holds iff $d\in\{1,2\}$.

For the sake of completeness, we give a short sketch of the proof
of Lemma \ref{lem:fullyoccupied}:

\begin{proof}
Part a) follows by standard arguments using $\det A^{-1} = 1/\det A$
and $\vol(A([0,1]^d))=|\det A|\,\vol([0,1]^d)$. The implication
$(v)\Rightarrow (vi)$ holds by $\sigma_d\subseteq [0,1]^d$. For
$d=1$, the implication $(vi)\Rightarrow (v)$ is clear by $\sigma_1 =
[0,1]^1$. For $d=2$, if $x$ is an integer non-vertex point in
$A([0,1]^2)$, then also $a_1+a_2-x$ is an integer non-vertex point
in $A([0,1]^2)$. But one of the two points is in $A(\sigma_2)$, so
that $(vi)\Rightarrow (v)$ holds. For $d\ge 3$, there are elementary
but not primitive simplices (see Examples \ref{ex:simpl} a and b
below).
\end{proof}

Here are two slightly different constructions leading to elementary
but not primitive simplices in dimension $d\ge 3$.

\begin{example} \label{ex:simpl}
{\bf a) } Let $d\ge 3$, fix an $m\in\N$, take $a_1:=e_1,\ a_2:=e_2,\
...\ ,\,a_{d-1}:=e_{d-1}\in\Z^d$ and $a_d:=(-1,...,-1,m)^\top$, and
let $A\in\Z^{d\times d}$ be the matrix with columns $a_1,...,a_d$.
Then
\[
\Sigma_{d,m}\ :=\ A(\sigma_d)\ =\ \conv\{0,a_1,...,a_d\}
\]
is primitive iff $m=\det A=1$. On the other hand, the number of
integer points in $\Sigma_{d,m}$ apart from its $d+1$ vertices is
equal to $k=\lfloor m/d\rfloor$ (integer division), and these $k$
other integer points are $1e_d,...,ke_d$. (To see this, look at the
projection of $\Sigma_{d,m}$ to the hyperplane spanned by
$e_1,...,e_{d-1}$.) So for $m\in\{2,...,d-1\}$ we have an elementary
but not primitive simplex.

{\bf b) } If we change the last column in the above example from
$a_d=(-1,...,-1,m)^\top$ to $a'_d:=(1,...,1,m)^\top$ with $m\in\N$
and call the new matrix $A'$ then, again, the simplex
\[
\Sigma_{d,m}'\ :=\ A'(\sigma_d)\ =\ \conv\{0,a_1,...,a_{d-1},a'_d\}
\]
is not primitive for $m=\det A'\ge 2$ but now it is elementary for
all $m\in\N$. So this simplex $\Sigma'_{d,m}$ can have arbitrarily
large volume $m/d!$ without containing any integer points other than
its vertices. The simplices $\Sigma'_{d,m}$ were first considered
(because of this property) by Reeve \cite{Reeve} in case $d=3$
and have since been termed {\sl Reeve simplices}. There are results
that relate the maximal volume of an elementary polytope in $\R^d$
to its surface area \cite{BHW} or its inradius \cite{Henk98}.
\end{example}

Given a full-dimensional lattice polytope $\Pi\subseteq\R^d$ and a
set $\T$ of full-dimensional lattice simplices $S_1,...,S_m\subseteq
\Pi$ with
\[
\bigcup_{i=1}^m S_i\ =\ \Pi\qquad\textrm{and}\qquad
S_i\cap S_j\ \textrm{is a face of both $S_i$ and $S_j$},\ \forall i,j,
\]
then the set $\T=\{S_1,...,S_m\}$ is called a {\sl triangulation} of
$\Pi$. The triangulation $\T$ is called {\sl elementary} or {\sl
primitive} if all its elements $S_i$ are, respectively, elementary
or primitive simplices.

Here is our main result on the equality \eqref{eq:2enlargements}:

\begin{proposition} \label{prop:main}
If a full-dimensional lattice polytope $\Pi\subseteq\R^d$ possesses
a primitive triangulation then equality \eqref{eq:2enlargements}
holds for all $n\in\N$.
\end{proposition}
\begin{proof}
Let $n\in\N$ and $x\in n*(\Pi\cap\Z^d)$. Then $x=p_1+...+p_n$ for
some $p_1,...,p_n\in \Pi\cap\Z^d$, so that $x\in\Z^d$ and $x=np$
with $p=(p_1+...+p_n)/n$. But $p\in \Pi$ by convexity of $\Pi$.

Now let $x\in (n\Pi)\cap\Z^d$, so $x=np$ is an integer point with
$p\in \Pi$. Let $\T=\{S_1,...,S_m\}$ be a primitive triangulation of
$\Pi$ and let $w_0,...,w_d$ be the vertices of a simplex $S_i$ that
contains $p$. Now there is a unique way to write $p$ as a convex
combination of $w_0,...,w_d$. So there are $\alpha_0,...,\alpha_d\in
[0,1]$ so that $p=\alpha_0 w_0+...+\alpha_d w_d$ and
$\alpha_0+...+\alpha_d=1$. Together with $x=np$ this implies
\begin{equation} \label{eq:convcomb}
\left(\begin{array}{cccc} |&|& & |\\
w_0&w_1&\cdots&w_d\\
| & |& & |\\[0.8ex] \hline \\[-2ex]1 & 1 &\cdots & 1
\end{array}\right)
\left(\begin{array}{c} n\alpha_0\\ n\alpha_1\\ \vdots\\
n\alpha_d\end{array}\right) \ =\ n\left(\begin{array}{c} |\\p\\
|\\[0.8ex] \hline \\[-2ex] 1\end{array}\right)
\ =\ \left(\begin{array}{c} |\\x\\
|\\[0.8ex] \hline \\[-2ex] n\end{array}\right).
\end{equation}
If we refer to the matrix in \eqref{eq:convcomb} as $M$ then, after
subtracting the first column from all the others and then expanding
by the last row,
\begin{eqnarray*}
\det M &=& \det \left(\begin{array}{cccc} |&|& & |\\
w_0&w_1-w_0&\cdots&w_d-w_0\\
| & |& & |\\[0.8ex] \hline \\[-2ex]1 & 0 &\cdots & 0
\end{array}\right)\\
& = & (-1)^{d+1} \det \left(\begin{array}{ccc} |& & |\\
w_1-w_0&\cdots&w_d-w_0\\
|& & |
\end{array}\right)\ \in\ \{\pm 1\}
\end{eqnarray*}
by Lemma \ref{lem:fullyoccupied} since $w_1-w_0,\,\cdots,w_d-w_0$
span the primitive simplex $S_i-w_0$. So $M^{-1}$ exists and is an
integer matrix. By \eqref{eq:convcomb}, it follows that
$\beta_0:=n\alpha_0,\,\cdots\,,\,\beta_d:=n\alpha_d$ are integers
since $M^{-1}$ and $x$ have integer entries. Summarizing, we get
that
\begin{equation} \label{eq:herewego}
\beta_0 w_0\,+\,\beta_1 w_1\,...\,+\,\beta_d w_d\ =\ x,
\end{equation}
where $\beta_0,...,\beta_d\in\{0,...,n\}$ and
$\beta_0+...+\beta_d=n$, so that \eqref{eq:herewego} is the desired
decomposition of $x$ into a sum of $n$ elements from $\Pi\cap\Z^d$.
\end{proof}

It is not clear to us whether the existence of a primitive triangulation
is necessary for equality \eqref{eq:2enlargements} to hold for all $n\in\N$.
(Is it possible that every $p\in\Pi$ is contained in a primitive simplex
$S(p)\subset\Pi$ without the existence of a ``global'' primitive triangulation
of $\Pi$?)

It is not hard to see that every lattice polytope $\Pi$ possesses an
elementary triangulation. (Existence of a triangulation can be shown
by induction over the number of vertices of $\Pi$, and every
non-elementary simplex $S_i$ can be further triangulated with
respect to its integer non-vertex points.) Existence of a primitive
triangulation however is a different question -- at least in
dimensions $d\ge 3$.

\begin{corollary}
If $\Pi$ is a full-dimensional lattice polytope in $\R^d$ with
$d\in\{1,2\}$ then equality \eqref{eq:2enlargements} holds for all
$n\in\N$.
\end{corollary}
\begin{proof}
Every lattice polytope has an elementary triangulation. In
dimensions $d\in\{1,2\}$, by Lemma \ref{lem:fullyoccupied} b), an
elementary triangulation is always primitive. Now apply Proposition
\ref{prop:main}.
\end{proof}

In dimension $d\ge 3$, it is generally a difficult question whether
or not a given lattice polytope $\Pi$ has a primitive triangulation
(see e.g. \cite{BetkeKneser,Kantor98}). The simplices $\Sigma_{d,m}$
in Example \ref{ex:simpl} a) with $m\in\{2,...,d-1\}$ and
$\Sigma_{d,m}'$ in \ref{ex:simpl} b) with $m\in\{2,3,...\}$ are
examples of lattice polytopes that have no primitive triangulation.
They are also examples, where \eqref{eq:2enlargements} is not valid
for general $n\in\N$. For example, $2\Sigma_{3,2}$ contains $e_3$,
which cannot be written as the sum of two integer points from
$\Sigma_{3,2}$. It is even possible to give examples $\Pi$ where,
for a given $k\in\N$, \eqref{eq:2enlargements} starts to fail at
$n>k$ while holding true for $n=1,2,...,k$. As an example, take
$\Pi=\Sigma_{2k+1\,,\,2}$.

The more specific question posed in the introduction is whether the
fact that $\Omega:=\Pi\cap \Z^d$ ~ (i) contains the origin, (ii) is
symmetric (i.e. $-x\in\ \Omega$ if $x\in \Omega$), and (iii)
generates $\Z^d$, i.e.
\begin{equation} \label{eq:generates}
\bigcup_{n\in\N} n*\Omega\ =\ \Z^d,
\end{equation}
guarantees equality \eqref{eq:2enlargements} for all $n\in\N$. But
also that has to be answered in the negative, as the example
$\Pi=\conv(\Sigma_{3,3}\,\cup\,-\Sigma_{3,3})=\conv\{\pm e_1\,,\,
\pm e_2\,,\, \pm(-1,-1,3)^\top\}\subseteq\R^3$ shows. Indeed,
$(-1,-1,1)^\top$ is in $2\Pi$ but not in $2*\Omega$ with
$\Omega=\Pi\cap\Z^3=\{\pm e_1\,,\, \pm e_2\,,\, \pm
e_3\,,\,\pm(-1,-1,3)^\top\,,\, 0\}$.

After all these examples, here are some results on the positive
side:

The symmetric hypercube $[-1,1]^d$ is the union of $2^d$ shifted
copies of $[0,1]^d$, each of which has a primitive triangulation.
The cross-polytope $\conv\{\pm e_1,...,\pm e_d\}$ triangulates into
$2^d$ unit simplices around the origin. Much more involved, there is
the following result by Kempf, Knudsen, Mumford and Saint-Donat
\cite{KMSD}:

\begin{lemma} \label{lem:triang}
For every lattice polytope $\Pi\subseteq\R^d$, there is an integer
$k\in\N$ such that $k\Pi$ possesses a primitive triangulation.
\end{lemma}

Consequently, for every lattice polytope $\Pi$, there is a $k\in\N$
such that $k\Pi$ satisfies \eqref{eq:2enlargements} in place of
$\Pi$ for all $n\in\N$.

\begin{remark}
The two conditions \eqref{eq:generates} with $\Omega:=\Pi\cap\Z^d$,
which says that $\Omega$ generates all of $\Z^d$, and
$0\in\inte(\Pi)$, which implies that $n*\Omega\subseteq
(n+1)*\Omega$, are connected with each other and with the validity
of \eqref{eq:2enlargements} for all $n\in\N$. Firstly, if
\eqref{eq:2enlargements} holds for all $n\in\N$ and $0\in\inte(\Pi)$
then
\[
\bigcup_{n\in\N} n*(\Pi\cap\Z^d)\ =\ \bigcup_{n\in\N} (n\Pi)\cap\Z^d
\ =\ \left(\bigcup_{n\in\N} n\Pi\right)\cap\Z^d\ =\ \Z^d
\]
 so that \eqref{eq:generates} holds.
On the other hand, from \eqref{eq:generates} it follows, by the
trivial inclusion ``$\supseteq$'' in \eqref{eq:2enlargements}, that
\[
\Z^d\ =\ \bigcup_{n\in\N} n*(\Pi\cap\Z^d)\ \subseteq\
\bigcup_{n\in\N} (n\Pi)\cap\Z^d \ =\ \left(\bigcup_{n\in\N}
n\Pi\right)\cap\Z^d\ \subseteq\ \Z^d,
\]
whence, by the convexity of $\Pi$, $\cup\, n\Pi=\R^d$ and hence
$0\in\inte(\Pi)$.
\end{remark}
\section{Boundaries of subsets of a group $\Gamma$} \label{sec:bdry}
Let $\Gamma$ be a finitely generated discrete group with identity
element $e$. We are going to introduce some notions of topological
type. Note that the standard topology on $\Gamma$ is the discrete
one; so every subset of $\Gamma$ is open with respect to this
topology.

Let $\Omega$ be a finite subset of $\Gamma$ which contains the
identity element $e$ and which generates $\Gamma$ as a semi-group,
i.e., if we set $\Omega_0 := \{e\}$ and if we let $\Omega_n$ denote
the set of all words of length at most $n$ with letters in $\Omega$
for $n \ge 1$, then $\cup_{n \ge 0} \Omega_n = \Gamma$. Note also
that the sequence $(\Omega_n)$ is increasing; so the operators
$P_{\Omega_n}$ can play the role of the finite section projections
$P_{Y_n}$ from the introduction, and in fact we will obtain some of
the subsequent results exactly for this sequence.

With respect to $\Omega$, we define the following
``algebro-topological'' notions. Let $A \subseteq \Gamma$. A point
$a \in A$ is called an {\sl $\Omega$-inner} point of $A$ if $\Omega
a := \{ \omega a : \omega \in \Omega \} \subseteq A$. The set
$\mbox{int}_\Omega A$ of all $\Omega$-inner points of $A$ is called
the {\sl $\Omega$-interior} of $A$, and the set $\partial_\Omega A
:= A \setminus \mbox{int}_\Omega A$ is the {\sl $\Omega$-boundary}
of $A$. Note that we consider the $\Omega$-boundary of a set always
as a part of that set. (In this point, the present definition of a
boundary differs from other definitions in the literature; see
\cite{Ada1} for instance.) One easily checks that
\begin{equation} \label{e030309.5}
\Omega_{n-1}\, \subseteq\, \mbox{int}_\Omega \Omega_n\, \subseteq\,
\Omega_n \quad \mbox{and} \quad
\partial_\Omega \Omega_n\, \subseteq\, \Omega_n \setminus \Omega_{n-1}
\end{equation}
for each $n \ge 1$.

Recall from \cite{Roch:FSMgroup} that there are at least two reasons
for the interest in the boundaries $\partial_\Omega \Omega_n$:
\begin{itemize}
\item
The sequence $(P_{\partial_\Omega \Omega_n})_{n \ge 1}$ belongs to
the $C^*$-algebra $\cS ({\sf Sh} (\Gamma))$ which is generated by
all finite sections sequences $(P_{\Omega_n} A P_{\Omega_n})_{n \ge
1}$ where $A$ runs through the operators on $l^2(\Gamma)$ of left
shift by elements in $\Gamma$ (i.e., they are given by the
left-regular representation of $\Gamma$ on $l^2(\Gamma)$), and it
generates the quasicommutator ideal of that algebra.
\item
There is a criterion for the stability of sequences in $\cS ({\sf
Sh} (\Gamma))$ which can be formulated by means  of limit operators,
and it turns out that it is sufficient to consider limit operators
with respect to sequences taking their values in the boundaries
$\partial_\Omega \Omega_n$.
\end{itemize}
For details, see \cite{Roch:FSMgroup}. In many instances one
observes that the ``algebraic'' boundary $\partial_\Omega \Omega_n$
coincides with the ``numerical'' boundary $\Omega_n \setminus
\Omega_{n-1}$; in fact, one inclusion holds in general as mentioned
in (\ref{e030309.5}). We will see now that the reverse inclusion can
be guaranteed if $\Gamma = \sZ^d$ and if $\Omega$ arises from a
lattice polytope $\Pi$ such that \eqref{eq:2enlargements} holds for
all $n\in\N$.
\begin{proposition} \label{01032010:p1}
Let $\Pi$ be a lattice polytope in $\sZ^d$ which satisfies
\eqref{eq:2enlargements} and set $\Omega := \Pi \cap \sZ^d$. Then
\begin{equation} \label{01032010:e2}
\partial_\Omega (n * \Omega) = (n * \Omega) \setminus ((n-1) * \Omega)
\end{equation}
holds for all positive integers $n$.
\end{proposition}
\begin{proof} As mentioned above, it is sufficient to show that
\[
(n * \Omega) \setminus ((n-1) * \Omega) \subseteq \partial_\Omega (n * \Omega).
\]
We start with working on the continuous level and check first the implication
\begin{equation} \label{01032010:e3}
\mbox{If} \quad x \in n \Pi \setminus (n-1) \Pi, \quad \mbox{then} \quad \Pi + x \not\subset n \Pi.
\end{equation}
Indeed, write $x$ as $t \omega$ with $\omega \in \partial \Pi$ (=
the usual topological boundary of $\Pi$) and $n-1 < t \le n$. Then
$x + \omega = (t+1) \omega$ with $t+1 > n$, whence $x + \omega
\not\in n \Pi$.

In the next step we show that $\omega$ can be chosen such that $x +
\omega$ becomes a grid point for $x$ a grid point. Indeed, let $x
\in \left( n \Pi \setminus (n-1) \Pi \right) \cap \sZ^d$. Consider
the points $x + \omega_i$ where the $\omega_i$, $i = 1, \ldots, k$,
run through the (integer) vertices of $\Pi$. If we would have $x +
\omega_i \in n \Pi$ for each $i$, then we would have
\[
x + \Pi = \conv \{x + \omega_i : i = 1, \ldots, k \} \subseteq n \Pi
\]
by convexity of $\Pi$, which contradicts (\ref{01032010:e3}). Hence,
for each $x \in \left( n \Pi \setminus (n-1) \Pi \right) \cap
\sZ^d$, there is a vertex $\omega_i$ of $\Pi$ such that
\[
x + \omega_i \in \left( (n+1) \Pi \setminus n \Pi \right) \cap \sZ^d.
\]
Employing the assumption \eqref{eq:2enlargements} we conclude that,
for each $x \in n* \Omega \setminus (n-1)* \Omega$ there is a
$\omega_i \in \Omega$ such that $x + \omega_i \in (n+1) * \Omega
\setminus n * \Omega$. Hence, $x$ is in the $\Omega$-boundary of $n
* \Omega$.
\end{proof}

We proceed with an example which shows that the generalized version
of (\ref{01032010:e2}),
\begin{equation} \label{01032010:e2a}
\partial_\Omega \Omega_n = \Omega_n \setminus \Omega_{n-1}\,,
\end{equation}
does not hold for general subsets $\Omega$ of a finitely generated
discrete group $\Gamma$ and $n\in\N$. Consider the matrices
\[
\omega_0 := \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}, \qquad
\omega_1 := \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \qquad
\omega_2 := \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix}
\]
and
\[
\omega_3 := \omega_2 \omega_1 = \begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix}, \qquad
\omega_4 := \omega_2^{-1} = \begin{pmatrix}
1 & -1 \\
0 & 1
\end{pmatrix}, \qquad
\omega_5 := \omega_4 \omega_1 = \begin{pmatrix}
-1 & 1 \\
1 & 0
\end{pmatrix}.
\]
Then $\Omega := \{\omega_i : i = 0, \ldots, 5 \}$ generates the
group $GL(2, \, \sZ)$ as a semi-group (clearly, $\Omega$ is not
minimal as a generating system: $\omega_0, \, \omega_1, \, \omega_2,
\, \omega_4$ already generate this group). One easily checks that
$\omega_0 \omega_1 = \omega_1, \; \omega_1 \omega_1 = \omega_0, \;
\omega_2 \omega_1 = \omega_3, \; \omega_3 \omega_1 = \omega_2, \;
\omega_4 \omega_1 = \omega_5$ and $\omega_5 \omega_1 = \omega_4$,
whence $\Omega \omega_1 \subseteq \Omega$. Thus, $\omega_1 \in
\Omega_1 \setminus \Omega_0$, but $\omega_1 \not\in \partial_\Omega
\Omega_1$. So, (\ref{01032010:e2a}) is violated already for $n = 1$.

Let us conclude with a curious consequence of the coincidence
(\ref{01032010:e2}) of the boundaries. We mentioned above that the
sequence $(P_{\partial_\Omega \Omega_n})_{n \ge 1}$ always belongs
to the $C^*$-algebra $\cS ({\sf Sh} (\Gamma))$ generated by the
finite sections sequences $(P_{\Omega_n} A P_{\Omega_n})_{n \ge 1}$
where $A$ is constituted by operators of left shift by elements of
$\Gamma$. Under the conditions of Proposition \ref{01032010:p1}, we
conclude that the sequence $(P_{\Omega_n} - P_{\Omega_{n-1}})$
belongs to $\cS ({\sf Sh} (\sZ^d))$. In particular, the sequence
$(P_{\Omega_{n-1}}) = (P_{\Omega_n}) - (P_{\Omega_n} -
P_{\Omega_{n-1}})$ belongs to $\cS ({\sf Sh} (\sZ^d))$.
Consequently, with each sequence $(P_{\Omega_n} A P_{\Omega_n})_{n
\ge 1}$, the sequence
\[
(P_{\Omega_{n-1}}) \, (P_{\Omega_n} A P_{\Omega_n}) \, (P_{\Omega_{n-1}})
= (P_{\Omega_{n-1}} A P_{\Omega_{n-1}})
\]
(with the operators $P_{\Omega_{n-1}} A P_{\Omega_{n-1}}$ considered
as acting on the range of $P_{\Omega_n}$) also belongs to $\cS ({\sf
Sh} (\sZ^d))$. In particular, the algebra $\cS ({\sf Sh} (\sZ^d))$
contains a shifted copy (hence, infinitely many shifted copies) of
itself. The same fact clearly holds for every algebra which is
generated by finite sections sequences $(Q_n A Q_n)$ and contains
the sequence $(Q_n - Q_{n-1})$. A less trivial example where this
happens is the algebra of the finite sections method for operators
in (a concrete representation of) the Cuntz algebra ${\sf O}_N$ with
$N \ge 2$ \cite{Roch:Cuntz}.


\section{Introduction}
\label{Intro}
Correlated systems are fundamental in developing and exploiting the possibilities arising from quantum mechanics. A premier role in such an endeavor is played by chains of interacting quantum spins~\cite{sugato}, in light  of their wide applicability to many physical settings and their ability to simulate the behavior of strongly correlated systems in condensed matter~\cite{heisen,abel}. They are a key ingredient in many distributed quantum computation and communication protocols~\cite{bose2}. If we are truly to benefit from the potential of such resources, a thorough understanding of the nature of such correlations is essential. In this respect, it is particularly important to gather information on the distribution and amount of genuine multipartite quantum correlations in one of such systems so as to be able to tailor specific protocols to the available structure of entanglement among the parties of a chain or lattice.

Entanglement and non-locality are the key figures of merit for non-classical correlations. Already a good understanding exists for simple bipartite entangled systems and there is much research in the direction of multipartite entangled states~\cite{HHHH}. Equally, non-locality is truly at the forefront of current research as the premier manifestation of true deviation  of the behavior of a system from classicality~\cite{popescu, exp, brunner}. Non-locality embodies a stronger type of correlation than entanglement and the development of tools to detect, quantify and characterize it is moving swiftly in the field. Of course one should be mindful of how any such system can be experimentally generated and utilized and of the fact that unavoidable interactions with a surrounding environment result in the loss of such strong quantum correlations. Methods to control the affect of these unwanted interactions (resulting in either losses or pure decoherence) have been proposed, ranging from bang-bang techniques~\cite{bang} to the active use of the quantum Zeno effect~\cite{zeno}. All such methods, to the best of our knowledge, require either heavy influences on the system at hand or expensive additional resources for entanglement protection. The generation of non-classical multipartite correlated states is already a formidable task: if loss/decoherence-tolerant methods for their production have to be in order, this would necessarily imply a frequently prohibitively escalating experimental cost. It should thus be evident that a scenario where highly quantum correlated states are produced as the ground state of a given many-body interacting model and {\it naturally} maintained while in the proximity of environmental effects, is endowed with extreme physical appeal.

In this paper we present a study of the Ising model of spin-1/2 particles (qubits) in a transverse magnetic field. This model has attracted wide attention due to its exactly solvable nature and the attractive natural properties of its ground states~\cite{buzek, kendon,iranians}. We go significantly further than the available literature on this model so far by addressing the case of genuine multipartite quantum correlations and their nature. We examine the zero temperature dynamics of entanglement and non-locality highlighting by means of an efficient hybrid approach for the determination of multi-site correlation functions. The finite temperature regime is then addressed and we show the ability to control and preserve the non-locality present in the system with minimal additional effort. We show the key ingredient in the model is the presence and manipulation of the magnetic field.

The remainder of the paper is organized as follows. In Sec.~\ref{Secmodel} we introduce the physical model for spin-spin coupling considered in this work and sketch the technique required for its full diagonalization. Sec.~\ref{Sectools} discusses the technical tools used in our investigation for the quantification of multipartite entanglement and quantum non-locality. Sec.~\ref{nonclassical} presents the dynamics of multipartite correlations at zero temperature while Sec.~\ref{TnonZero} we consider how entanglement and non-locality are affected in the finite temperature case. Finally Sec.~\ref{conclus} summarizes our findings.

\section{The Model}
\label{Secmodel}

We begin our study by discussing features and properties of an isotropic Ising model in a transverse magnetic field. For $N$ coupled spins the Hamiltonian model reads
\begin{equation}
\label{model0}
\hat{\cal H}=-{\cal J}\sum^{N}_{n=1} \hat\sigma^{x}_{n} \otimes \hat\sigma^{x}_{n+1} + {\cal B} \sum^{N}_{n=1} \hat\sigma^{z}_{n}
\end{equation}
with ${\cal J}$ the inter-spin coupling strength and ${\cal B}$ the global magnetic field for each qubit and $\sigma^i_n$ ($i=x,y,z$) the $i$ Pauli spin operators for spin $n=1,..,N$. In order to strip our analysis from unnecessary complications, we work with a dimensionless version of the Hamiltonian defined by assuming ${\cal J}=J{\cal E}$ and ${\cal B}=B{\cal E}$ with ${\cal E}$ a common order-of-magnitude factor and $J,B$ are dimensionless parameters. We thus consider the dimensionless Hamiltonian
\begin{equation}
\label{model}
\hat{H}\equiv\frac{\hat{\cal H}}{\cal E}=-J\sum^{N}_{n=1} \hat\sigma^{x}_{n} \otimes \hat\sigma^{x}_{n+1} + B \sum^{N}_{n=1} \hat\sigma^{z}_{n}
\end{equation}
We can readily determine the thermal state associated with this Hamiltonian using
\begin{equation}
\label{thermalstate}
\varrho(T)= e^{-\frac{\beta}{T}\hat{H}}/{{\cal Z}}
\end{equation}
where ${\cal Z}$ is the partition function and $\beta=\hbar/k_B$. Clearly each spin only interacts with its nearest neighbors and we shall assume a cyclic boundary condition such that the last spin interacts with the first, thus forming a spin-ring. It is also worth noting that we have taken the magnetic field to act orthogonal to the interaction. This particular model is very attractive due to its exactly solvable nature. The solution is presented in~\cite{buzek,mattis,osborne}, and so for completeness we simply comment on the methods employed. In order to diagonalize the Hamiltonian we use the Jordan-Wigner transformation~\cite{JW}. Firstly, we transform the Hamiltonian using a new variable $\sigma^{\pm}=1/2(\sigma^x_n \pm i\sigma^y_n)$. We can then easily move into the fermionic picture by introducing a further transformation
\begin{equation}
\hat c^\dag_n=(\hat c_n)^\dag=\otimes^{n-1}_{j=1}(-\hat{\sigma}^z_j)\hat{\sigma}^+_n~~~~\{c_n,c^\dag_m\}=\delta_{nm}
\end{equation}
with $\delta_{nm}$ the Kronecker delta, which is $1$ for $n=m$ and $0$ otherwise.
These fermionic variables then allow the Hamiltonian to be written in the form
\begin{equation}
\hat H{=}-J\sum^{N}_{n=1}(\hat c^{\dagger}_n-\hat c_n)(\hat c^{\dagger}_{n+1}+\hat c_{n+1})+2B \sum^{N}_{n=1}(\hat c_n^\dagger \hat c_n -\frac{1}{2}).
\end{equation}
Using a Fourier transform we pass to the momentum representation. The resulting Hamiltonian can then be diagonalized by means of a Bogoliubov transformation introducing new fermionic operators $\{\hat b_k,\hat b^\dag_k\}$. The latter are related to the Fourier-transformed fermionic operators $\hat{c}_{\pm k}$ through $\hat{b}_{k}=\cos(\vartheta_k/2)\hat{c}_k-i\sin(\vartheta_k/2)\hat{c}^\dag_{-k}$ and $\hat{b}^\dag_{k}=\cos(\vartheta_k/2)\hat{c}^\dag_k+i\sin(\vartheta_k/2)\hat{c}_{-k}$. This brings it into the free-fermion form
\begin{equation}
\label{free}
\hat{H}_{\text{ff}}=\sum_{k}\epsilon_k\hat{b}^\dag_k\hat{b}_k- \sum_k\epsilon_k,
\end{equation}
where ${\epsilon_k=\sqrt{J^2+B^2-2JB\cos \phi_k}}$, $\tan\vartheta_k=(-B+J\cos\phi_k)/(J\sin\phi_k)$ and ${\phi_k=\pi(2 k+1)/N}$ with $k=-N/2,..,N/2-1$ in the subsector with an even number of fermions and a slightly different definition for the odd-number case. However, in what follows we will be mostly concerned with the ground state of the Hamiltonian, which lives in the even-number sector (the appropriate expression will be used when leaving the ground state).  The constant term in Eq.~(\ref{free}) is related to $\sum_k\epsilon_k$. The ground state of the system is then the state with no population in any of the decoupled $\hat{b}_{k}$ modes, {\it i.e.} the state solves the set of conditions $\hat b_k\ket{\text{gs}_N}=0$ simultaneously, for any allowed value of ${k}$. A standard calculation (see for details Ref.~\cite{rico}) then leads to the ground state
\begin{equation}
\ket{\text{gs}_N}=\bigotimes_{k}[\cos\frac{\vartheta_k}{2}\ket{00}_{k,-k}+i\sin\frac{\vartheta_k}{2}\ket{11}_{k,-k}]
\end{equation}
with $\ket{0}_{\phi_k}$ ($\ket{1}_{\phi_k}$) the state with 0 (1) fermions with momentum $\phi_k$. This  approach is particularly useful when a ring with a large number of particles is considered. In this case, the direct diagonalization of Eq.~(\ref{model}), in fact, becomes computationally demanding.
Regardless of the dimension of the ring at hand, the magnetic field proves to be key to the behavior of entanglement and non-locality within our system. In fact, when no magnetic field is present, from Eq.~(\ref{free}) we see that $\epsilon_k=J~\forall{k}$, so that the ground state has energy $-NJ$, which is the same for $\otimes^N_{n=1}\ket{+}_n$ and $\otimes^N_{n=1}\ket{-}_n$ with $\ket{\pm}_n=(\ket{0}_n\pm\ket{1}_n)/\sqrt 2$ being the eigenstates of $\hat\sigma^x_n$. In these expressions, $\{\ket{0}_n,\ket{1}_n\}$ is the computational logic basis of spin $n=1,..,N$. As Stelmachovic and Buzek proved in Ref.~\cite{buzek}, the proper ground state of the ring in these conditions can be taken as a Greenberger-Horne-Zeilinger state~\cite{GHZ} in the basis built from the tensor product of eigenstates of $\hat\sigma^x_n$. That is
\begin{equation}
\ket{\text{GHZ}^x_N}=(\ket{++..+}+\ket{--..-})/\sqrt 2,
\end{equation}
which, when written in the $\otimes^N_{n=1}\hat\sigma^z_n$ eigenbasis, always has an even number of spins in their $\ket{1}$ state. In what follows, the energy of the ground state on a system of $N$ spins is indicated as
\begin{equation}
\Lambda_N=-\sum_k\epsilon_k.
\end{equation}

\section{Tools for studying multipartite entanglement and non-locality}
\label{Sectools}

In order to examine the dynamics of non-locality and entanglement, we need appropriate instruments to quantify them. Bipartite entanglement can be quantified by a number of well-defined measures. However for mixed, multipartite states no such measure exists~\cite{HHHH}. Here we we shall employ a suitable generalization of negativity, an entanglement measure based on the well-known Peres-Horodecki criterion~\cite{pereshorodecki}, which is necessary and sufficient for $2 \times 2$, $2 \times 3$ and $\infty \times \infty$ systems., together with an extension of Wootters concurrence~\cite{Wconc} to the case of a multipartite register of an even number of elements.

For a two-qubit system, negativity is defined as
\begin{equation}
\label{bipartneg}
{\cal N}_2=-2\text{max} [0,\lambda_{\text{neg}}],
\end{equation}
where $\lambda_{\text{neg}}$ is the single negative eigenvalue of the partially transposed version of the density matrix describing the state of the qubits. This measure can be extended to tripartite systems by simply considering the partial transposition of the density matrices associated with appropriate bipartitions of a given system~\cite{sabin}. Let us consider three spins, $A, B$ and $C$. The entanglement of spin $A$ with spins $B$ {\it and} $C$ is
\begin{equation}
{\cal N}_{A(BC)}=-2\text{max} [0,\sum_j \lambda^{A(BC)}_{\text{neg},j}].
\end{equation}
with $\lambda^{A(BC)}_{\text{neg},j}$ the negative eigenvalues of the partially transposed density matrix describing the bipartition $A-(B,C)$. We can then determine the genuine tripartite entanglement content of the tripartite state by taking a geometric average of all the bipartitions, thus ensuring that the corresponding measure remains a true entanglement monotone. Therefore
\begin{equation}
\label{tripartneg}
{\cal N}_3= [{\cal N}_{A(BC)}{\cal N}_{B(AC)}{\cal N}_{C(AB)}]^{\frac{1}{3}}.
\end{equation}
Unfortunately, the extension of this measure to larger systems is not as straightforward and one has to resort to other tools for the quantitative analysis of multipartite entanglement. A rather handy one is embodied by $N$-concurrence~\cite{nconc}, which is the direct generalization of bipartite concurrence to the case of an {\it even} number of qubits. That is, one has to consider the eigenvalues $\eta_j$ of the $N$-spin flipped matrix
\begin{equation}
\rho(\otimes^N_{j=1}\hat\sigma^y_j)\rho^*(\otimes^N_{j=1}\hat\sigma^y_j)
\end{equation}
with $\rho$ the density matrix describing the state of the system and $\rho^*$ its complex conjugate. The $N$-concurrence $NC$ is then given by $NC=\max[0,\sqrt\eta_1-\sum^{N^2}_{j=2} \sqrt{\eta_j}]$ with $\eta_1\ge\eta_j~\forall{j}$. This measure is appealing due to its computationally handy form and its sensitivity to global entanglement. In fact, it is exactly zero if any qubit is separable from the rest of a system. Its non-nullity, though, is only a sufficient condition for multipartite entanglement as there are multipartite entangled states with vanishing $N$-concurrence, such as the N-qubit $W$-states.  According to the definition given by Wong and Christensen~\cite{WC}, in pure states N-concurrence quantifies the overlap between a given state and the one obtained upon application of an anti-unitary time-reversal operation. For qubit states, such operation is embodied by a collective spin-flip operation. $N$-concurrence is thus a natural choice when the state to study is the eigenstate of a Hamiltonian respecting time-reversal symmetry, such as those that can be written as the sum of tensor products of an even number of non-identity Pauli matrices~\cite{nconc}. The Ising Hamiltonian satisfies such condition when no magnetic field is applied, thus justifying our choice and our efforts here can well be seen as the quantification of the robustness of the ground state of a Ising-like model to the perturbation induced by the magnetic field.

There is no formal way to quantify the degree of non-locality of a given state. Nevertheless, the degree of violation of Bell-type inequalities can be taken as a an intuitive means of ascertaining the amount of non-locality~\cite{bell}. A more pragmatic viewpoint to quantify non-locality, on the other hand, is to consider the amount of noise required by a given state to stop violating a Bell-like inequality. Moreover, in the multipartite setting, a few ambiguities still remain in deciding which type of inequality to utilize. In fact, as very clearly pointed out by Cereceda~\cite{cereceda} by reprising a point originally made by Svetlichny~\cite{svetlichny}, $n$-particle entanglement is not physically equivalent to $n$-particle non-locality. In this respect, the violation of an $n$-particle Bell-like inequality of some sort by an $n$-particle entangled state is not enough, {\it per se}, to prove genuine multipartite non-locality.   When one moves into the multipartite setting Bell-type inequalities exist that probe various different types of correlation. Technically, this issue has produced considerable theoretical effort directed towards the formulation of appropriate Bell-like inequalities able to capture the genuine multipartite non-local nature of a given state. Noticeably, Svetlichny has derived an inequality for the tripartite case that, while obeyed by models assuming two-particle non-locality, is violated by quantum mechanical states that are genuinely three-particle entangled~\cite{svetlichny}. As discussed by Cereceda~\cite{cereceda}, a Svetlichny inequality (SI) is a righteous Bell inequality for the tripartite case and emerges as a valuable tool for the unambiguous assertion of the existence of genuine tripartite entanglement and tripartite non-locality in any three-particle state. To this task, the use of a standard Mermin's inequality~\cite{mermin} is not sufficient: on the contrary, quantum correlations that violate SI are strong enough to maximally violate Mermin's inequality as well. The inequality by Svetlichny has been independently extended to the $n$-partite scenario in Refs.~\cite{gisin,svet}. In particular, Collins {\it et al.}~\cite{gisin} have provided an iterative way to construct multipartite Bell inequalities for dichotomic observables. We shall call $o_j$ and $O_j$ the two outcomes of a local measurement performed over one of the particles. It should be noted, the explicit form of the observable can be dependent on the state to be tested. For instance, measuring along the incorrect axis for a particular state can lead to a failure to see any violation. In the proceeding section we will fix the appropriate observable for our model. Then, by setting $m_1=o_1$ ($M_1=O_1$), one constructs the polynomials
 \begin{equation}
 \label{theory}
 \begin{aligned}
m_n&=\frac{1}{2}m_{n-1}(o_n+O_n)+\frac{1}{2}M_{n-1}(o_n-O_n),\\
M_n&=\frac{1}{2}M_{n-1}(o_n+O_n)+\frac{1}{2}m_{n-1}(O_n-o_n).
\end{aligned}
 \end{equation}
Quantum mechanically, we shall interpret the polynomials $m_n$ and $M_n$ as sums of $n$-particle correlation functions for measurements having outcomes $o_j$ and $O_j$. We can then define the generalized Svetlichny polynomials
\begin{equation}
\label{polys}
{\cal S}_n=\left\{
\begin{aligned}
&m_n~~~(n~\text{even})\\
&(m_n+M_n)/2~~~(n~\text{odd})
\end{aligned}
\right.
\end{equation}
The bound imposed to ${\cal S}_n$ by local hidden-variable models is $1$, while quantum mechanically an $n$-qubit GHZ state achieves the maximum value of $\sqrt{2^{n-1}}$ and $\sqrt{2^{n-2}}$ for an even and odd number of particles, respectively. Armed with such tools, we now tackle the main point of our study.

\section{Behavior of non classical correlations: zero-temperature case}
\label{nonclassical}

We begin our analysis considering the dynamics of the correlations in the simplest case of just two coupled spins. By assuming that both the spins are at zero temperature, the ground state  of Eq.~(\ref{model}) has the form
\begin{equation}
\label{gs2}
\ket{\text{gs}_2}=\frac{-(B+\Lambda_2/2)\ket{00}_{12}+J\ket{11}_{12}}{\sqrt{J^2+(B+\frac{\Lambda_2}{2})^2}}\end{equation}
with $\Lambda_2=-2\sqrt{B^2+J^2}\ge{B}$. Clearly, for $B\rightarrow{0}$, $\ket{\text{gs}_2}\rightarrow(\ket{00}+\ket{11})/\sqrt 2\equiv\ket{\text{GHZ}^x_2}$, in agreement with the expectations for the ground state of the model at hand. The calculation of negativity is therefore straightforward and leads to
\begin{equation}
{\cal N}_{2}=\frac{J}{\sqrt{B^2+J^2}},~~~~(B>0).
\end{equation}
The behavior against $J$ and $B$ is shown in Fig.~\ref{zerotempentanglement} {\bf (a)}. Clearly, as $B\rightarrow0$ the entanglement achieves a full ebit, while as we increase the amplitude of the magnetic field, the amount of zero-temperature entanglement decreases, vanishing asymptotically. This behavior is in agreement with what has been shown in Refs.~\cite{buzek,kendon}, where concurrence was considered.
\begin{figure}[t]
{\bf (a)}\hskip4cm{\bf (b)}
\psfig{figure=bipartitenegativity-zerotemp.eps,width=4cm,height=2.7cm}~~~\psfig{figure=tripartitenegativity-zerotemp.eps,width=4.3cm,height=3.3cm}
\caption{(Color online). Behavior of entanglement against $B$ and $J$ for {\bf (a)} bipartite spin ring and {\bf (b)} tripartite spin ring. For small values of $B$ the ground state is highly entangled. As the field strength is increased we see the entanglement asymptotically approach zero.}
\label{zerotempentanglement}
\end{figure}
We can extend our study to the case of tripartite entanglement. The zero-temperature tripartite negativity of the ground state of a three-spin ring can be calculated using Eq.~(\ref{tripartneg}) to be
\begin{equation}
{\cal N}_3=\frac{1}{3} \sqrt{\frac{3 J^2}{B^2-B J+J^2}-\frac{4 B-2 J}{\sqrt{B^2-B J+J^2}}+4}.
\end{equation}
Fig.~\ref{zerotempentanglement} {\bf (b)} shows its behavior. Analogously to the entanglement of the two-spin ring, we see that as the magnetic field is increased there is an asymptotic decay to zero. While at $B\rightarrow0$ the state is the maximally entangled GHZ state, interestingly as we increase the magnetic field strength we find the reduced state of any two spins becomes entangled, thus stating the departure of the ground state of the system from the GHZ class of entangled states. As discussed in Sec.~\ref{Sectools}, the case of an arbitrarily sized ring is not easily managed because of the lack of available measures for genuine multipartite entanglement. Yet, partial information on the distribution of quantum correlations within an even-parity length ring  can be effectively gathered by means of the $N$-concurrence. We have thus evaluated the $NC$ for the significant cases of $N=4$ and $6$ and compared it with the entanglement measured by Wootters concurrence evaluated for a two-spin chain. The results are shown in Fig.~\ref{Nconc} against $J$ and $B$ and reveal an increasing fragility of multipartite entanglement as $N$ grows. As it should be evident, at non-zero $J$ entanglement only asymptotically decreases as $B$ grows showing that, although the ground state of the ring progressively deviates from a GHZ state, a degree of multipartite entanglement is nevertheless kept by the spins.

However, the inability to discuss the non-classical features of correlations in the odd-parity scenario and the lack of a ``homogeneous" way to quantify entanglement (two different measures had to be used in the above) is certainly an unsatisfactory feature. Therefore, in order to provide a much more self-consistent characterization of the correlation properties within the ground states of the Ising ring, we move to the study of  non-locality. In such a two-spin setting, the well-known Closer-Horne-Shimony-Holt (CHSH) inequality~\cite{bell} can be used, which is encompassed in the formulation of the general multipartite problem given in Sec.~\ref{Sectools}. In order to run the CHSH inequality test, we thus project the state of the two-spin system onto the eigenstates of the local observable
\begin{equation}
R_i(\theta_i)=\left(
\begin{array}{cc}
 \cos\theta_i & -i\sin\theta_i \\
 i\sin\theta_i & -\cos\theta_i
\end{array}
\right)~~(i=1,2)
\end{equation}
and calculate the correlation function among the various combinations of outcomes. This is equivalent to
\begin{equation}
C(\theta_1,\theta_2)=\bra{\text{gs}_2}\hat{R}_1(\theta_1)\otimes\hat{R}_2(\theta_2) \ket{\text{gs}_2}
\end{equation}
with $\theta_1$ and $\theta_2$ the angles defining the orientation of the observable $\hat{R}_i$. We then construct the polynomial $m_2$ by identifying each term appearing in the first of Eqs.~(\ref{theory}) with the correlation function obtained by using two different sets of angles, $(\theta_1,\theta_2)$ and $(\theta'_1,\theta'_2)$. We can then construct the CHSH function as
\begin{equation}
\label{CHSH}
{\cal S}_{2}=\frac{1}{2}[C(\theta_1,\theta_2)+C(\theta_1,\theta'_2)+C(\theta'_1,\theta_2)-C(\theta'_1,\theta'_2)].
\end{equation}
Classically this function is bounded by $1$, while there are quantum mechanical states that violate this bound by up to a factor $\sqrt{2}$. In Fig~\ref{zerotempnonlocality} {\bf (a)} we see how a typical CHSH function $|{\cal S}_2|$ behaves against the amplitude of the transverse field $B$, for a set value of $J$. For nearly vanishing values of $B$ we have a maximally entangled state which maximally violates the bound set by local realistic theories. As the amplitude of $B$ increases, we see ${\cal S}_2$ asymptotically approaching the local realistic bound of $1$.

\begin{figure}[b]
\psfig{figure=NConc,width=7cm,height=4cm}
\caption{(Color online). Comparison of the $N$-concurrence $NC$ corresponding to the ground state of a ring of $N=2,4$ and $6$ spins (curves from top to bottom one) at zero temperature. The $N$-concurrence is plotted against $B$ and $J$.}
\label{Nconc}
\end{figure}

\begin{figure}[t]
{\bf (a)}\hskip3cm{\bf (b)}
\psfig{figure=PariNonLoc.eps,width=4.5cm,height=3.0cm}~\psfig{figure=DispariNonLoc.eps,width=4.5cm,height=3cm}
\caption{(Color online). {\bf (a)} Multipartite non-locality of the ground state of a ring of an even number of spins against the amplitude of the magnetic field $B$ and at $J=5$. The solid line is for $N=6$, the dashed line for $N=4$ and, finally, the dotted one is for $N=2$. {\bf (b)} Same as in panel {\bf (a)} but for an odd number of spins. The dotted line is for $N=3$, the dashed for $N=5$ while the squares show the results for $N=7$.}
\label{zerotempnonlocality}
\end{figure}

Despite the apparent simplicity of the case studied above, the results achieved in the two-spin ring serve as useful milestones in the analysis to follow. In fact, we can extend our investigation to the genuinely multipartite scenario by using the information on the ground state of the Ising ring under scrutiny that can be gathered by means of the approach described in Sec.~\ref{Secmodel} and applying it to the formalism for multipartite Bell inequalities given in Sec.~\ref{Sectools}. Our approach goes as follows: although certainly possible, the calculation of the correlation function in terms of the fermionic operators putting  $\hat H$ into the form $\hat{H}_{\text{ff}}$ given in Eq.~(\ref{free}) is not straightforward, especially under the scalability viewpoint. Indeed, as significantly remarked by Osborne and Nielsen~\cite{osborne}, calculating correlation functions for more than two particles is, in general, a daunting problem. One should, in fact, work out the expression of $\hat{R}_{i}(\theta_i)$ in the fermionic representation, keep in consideration the anti-commutation rules obeyed by fermions and re-cast the tensor product of $N$-particle local observables in terms of the operators $\hat{b}_k$ and $\hat{b}^\dag_{k}$. Expectation values over the ground state $\ket{\text{gs}_N}$ of an $N$-spin ring could be finally evaluated. Instead, we decided to use the information on the energy $\Lambda_N$ obtained by means of the apparatus set up in Sec.~\ref{Secmodel} so as to determine the form of the ground state of the system as expressed in terms of the elements of the computational basis. This is easily and computationally very conveniently done by solving the set of equations
\begin{equation}
\label{eig}
(\hat{H}-\Lambda_N\openone_{2^N})\ket{gs_N}={\bm 0},
\end{equation}
where $\Lambda_N$ is now a known quantity, $\hat\openone$ is the identity matrix and ${\bm 0}$ is the identically null vector. Once the structure of the ground state is determined, the evaluation of the necessary correlation functions is largely simplified. To the best of our knowledge, our study is the first to address multi-spin correlations over the ground state if an Ising ring. We have thus studied the behavior of the Svetlichny function against the amplitude of the magnetic field $B$ at a set value of $J$ (we remind that the important parameter, in this problem, is the ratio $J/B$). In Figs.~\ref{zerotempnonlocality} {\bf (a)} and {\bf (b)} we show the results of our investigation for rings of up to $7$ spins. It is worth stressing that larger rings can well be studied by our method. Given the determination of the ground state has been reduced to the solution of the eigen-equation Eq.~(\ref{eig}) we require only the solution of $2^N$ linear equations. It is in the evaluation of the multipoint correlation functions that care must be taken. Here the number of correlation functions to be evaluated scales with $2^N$, and we must then optimize the generalized Svetlichny polynomials Eq. (\ref{polys}) with respect to $2N$ parameters. This leads to escalating computational cost making rings larger than $N=7$ intractable. We have split the behavior of the ${\cal S}_{n}$ functions against the parity of the length of a ring, thus defining ${\cal S}_{even}$ and ${\cal S}_{odd}$.  As it will be shown later, parity appears to be relevant in the investigation of the features here under scrutiny.

The dependence of the the two parity-related Svetlichny functions on $B$ is mutually fairly similar and overall consistent with the behavior revealed for ${\cal S}_{2}$: at large values of $J/B$, the ground state of the ring exhibits strong non-locality, up to the maximum allowed value set by the number of spins being considered. As $B$ grows (for a set value of $J$), the non-local nature of $\ket{\text{gs}_n}$ deteriorates. Remarkably, though, while the even-parity case approaches the local realistic bound only asymptotically with $B$, a finite value of its amplitude is sufficient to set $|{\cal S}_{odd}|=1$ exactly. Although it is difficult to establish an explicit relation at this level, we believe that such a parity-induced effect is related to the {\it differences} in the way the ground state of an Ising ring has to be considered in the even- and odd-number of spins cases, as highlighted in Sec.~\ref{Sectools} when the ring Hamiltonian was diagonalized. However, it should be remarked that longer chains of an odd number of spins  require a larger $B$ to reach the local realistic bound, while in a long chain, the addition of a single spin would make a very small difference, as it is intuitively reasonable to expect. Therefore, one can extrapolate to the thermodynamical limit the behavior of non-locality and claim that the parity effect highlighted above would eventually disappear. As a side remark, the reduced state of any two-spin subsystem, regardless of the length of a ring, despite being entangled do not violate Eq.~(\ref{CHSH}).

\begin{figure}[b]
{\bf (a)}\hskip3.0cm{\bf (b)}\hskip3.0cm{\bf (c)}
\psfig{figure=spectra.eps,width=9.0cm,height=2.0cm}
\caption{(Color online). Energy spectrums for a spin ring of $N=2$ [panel {\bf (a)}], $3$ [panel {\bf (b)}] and $4$ [panel {\bf (c)}] against $B$. We have taken $J=5$.}
\label{spectrums}
\end{figure}

\begin{figure*}[t]
{\bf (a)}\hskip3cm{\bf (b)}\hskip4cm{\bf (c)}
\psfig{figure=2qubit-thermalnonlocality.eps,width=4.7cm,height=3.5cm}~~\psfig{figure=3qubit-thermalnonlocalityN.eps,width=4.5cm,height=3.6cm}~~\psfig{figure=4qubit-thermalnonlocality.eps,width=4.6cm,height=3.5cm}
\caption{(Color online). Behavior of Svetlichny functions ${\cal S}_{N}$ for the state of a ring with $J=5$ and $T\neq0$. We address the case of $N=2,3$ and $4$ (panel ${\bf (a)}$, ${\bf (b)}$ and ${\bf (c)}$ respectively) as representative of the general trend exhibited by arbitrary $N$ values. Each curve in the same plot is for for a specific choice of $B\in[1,10]$. Regardless of $N$, as $B$ increases the degree of violation of the appropriate Bell-like inequality decreases. However, the resilience to spoiling effects due to temperature increases.}
\label{thermaleffects}
\end{figure*}

\section{Behavior of non classical correlations: thermal effects}
\label{TnonZero}

We now investigate the behavior of non-classical correlations in the non-zero temperature case. Using Eq.~(\ref{thermalstate}) we can determine the thermal state of the Hamiltonian. In general, this necessitates of the full spectrum of eigenstates the Hamiltonian, whose determination for a large number of spins becomes a very difficult task to achieve. However, very useful information comes from the study of the energy spectrum of $\hat H$. Fig.~\ref{spectrums} shows the spectrum of a ring of  two, three and four spins for $J=5$ and against increasing field $B$~\cite{nota}. It is seen that the splitting between the two lowest energy states increases with the strength of the magnetic field. In the region of  $B\ll J$, the ground and first excited energy levels are almost degenerate and a much larger energy gap separates the quasi-degenerate doublet from the rest of the spectrum. This implies that an excellent approximation of the thermal state of the ring comes from considering an admixture of $\ket{\text{gs}_N}$ and $\ket{\text{first}_N}{=}\hat{b}^\dag_k\ket{\text{gs}_N}$ with coefficients determined by the Boltzmann factor corresponding to the energies $\Lambda_N$ and $\Lambda^1_N{=}\bra{\text{first}_N}\hat{H}_{\text{ff}}\ket{\text{first}_N}$ such as
\begin{equation}
\label{appro}
\varrho(T){\simeq}\left(e^{-\frac{\beta\Lambda_N}{T}}\ket{\text{gs}_N}\bra{\text{gs}_N}+e^{-\frac{\beta\Lambda^1_N}{T}}\ket{\text{first}_N}\bra{\text{first}_N}\right)/{\cal Z}.
\end{equation}
Clearly, the approximation is legitimate as long as the state~(\ref{appro}) remains quasi-normalized, that is $(\exp[{-\frac{\beta\Lambda_N}{T}}]+\exp[{-\frac{\beta\Lambda^1_N}{T}}])/{\cal Z}\simeq{1}$. The proximity of the first two energy levels for small values of $B$ implies that even a small amount of thermal energy is enough to fully mix the two lowest eigenstates, which might well result in the wash-out of the quantum correlations within the system. On the other hand, for increasing values of $B$, a larger temperature is required to mix the eigenstates.

We have examined the entanglement properties for even values of $N$ {\it and} $N=3$ (which, we remark, is the only odd case treatable through tripartite negativity) against the temperature. For $B\ll J$, entanglement suddenly disappears at finite values of $T$. On the other hand, for $B\approx J$, quantum correlations appear to cope much better against the effects of a non-zero temperature: as $B$ gets larger, the state of the ring is able to withstand larger values of $T$ before we starting to decay. Yet, this effect is accompanied by a reduction in the amount of entanglement shared by spins.  Again, in order to bypass the problems inherent in entanglement measures for mixed multipartite states, we use the tools provided by multipartite non-locality tests. Figs.~\ref{thermaleffects} {\bf (a)-(c)} show the behavior of the generalized Svetlichny functions for a few significant cases. Evidently, as $B$ grows, while paying in terms of degree of violation of the appropriate Bell-like inequality, we significantly enlarge the range of $T$ within which the correlations within the state of a ring cannot be described by a local realistic theory.

\begin{figure}[b]
\psfig{figure=2qubitMagn.eps,width=6.5cm,height=5cm}
\caption{(Color online). Two-spin negativity against magnetization and temperature for $J=5$. We have taken $T=1,..,20$ (increasing at unit steps). The entanglement for magnetization going toward $1$ stays at finite non-zero values only because we have considered $B\in[0,100]$. By enlarging this range, smaller final values of negativity would be reached. In fact, we have ${\cal N}_2\rightarrow0$ as $\langle\hat{M}_2\rangle\rightarrow1$.}
\label{fig5}
\end{figure}

This {\it protection effect} can be understood as follows: the increasing magnetic field tends to align the spins in the ring, thus affecting the strength of the spin-spin correlations. On the other hand, the tendency to mutual alignment of the spins makes them less sensitive to the fluctuations induced by thermal effects. At the same time, a growing $B$ makes the spectral gap between ground and first excited state of the ring wider, thus making the admixture between the ground state and the rest of the spectrum less likely to occur. {We can link the effects analyzed above to the magnetization of the ring so as to provide a much neater connection between the correlation properties of the system and its intrinsic macroscopic features, which is certainly a very interesting point to assess. Let us concentrate on the two-spin ring case, which is non-trivial enough to encompass the general features we are interested in.  We consider the expectation value of the global magnetization operator $\hat{\cal M}_2=\sum^2_{j=1}\hat{\sigma}^z_j$ over the state of a two-spin ring at temperature $T$. Clearly, the modulus of this function grows monotonically from zero to one at a rate slowed down by the increasing temperature. Here, we are interested in the connection between non-classical correlations and the global magnetization $\langle\hat{\cal M}_2\rangle$. Due to the dimension of the problem at hand, we can equally use negativity or the Bell-like function ${\cal S}_2$ and we focus on the former simply for easiness of calculations. In Fig.~\ref{fig5} we plot ${\cal N}_2$ against $\langle\hat{\cal M}_2\rangle$ for a range of values of the temperature. At $T=0$, small values of the magnetization (corresponding to large values of the ratio $J/B$) correspond to a very large degree of entanglement, in agreement with the analysis so far~\cite{buzek}. A growing magnetization sees ${\cal N}_2$ decreasing to zero. The behavior changes abruptly for even small non-zero values of $T$, when  we start from an initially maximally entangled state and as we increase the magnetization of the system we see a decrease in non-classical correlations. For the bipartite case we see the correlations become purely classical only when the system is fully saturated by the magnetization, i.e. when $M=1$ or equivalently $B\rightarrow\infty$. Qualitatively the results hold in the higher dimensional cases. While the study of this figure against temperature reinforces the features revealed above (entanglement decays more slowly at larger values of $B$, although the maximum achievable degree of negativity decreases), we can also infer that larger values of the magnetic field are required in order to optimize the entanglement within the ring, if $T$ grows. This is a clear manifestation of the {\it correlation-by-alignment} effect discussed before: the thermal randomness is counteracted by the order induced by a strong magnetic field. However, if the alignment induced by $B$ exceeds the threshold determined by the degree of thermal disorder, quantum correlations start to vanish as the spins tend to be only classically correlated (as expected in an Ising ring beyond the critical point). A qualitatively similar analysis holds for longer rings. By controlling the macroscopic features of magnetization, one could thus determine the degree of quantum correlations set among the elements of the ring.

\section{conclusions}
\label{conclus}
We have studied the non-classical correlations naturally occurring in the ground and thermal state of an Ising ring in transverse magnetic field. The focus of our analysis was the behavior of multipartite entanglement and non-locality as quantified by the violation of {\it ad hoc} many-spin Bell-like inequalities~\cite{gisin}. At zero temperature, while even-length rings cannot be described by local realistic theories even at the limit of asymptotically infinite strength of the magnetic field, odd-length ones behave differently: there is always a finite choice of $B$ beyond which the bound imposed by local realism is satisfied. We have been able to exactly tackle multipartite correlation functions in such a many-body problem, therefore going significantly far from the standard analysis of the behavior of quantum correlations in interacting many-body models. At non-zero temperature, we have found that the thermal randomness that would destroy the non-classical correlations found in the ground state of the model can be counteracted simply by setting a large enough magnetization of the ring: a large magnetic field, in fact, while decreasing the degree of violation of Bell-like inequalities, will make the generalized Svetlichny functions used for non-locality test resilient against thermal spoiling effects. This suggests that  a genuinely multipartite quantum-correlated state in an Ising ring can be generated simply by controlling the macroscopic properties of the system, i.e. magnetization and temperature. We believe our results will trigger further interest in examining entanglement and non-locality in many-body interacting systems.

\acknowledgments
We thank Dr. Nicolas Brunner, Dr. Tony Apollaro, Prof. Sougato Bose and Prof. Vlatko Vedral for invaluable discussions. We acknowledge financial support from DEL and the UK EPSRC (EP/G004579/1),