\section{Introduction}
The impressive progress in atomic quantum gases can be largely
attributed to technical advances in the control of atomic quantum
states utilizing electromagnetic interactions. While laser cooling
with near resonant light narrows down atomic momentum distribution,
forced evaporation on trapped atoms favors spatial confinement into
the quantum degenerate regime. Since the first successful experiment
on atomic Bose-Einstein condensation, or the controlled production
of many bosonic atoms into the same quantum ground state of a
trapping potential, the field of atomic quantum gases has blossomed
into one of the leading frontiers in physics, attracting ever increasing
interest. The ultimate goal along this line of thought
is to control the state of
an arbitrary assembly of an interacting many body system, and either use
it to simulate other physical systems or to perform quantum
computations.

Significant efforts have been directed toward the above ambitious goal,
with the immediate pursuit from more recent endeavors targeted
at the generation, control, and
study of cold molecules. These endeavors include important successes
such as the BCS to BEC cross-over due to tunable interactions
induced through B-field magnetic Feshbach resonances and the
production of quantum degenerate heteronuclear molecules in
specific low lying ro-vibrational states of
their electronic ground states \cite{ni08}. The latter advance
is especially illuminating, highlighting the
prospects for observing and utilizing
anisotropic and long range interactions from the permanent electric dipoles.

Molecule quantum fluids bring in new possibilities due to their
richer internal degrees of freedom, such as alignment and rotation
\cite{friedrich95}, even after their hyperfine spins like those of
atomic spinor quantum gases are neglected. In this short note, we
propose a simple mechanism to generate a vortex state, relying on
the internal degrees of freedom of a heternuclear molecule, through
adiabatically flipping the axial bias of an external E-field
Ioffe-Pritchard trap (IPT) \cite{pritchard83}. The
physics of our proposal can be analogously mapped onto the
successful protocol first proposed \cite{isoshima00} and
demonstrated for magnetically trapped atomic condensates
\cite{leanhardt02,leanhardt03,isoshima07}. Our theory hopefully will
lead to parallel experiments that begin to address new opportunities
afforded by the molecular internal degrees of freedom.

This paper is organized as follows. First, we describe our model for
dc E-field trapping of cold heteronuclear molecules based on the interaction
between their permanent electric dipoles with a spatially varying static
E-field IPT. This is analogous to the operation of the
famous IPT for magnetic dipoles with
an inhomogeneous B-field \cite{pritchard83}. The heteronuclear diatomic molecule is
modeled as a 3D rigid rotor \cite{friedrich95}. We then discuss the eigen structure of
a heteronuclear molecule inside a dc E-field and identify states capable
of trapped by a local E-field minimum and their associated quantum
numbers. Next, the rotational properties of such eigen states are
studied, which helps to illustrate the proposed vortex pump mechanism based
on the flipping of the axial bias E-field
\cite{isoshima00,leanhardt02,leanhardt03,isoshima07,mottonen07,xu08}. Finally we conclude.

It is imperative to present
this study because this analogy between E-field and B-field
and between molecular E-dipole and atomic B-dipole is not a simple one to one map.
The magnetic dipole of an atom contains both an electronic and
a nuclear component, respectively, proportional to the electronic spin ($\vec S$) and
the nuclear spin ($\vec I$). Its projection onto the local B-field is
approximately proportional to the projection of its hyperfine spin $\vec F=\vec S+\vec I$
when the B-field is weak. In contrast, the
molecular dipole is a fixed constant and pointed along the molecule
axis connecting the two nuclei. The internal
rotational angular momentum of the molecule stays in the perpendicular
plane of the molecule axis in the absence of an E-field. Despite
these differences, our study finds that the Isoshima, {\it et al.},
protocol \cite{isoshima00,leanhardt02,leanhardt03},
 originally developed for B-field trapped atoms or
molecules, remains effective for heteronuclear molecules in a
E-field IPT. Much of these comparative studies and analogies will
become clear from the materials presented in the third and fourth
sections.

\section{Trapping a heteronuclear molecule with static electric field}

We consider a diatomic molecule composed of two different atomic
nuclei and described in terms of their center of mass $\vec R$ and
relative coordinate $\vec r$. Neglecting the nuclear and electronic
spins and their associated interactions, or assume we consider an eigen
state of the above internal degrees of freedom, the remaining
Hamiltonian reads
\begin{eqnarray}
  \mathcal{H}&=&\frac{\mathbf{P}_{R}^2}{2M}+\frac{\mathbf{P}^2_{r}}{2\mu}
  -\vec{D}(\vec r)\cdot\vec{E}(\vec{R})
  \nonumber\\
  &=&\frac{\mathbf{P}_{R}^2}{2M}-\frac{\hbar^2}{2\mu r^2}\frac{\partial}{\partial r}
  \left( r^2\frac{\partial}{\partial r} \right)+\frac{\mathbf{J}^2(\theta_r,\phi_r)}{2\mu r^2}
  -\vec{D}(\vec r)\cdot\vec{E}(\vec{R}),
  \label{ham}
\end{eqnarray}
where $\vec{E}(\vec{R})$ denotes the inhomogeneous static E-field,
which is slowly varying over the molecule size of $r$, and
$\vec{D }(\vec r)$ is the permanent electric dipole moment operator
for this specific electronic and internal spin eigen-state \cite{friedrich95,cohen84,micheli07}.
$M=M_1+M_2$ and $\mu=M_1M_2/(M_1+M_2)$ are, respectively, the total
and the reduced mass of the two atoms making up the molecule.

We adopt the Bohn-Oppenheimer approximation to study the effective
trapping of the molecule center of mass motion. Within this
approximation, the molecular wave function for $\vec R$ and $\vec r$
is decomposed into the form
$\Psi(\vec{r},\vec{R})=\Phi(\vec{r})\psi(\vec{R})$, and we find
\begin{eqnarray}
  \fl\qquad\left[-\frac{\hbar^2}{2\mu r^2}\frac{\partial}{\partial r}
  \left( r^2\frac{\partial}{\partial r} \right)+\frac{\mathbf{J}^2(\theta_r,\phi_r)}{2\mu r^2}
  -\vec{D}(\vec r)\cdot\vec{E}(\vec{R})\right]\psi_{M_J}(\vec{r})
  &=&\mathcal{V}_{M_J}(\vec{R})\psi_{M_J}(\vec{r}), \label{rot}\\
  {\hskip 112pt \left(\frac{\mathbf{P}_{R}^2}{2M}+\mathcal{V}_{M_J}(\vec{R})\right)
  \Phi_n(\vec{R})}&=&E_{M_J}^{(n)}\Phi_n(\vec{R}).
\end{eqnarray}

Now consider for a specific vibrational state, the Eq. (\ref{rot})
above then reduces to the simple form of a 3D rotator in a dc E-field
with a rotational constant $B=\langle\hbar^2/2\mu r^2\rangle$. If we
further assume the local E-field direction to be the quantization
z-axis, then the Schr\"odinger equation for the internal part of the
diatomic molecule becomes
\begin{eqnarray}
  \left[BJ^2-DE(\vec R)\cos\theta_r\right]\psi_{M_J}(\theta_r,\phi_r)
  =\mathcal{V}_{M_J}(\vec R)\psi_{M_J}(\theta_r,\phi_r),
\end{eqnarray}
where we have neglected the dependence of the permanent dipole moment on the
internuclear distance $\vec r$.

Some intuition can be gained for weak E-fields if a simple perturbation
theory is adopted as is done in Ref. \cite{micheli07}. We take the
unperturbed Hamiltonian to be $H_0=BJ^2$ with
$H_0|J,M_J\rangle=E_{JM_J}|J,M_J\rangle$, where the eigen-energy and
eigen-function are, respectively, $E_{JM_J}=BJ(J+1)$
and $|J,M_J\rangle$, with spherical harmonics $|J,M_J\rangle$ being
functions of $\hat r$.
The dipole interaction $-DE(\vec R)\cos\theta_r$
is treated as a perturbation.
To first order the wave function becomes
\begin{eqnarray}
  \psi_{M_J}&=&|J,M_J\rangle+\frac{DE}{B}\frac{1}{2(J+1)}
  \sqrt{\frac{(J+1)^2-M_J^2}{(2J+1)(2J+3)}}\,|J+1,M_J\rangle \nonumber\\
   && -\frac{DE}{B}\frac{1}{2J}\sqrt{\frac{J^2-M_J^2}{(2J-1)(2J+1)}}\,|J-1,M_J\rangle,
   \label{psifo}
\end{eqnarray}
and the corresponding eigen-energy to second order in the external E-field
becomes
\begin{eqnarray}
  \mathcal{V}_{M_J}&=&BJ(J+1)+\frac{D^2E^2}{B}\frac{1-3M_J^2/J(J+1)}{2(2J-1)(2J+3)}.
\end{eqnarray}

For this model system, $M_J$ is a conserved quantity.
Assuming the center of mass motion is adiabatic with respect to the
internal state, the above eigen energy from the internal state
$\mathcal{V}_{M_J}$ then clearly acts as an effective potential
due to its dependence on $\vec R$. In particular, we find any state with
$1-3M_J^2/J(J+1)>0$ is a weak field seeking state, {\it i.e.}, free
space E-field traps analogous to the B-field IPT can be constructed.
These states, of course, are meta-stable, and trapping is
possible because of the dynamic stability from the Larmor-like
precession of the permanent electric dipole along the direction of
a local E-field. On the other hand, the strong field seeking states with
$1-3M_J^2/J(J+1)<0$ cannot be used to spatially confine
molecules with static fields, because it is impossible to construct a local
E-field maximum.

The significant progress gained over the years
in trapping and manipulating neutral atoms with static B-fields \cite{TB}
provide important enabling technologies for the experimental successes of
cold atom physics research. A variety of B-field traps have been implemented
for various applications \cite{pritchard83,TB}.
Despite the analogy described earlier
of the B-field confinement of magnetic dipoles with
the E-field confinement of electric dipoles (of polar molecules),
experimental efforts of trapping and manipulating polar molecules
with electrostatic E-fields become an active research topic
only in recent years with the interests on cold atoms broadening into
cold molecules. The Meijer's group pioneered the research of cooling and
 trapping of molecules with (3D quadrupole) E-fields \cite{GM00}.
They also demonstrated the guiding of
cold molecules with a torus shaped (2D) E-field hexapole \cite{GM01}.
With suitable modifications, an analogous IPT of E-field can be
constructed for polar molecules \cite{pritchard83,TB,ab,GM}.

In Fig. \ref{fig1}, we show the calculated dc Stark shift for
the $M_J=1$ state of heteronuclear molecule KRb
that is connected to the $J=2$ manifold when the E-field is absent.
We adopt the
parameters as from its ground state $X^1\sum{\nu=0}$ as
measured recently \cite{ni08} with the electric dipole
moment $D=1.36\times 10^4$\,$\rm Hz\cdot m/V$ and the
rotational constant $B=1.1139\, \rm GHz$.
The calculation is accomplished by a numerical diagonalization of the
Hamiltonian over a truncated basis of common eigen states for
operators $\mathbf{J}^2$ and $J_z$. The diagonalization
is carried out in the subspace of a conserved
$M_J$ \cite{cohen84}, and the procedure is found to be rapidly
convergent. At the E-field strength we consider, the
perturbation result $\psi_{M_J}$ in Eq. (\ref{psifo}) with $M_J=1$ and $J=2$
is found to consists of an excellent approximation. This particular state
as illustrated in Fig. \ref{fig1} is clearly a weak field seeking state.
The maximum trap height can become as high as $173\,\rm MHz$.

\begin{figure}[H]
\centering
\includegraphics[width=4.5in]{fig1.eps}
\caption{The Stark shift of the $M_J=1$ state for the polar molecule KRb in the
electronic ground state $X^1\sum{\nu=0}$, for realistic electrostatic
E-trap field strength \cite{GM,ab}. }
\label{fig1}
\end{figure}

\section{Rotational transformation of the wave function $|J,M_J\rangle$}
Before we extend the Isoshima, {\it et al.}, protocol \cite{isoshima00}
from B-field trapped atoms to E-field trapped polar
molecules, we will briefly review the rotational properties of the
spherical harmonics $|J,M_J\rangle$. Under a rotation
along a unit vector $\hat{n}$ by an angle $\theta$,
the wave function changes to
\begin{eqnarray}
  R(\theta\hat{n})|J,M\rangle=e^{-i\theta\hat{n}\cdot\mathbf{J}}|J,M\rangle.
\end{eqnarray}
Expressed in terms of the Euler angles, the most general 3D rotation
takes the following form
\begin{eqnarray}
  R(\alpha,\beta,\gamma)=\exp(-i\alpha J_z)\exp(-i\beta J_y)\exp(-i\gamma J_z).
\end{eqnarray}
According to the representation theory of rotation, we find
\begin{eqnarray}
  R(\alpha,\beta,\gamma)|J,M\rangle&=&\sum\limits_{M'}D^{J}_{M'M}(\alpha,\beta,\gamma)
  |J,M'\rangle\nonumber\\
  &=&\sum\limits_{M'}\exp[-iM'\alpha]d^{J}_{M'M}(\beta)\exp[-iM\gamma]|J,M'\rangle,
\end{eqnarray}
where
\begin{eqnarray}
  d^{J}_{M'M}(\beta)&=&\langle J,M'|\exp[-i\beta J_y]|J,M\rangle\nonumber\\
  &=&[(J+M)!(J-M)!(J+M')!(J-M')!]^{1/2}\nonumber\\
  &&\times\sum\limits_{\nu}[(-1)^{\nu}(J-M'-\nu)!(J+M-\nu)! (\nu+M'-M)!\nu!]^{-1}\nonumber\\
  &&\times\left(\cos\frac{\beta}{2}\right)^{2J+M-M'-2\nu}\left(-\sin\frac{\beta}{2}\right)^{M'-M+2\nu}.
\end{eqnarray}
The value of $\nu$ needs to ensure that the numbers in the factorials
stay non-negative. For example, when $\beta=\pi$ and
$\cos\frac{\beta}{2}=0$, for $d^{J}_{M'M}$ to be nonzero, we need to
let $2J+M-M'-2\nu=0$. This gives
\begin{eqnarray}
  (J-M'+\nu)!=\left[ -\frac{1}{2}(M'+M) \right]!,\\
  (J+M-\nu)!=\left[ \frac{1}{2}(M'+M) \right]!.
\end{eqnarray}
In order to assure that both $J-M'-\nu$ and $J+M-\nu$ are
non-negative, we need to make $M'+M=0$. As a result we find
\begin{eqnarray}
  d^{J}_{M'M}(\pi)=(-1)^{J-M'}\delta_{M',-M},
\end{eqnarray}
which gives
\begin{eqnarray}
  R(\alpha,\pi,\gamma)|J,M\rangle=(-1)^{J+M}\exp[iM\alpha]\exp[-iM\gamma]|J,-M\rangle.
\end{eqnarray}

We want to stress at this point that the angles $\alpha$, $\beta$,
and $\gamma$ are parameters used to specify arbitrary rotations, and
they have nothing to do with the internal state angles $\theta_r$
and $\phi_r$. To understand the physics behind the vortex generation protocol,
which is directly related to the geometrical properties of the
static E-field, we first consider several special cases of
unit vectors as rotation axes: for
$\hat{n}=\hat{X}\cos\phi+\hat{Y}\sin\phi$,
$\hat{n}_{\perp}=-\hat{X}\sin\phi+\hat{Y}\cos\phi$, and
$\hat{n}_q=\hat{X}\sin\phi+\hat{Y}\cos\phi$, respectively, with $\hat{X}$ and
$\hat{Y}$ the unit vectors along the Cartesian $x$- and
$y$-directions. $\theta$ and $\phi$ denote the polar and
azimuthal angles of the center of mass coordinate $\vec R$. The last
case of $\hat{n}_q$ corresponds simply to the direction of a IPT
E-field in the plane of $Z=0$. Using the
relationship derived above, we find that
\begin{eqnarray}
  R_{\hat{n}}(\pi)|J,M_J\rangle&=&\exp(-i\pi\hat{n}\cdot \mathbf{J})|J,M_J\rangle\nonumber\\
  &=&R_{\hat{z}}(\phi-\pi/2)R_{\hat{y}}(\pi)R_{\hat{z}}(\pi/2-\phi)|J,M_J\rangle\nonumber\\
  &=&-(-1)^{J+M_J}\exp[2iM_J\phi]|J,-M_J\rangle, \\
    R_{\hat{n}_{\perp}}(\pi)|J,M_J\rangle&=&\exp(-i\pi\hat{n}_{\perp}\cdot \mathbf{J})|J,M_J\rangle\nonumber\\
  &=&R_{\hat{z}}( (\pi/2+\phi)-\pi/2)R_{\hat{y}}(\pi)R_{\hat{z}}(\pi/2-(\pi/2+\phi))|J,M_J\rangle\nonumber\\
  &=&(-1)^{J+M_J}\exp[2iM_J\phi]|J,-M_J\rangle,\\
  R_{\hat{n}_q}(\pi)|J,M_J\rangle&=&\exp(-i\pi\hat{n}_q\cdot \mathbf{J})|J,M_J\rangle\nonumber\\
  &=&R_{\hat{z}}( (\pi/2-\phi)-\pi/2)R_{\hat{y}}(\pi)R_{\hat{z}}(\pi/2-(\pi/2-\phi))|J,M_J\rangle\nonumber\\
  &=&(-1)^{J+M_J}\exp[-2iM_J\phi]|J,-M_J\rangle.\label{rotipt}
\end{eqnarray}

These properties of rotational transformation show that by
enforcing a rotation of the internal state, the wave function gains
an appropriate topologically phase specified by the
azimuthal angle coordinate of the center of mass (or the molecule) coordinate.
Provided the internal state flipping of
the permanent dipole moment and the center of mass motion are both
adiabatic, the above results show that different vortical phases are generated
as in the protocol of Isoshima, {\it et al.}, for B-field trapped
atomic spinor condensates \cite{isoshima00,leanhardt02,leanhardt03}.

\section{Vortex creation in a condensate of heteronuclear molecules}

In this section, we will confirm numerically the vortex creation
protocol for a condensate of heteronuclear molecules in an E-field
IPT. Basically, we will simulate the axial E-field bias flip and
check for the adiabatic conditions of the associated internal state.
There are two main points we need to watch for. First we need to
create the proper vortical phase structure on a condensate during
the time evolution of the E-field. Secondly we must maintain
adiabaticity during the flip of the internal state of a polar
molecule.

Considering the first question, assume the initial state
corresponds to a polar molecule locally aligned along the E-field
direction of a E-field IPT, which is essentially along the axial $z$-axis direction
close to the $z$-axis, its local internal state $\phi_{M_J}(\theta_r,\phi_r)$
can then be described approximately as the
eigen state of the system in the local E-field. Since
$M_J$ is a conserved quantity, this eigen state $\psi_{M_J}(\theta_r,\phi_r)$
can be expanded by a series of $|J,M_J\rangle$ in $J$,
\begin{eqnarray}
  \psi_{M_J}(\theta_r,\phi_r)=\sum\limits_J C_J|J,M_J\rangle.
  \label{psimj}
\end{eqnarray}
The flipping of the E-field bias adiabatically corresponds then
to nothing but a rotation of the initial state along
the unit vector $\hat{n}_q(\theta,\phi)$ in the transverse plane
by an angle of $\pi$. According to the rotation properties
we discussed before, this flipping of the bias then gives
\begin{eqnarray}
  R_{\hat{n}_q}(\pi)\psi_{M_J}(\theta_r,\phi_r)=\sum\limits_J (-1)^{J+M_J}
  C_J\exp[-2iM_J\phi]|J,-M_J\rangle,
  \label{rotpsimj}
\end{eqnarray}
which gains clearly a vortex with a winding number $-2M_J$.

Turning to the second question concerning the adiabaticity, we simply can
propagate the initial wave function $\psi_{M_J}$ in real time,
simulating the complete process of the bias flip. To test the level
of adiabaticity and the validity of Eq. (\ref{rotpsimj}), we
consider points $(X_0, Y_0, 0)$ along a circle in the $X$-$Y$ plane.
The absolute value for the E-field along the circle is then a
constant, although their local directions are all different. In
fact, for the E-field IPT being considered here, the E-field can be
written as $\vec{E}=E'(X,-Y,L)$ with $L$ a constant. Choosing the
weak field seeking state $\psi_{M_J=1}$ with $J=2$ at $\vec{E}=3\,
\rm kV/cm\ \hat{Z}$ as considered in Fig. \ref{fig1}, we simulate
the flip protocol over various time intervals. The time evolution of
the bias E-field is taken to be the same as that in the optimal
B-field protocol \cite{xu08}, and the E-field in the transverse
$X$-$Y$ plane is treated as a constant.

\begin{figure}[H]
\centering
\includegraphics[width=4.5in]{fig2.eps}
\caption{The topological phase gained after numerically flipping the
bias E-field, that is calculated by taking the inner product of the
time evolved state after flip with the nominally down polarized
($M_J=-1$) state. $\fullsquare$ denotes numerically computed data,
and the red solid curve represents a linear fit. Both axes are
dimensionless in units of $2\pi$.  }
\label{fig2}
\end{figure}

In our simulation, we take the E-field at $t=0$ as $E'L=3\, \rm
kV/cm$ and $E'\sqrt{X_0^2+Y_0^2}=0.3\, \rm kV/cm$. After the flip of
the E-bias over $10^5/B$ ($\sim 0.1$ ms at $B=1.1139\, \rm GHz$), which is
a reasonably
fast time scale, we calculate the inner product of the final state
with the eigen-state $\psi_{M_J=-1}$ (of $J=2$) when the E-field
$\vec{E}=-3 \rm\,kV/cm\,\hat{Z}$ is pointed in the downward
direction. Using these parameters, the adiabaticity is found to be
fully satisfied, and the state overlap is essentially 100\% except
for the extra topological phase gained as shown in Fig. \ref{fig2}.
A numerical fit gives precisely the slope for the phase over $\phi$
being exactly equal to $-2$, which confirms the high fidelity
operation of our vortex pump proposal.

Before concluding, we point out that, like the system of
magnetically trapped atomic spinors inside a B-field IPT
\cite{peng07,xu08}, the quantity of $J_z-L_z$ is found to commute
with the Hamiltonian $B\mathbf{J}^2-\mathbf{D}\cdot E'(X,-Y,L)$,
where $E'$ is the spatial gradient of the E-field IPT, and $L_z$ is
the mechanical angular momentum of the heteronuclear molecule. $\vec
J$ is the rotational angular momentum of the molecule. If the flip
of the E-bias is indeed adiabatic, the respective quantum numbers
conserve the combination $M_J-L_z$. After going through the flip,
the internal rotational state is changed from $M_J$ to $-M_J$, which
then must be accompanied by an increase of $2M_J$ to its mechanical
angular momentum. Thus we see the appearance of a vortex state.

In conclusion, by analogy with B-field trapping of neutral atoms
with magnetic dipoles, we study and identify weak field trapping
states of polar molecules inside a spatially inhomogeneous dc
E-field. Further, we suggest that an effective and efficient vortex
pump protocol can be envisioned for condensates of polar molecules
\cite{mottonen07,xu08}, based on the flipping of the axial bias
field, originally suggested \cite{isoshima00} and experimentally
demonstrated for atomic spinor condensates inside a B-field IPT
\cite{leanhardt02,leanhardt03,isoshima07}. We have confirmed that
the E-field bias flip remains effective, and the vortex state created
maintains a vorticity proportional to the $M_J$ quantum number of
the trapped molecule state. When a diatomic molecule is placed
inside a homogeneous E-field, $M_J$ is a good quantum number. The
weak field trapping states can be associated with very large values
of $M_J$, thus the amount of vorticity created could become very
significant even after a single bias flip. This vorticity possibly could open
up a practical approach to reach the rapid rotation limit of
atomic/molecular quantum gases.

More generally, we find that this protocol for vortex creation
remains effective if the diatomic molecule is taken as a symmetric
top \cite{herzberg}. Interestingly we find the E-bias flip also
works for the strong field trapping states provided the polar
molecules are confined through other means not relying on its
permanent electric dipole. Finally, other improvements to the
 B-field bias flip protocol \cite{isoshima00}, such as
those developed for cyclically operated continuous vortex pumping schemes can
be analogously extended to the case of heteronuclear molecules
\cite{mottonen07,xu08}.

\section{Acknowledgement}
This work is supported by US NSF, NSF of China under Grant
10640420151, and NKBRSF of China under Grants 2006CB921206 and
2006AA06Z104.

\section*{References}


\section{Introduction}

The transition between the microscopic and macroscopic worlds is a
fundamental issue in quantum theory both from the point of view of
foundations of physics and of the application to quantum computation \cite{giulini,zurek}.

Spontaneous symmetry breaking (SSB) indicates a situation where,
given a symmetry of the Hamiltonian, there are eigenstates which
are not invariant under the action of this symmetry, unless a term
is added which explicitly breaks the symmetry. Usually, when the
control parameter reaches a critical value, the lowest energy
eigenstate keeping the Hamiltonian symmetry is no longer stable in
the presence of infinitely small perturbations, and new stable
solutions appear which are not symmetric. SSB leads naturally to a
degenerate manifold of ground states.

Symmetry breaking usually occurs in the thermodynamic limit,
when superselection destroys quantum coherence. Important exceptions are the
XXX Heisenberg model and the XY model in a transverse magnetic field at the
particular value of the field where ground state factorization occurs \cite{kurmann}. In these cases, ground state degeneracy occurs for any size of the system,
and it is therefore possible to explicitly study the transition from quantum
to classical behavior. How entanglement is affected, in the thermodynamic limit, by the presence of a term which explicitly breaks the symmetry, ha been discussed by Sylju\aa{}sen \cite{syl} and by Osterloh \textit{et al.} \cite{osterloh}. Here, we face the problem from another point of view, starting from small systems, and then increasing the size until the thermodynamic limit is reached.

The XY model in a transverse field has been introduced in the
early sixties and solved by Katsura \cite{katsura}, by means of
the Jordan-Wigner transformation, formerly introduced by Lieb,
Schultz, and Mattis \cite{lieb}. Subsequently, the correlation
functions were investigated in great detail by Barouch and MacCoy
\cite{barouch}, who found the existence of a second critical value
of the transverse field separating qualitatively different
behaviors of the correlation functions. Later on, Kurmann, Thomas and Muller \cite{kurmann}
discovered ground state factorization for a large class of spin
models. In the particular case of the XY model, the field at which
factorization occurs is exactly the critical field of Barouch and
MacCoy. Recent interest has been devoted to the study of
entanglement properties of many-body systems undergoing a quantum
phase transition \cite {osborne,fazio,vidal,vedral}. As shown in
Ref. \cite{amico}, the critical point turns out to separate two
regions with qualitatively different bipartite entanglement. It has been shown in Ref. \cite{baroni} that, in the vicinity of the factorizing field, the range of concurrence diverges, and that such divergence
corresponds the appearance of a characteristic length scale in the system.
Recently, the conditions for the existence of the factorizing field for models with long-range interaction have been determined \cite{giampaolo}, and the study of this effect has been extended to dimerized chains \cite{gianluca}.

The two systems we wish to investigate belong to different classes of
symmetry. While the XXX Heisenberg model has the SU(2) continuous symmetry,
i.e. the Hamiltonian commutes with the total spin along any possible direction, the XY
model is invariant under parity transformations and possesses the discrete $Z_{2}$
symmetry.

It is commonly accepted that purely quantum effects are not
observable on the macroscopic scale, except for superconductivity,
superfluidity. On the other hand, quantifying entanglement
(perhaps the most genuine manifestation of quantum properties) as
a function of the system size represents a fundamental issue
\cite{vednat}. Here, we wish to investigate in detail the two-spin
entanglement dependence on the total number of spins for these
models. In particular, we shall derive the difference in the size
effects due to the difference in the system symmetry. To be more
specific, in the case of a discrete symmetry there is an
exponential entanglement decrease, while with a continuous
symmetry entanglement shrinks linearly with the growth of the
system size.

Since the XY model has been studied through the last four decades, and
results are scattered over a vast literature, for convenience we shall give here a brief survey
of the main results, with the primary aim of focusing on the existence of the
factorizing field and its independence from the system size.

The paper is organized as follows. In Sec. \ref{maro} the XY model
in a transverse field is discussed. We give special emphasis to
the finite size solution with the scope of enlightening the
emergence of the factorizing field as a size-independent
degeneracy point. Furthermore, by means of the finite size
picture, we are able to explain in a simple way the appearance of
spontaneous symmetry breaking in the thermodynamic limit. In Sec.
\ref {XXX} we describe briefly the structure of the ground state
for the isotropic Heisenberg (XXX) model. Even if, for any finite
number of spins the ground state manifold has finite dimension, an
over-complete set of states can be introduced that allows to study
the microscopic-to-macroscopic transition. In Sec. \ref{2sp} we
derive the value of the concurrence for pairs of spins and the
order parameter fluctuation in a superposition state as a function
of the size systems both for the XY model and the XXX model.
Finally, in Sec. \ref{disc}, results are discussed. In particular,
we will focus on the influence of the symmetry in the different
behaviors.

\section{XY Model\label{maro}}

Let us consider a chain of $N$ spins
\begin{equation}
H=\sum_{l}\left[ J\frac{\left( 1+\gamma \right) }{2}\sigma _{l}^{x}\sigma
_{l+1}^{x}+J\frac{\left( 1-\gamma \right) }{2}\sigma _{l}^{y}\sigma
_{l+1}^{y}+h\sigma _{l}^{z}\right] ,
\end{equation}
where $\sigma ^{\epsilon }$ are the three Pauli matrices $\left( \epsilon
=x,y,z\right) $, and periodic boundary conditions ($\sigma _{N+1}^{\epsilon
}=\sigma _{1}^{\epsilon }$) are assumed. In the following we will assume $J=-1
$ (ferromagnetic coupling). The above Hamiltonian is invariant under the
Z_{2}$ group of the rotations by $\pi $ about the $z$ axis, since
it commutes with the parity operator $P=\prod_{l}\sigma _{l}^{z}$.
Due to this symmetry, eigenstates are classified depending on
parity eigenvalue. This system is known to undergo a quantum phase
transition at the critical point $h_{c}=1$. Below this value, in
the thermodynamic limit, spontaneous magnetization along the $x$
axis appears.

Since the work of Ref. \cite{lieb}, the Jordan-Wigner transformation,
defined through $\sigma _{l}^{z}=1-2c_{l}^{\dagger }c_{l}$, $\sigma
_{l}^{+}=\prod_{j<l}\left( 1-2c_{l}^{\dagger }c_{l}\right) c_{l}$, $\sigma
_{l}^{-}=\prod_{j<l}\left( 1-2c_{l}^{\dagger }c_{l}\right) c_{l}^{\dagger }$
is introduced to map spins in spinless fermions. The transformed Hamiltonian
is $H=H_{0}-PH_{1}$ with
\begin{eqnarray}
H_{0} &=&-\sum_{l=1}^{N-1}\left[ \left( c_{l}^{\dagger
}c_{l+1}-c_{l}c_{l+1}^{\dagger }\right) +\gamma \left( c_{l}^{\dagger
}c_{l+1}^{\dagger }-c_{l}c_{l+1}\right) -h\left( 1-2c_{l}^{\dagger
}c_{l}\right) \right] , \\
H_{1} &=&-\left[ \left( c_{N}^{\dagger }c_{1}-c_{N}c_{1}^{\dagger }\right)
+\gamma \left( c_{N}^{\dagger }c_{1}^{\dagger }-c_{N}c_{1}\right) \right] .
\end{eqnarray}
Since $\left[ H,P\right] =0$, all eigenstates of $H$\ have
definite parity, and we can proceed to a separate diagonalization
of $H$ in the two subspaces labelled by to $P=\pm 1$. Then, the
complete set of eigenvectors of $H$ will be given by the even
eigenstates of\ $H^{+}=H_{0}-H_{1}$ and the odd eigenstates of
$H^{-}=H_{0}+H_{1}$. Both for $H^{+}$\ and $H^{-}$ the
diagonalization is obtained by first carrying out the space
Fourier transform
\begin{equation}
c_{k}=\frac{1}{\sqrt{N}}\sum_{k}e^{-i\frac{2\pi }{N}kl}c_{l},
\end{equation}
where $k=0,1,\ldots ,N-1$ in $H^{-}$, and $k=1/2,3/2,\ldots ,N-1/2$ in $H^{+}
$, and then making the Bogoliubov transformation
\begin{equation}
c_{k}=\cos \vartheta _{k}\eta _{k}+i\sin \vartheta _{k}\eta
_{-k}^{\dagger },
\end{equation}
with $\vartheta _{k}=-\vartheta _{-k}$. Here, $\eta _{-k}$ stands for $\eta _{N-k}$.
The diagonalization condition implies for $\vartheta _{k}$
\begin{equation}
\tan 2\vartheta _{k}=-\frac{\gamma \sin k}{h-\cos k}.
\end{equation}
Eventually, we end up with the quasi-particle Hamiltonians
\begin{eqnarray}
H^{+} &=&\sum_{k=1/2}^{N-1/2}\Lambda _{k}\left( \eta _{k}^{\dagger }\eta
_{k}-\frac{1}{2}\right) , \\
H^{-} &=&\sum_{k=0}^{N-1}\Lambda _{k}\left( \eta _{k}^{\dagger }\eta _{k}
\frac{1}{2}\right) ,
\end{eqnarray}
where the eigenvalues are given by
\begin{equation}
\Lambda_{k}=2\sqrt{\left(h-\cos \frac{2\pi}{N}k\right)^{2}+\gamma^{2}\sin^{2}\frac{2\pi}{N}k}.
\end{equation}
The ground states of $H^{+}$ and $H^{-}$
are the corresponding vacuum states with eigenvalues
$E_{0}^{+}=-\sum_{k=1/2}^{N-1/2}\Lambda _{k}$ and
E_{0}^{-}=-\sum_{k=0}^{N-1}\Lambda _{k}$.

The vacuum in the generic $k$ mode is determined by $\eta _{k}\left| 0^{\pm }\right\rangle =0$.\
While for every $k\neq 0$ the Bogoliubov vacuum corresponds to an even state
(the absence of quasi-particles implies zero or two particles), the mode $k=0
$ plays a special role. In fact, the correspondent Bogoliubov transformation
reads
\begin{equation}
\eta _{0}=\frac{1}{2}\left( 1+\frac{h-1}{\left| h-1\right| }\right) c_{0}
\frac{i}{2}\left( 1-\frac{h-1}{\left| h-1\right| }\right)
c_{0}^{\dagger },
\end{equation}
with the important consequence that the quasi-particle vacuum corresponds to a
zero-particle state for $h<1$ and to one-particle state for $h>1$. The
presence or the absence of the particle in the $k=0$ mode changes the parity
of the state. Thus, for $h>1$, the vacuum of $H^{-}$, because of its
symmetry, does not belong to the set of eigenstates of $H$, while for $h<1$
it becomes an eigenstate of physical interest. Above the $h=1$, the odd
state of lowest energy is obtained by adding one quasi-particle
corresponding to the bottom of the energy band with energy $\Lambda _{\min
}=2\left( h-1\right) $. This energy gap prevents the degeneracy even in the
thermodynamic limit.

\subsection{Quantum phase transition and ground state factorization}

The change of symmetry of the vacuum of H$^{-}$ is the very cause of the
phase transition in the thermodynamic limit. Indeed, on the macroscopic
scale the sum over $k$ becomes an integral yielding $E_{0}^{+}=E_{0}^{-}$. Then,
below the critical point $h_{C}=1$ the odd and the even lowest eigenstates
are degenerate, and the Hamiltonian symmetry is spontaneously broken, while,
for $h>h_{C}$, \ due to the existence of the energy gap $\Lambda _{\min }$,
the ground state keeps its parity (even). For $h<h_{C}$, because of
superselection rules, the system is necessarily found in symmetry-broken
states.

As pointed out in Ref. \cite{barouch}, below the critical point there are
two different regions where two-body correlation functions can decrease
monotonically or oscillate as a function of the spin distance, depending on
the Hamiltonian parameters. These regions are separate, in the $\{h,\gamma \}
$ diagram, by the set of points satisfying $h_{F}^{2}+\gamma ^{2}=1$.
More recently, it has been shown that on this border line the ground state
factorizes \cite{verrucchi}, i.e. it can be written as $\left| \Psi
_{F}^{\pm }\right\rangle =\otimes _{l}\left| \Psi _{F,l}^{\pm }\right\rangle
$, with $\left| \Psi _{F,l}^{\pm }\right\rangle =\left( \cos \alpha \left|
\uparrow _{l}\right\rangle \pm \sin \alpha \left| \downarrow
_{l}\right\rangle \right) $, where $\cos 2\alpha =\left[ \left( 1-\gamma
\right) /\left( 1+\gamma \right) \right] ^{1/2}$.

The existence of the factorizing field, originally derived by
requiring only size-independent degeneracy between the lowest odd
and even eigenvalues \cite {kurmann}, can be studied within the
general solution of the model. By analyzing lowest odd and even
eigenvalues of $H$ in the symmetry broken region for finite $N$ as
a function of the transverse field, we observe a series of $N/2$
level crossings for $h=h_{i}$ (see Fig. \ref{energie}). In
correspondence of each $h_{i}$ the ground state changes its
symmetry. The existence of such points has been discussed in Ref.
\cite{hoeger} and more recently in Ref. \cite{rossignoli}, and is
responsible for the magnetization jumps reported in Ref.
\cite{puga}. In the thermodynamic limit, this kind of structure
implies two different symmetry breaking mechanisms. For
$0<h<h_{F}$, as $N\rightarrow \infty $, the set $\left\{
h_{i}\right\} $ of the degeneracy points becomes a denumerable
infinity, while for $h_{F}<h<1$ there is a the usual symmetry
breaking due to the vanishing of the gap.
An interesting problem would be to check whether this is the
microscopic mechanism responsible for the qualitative change in
the behavior of the correlation functions above and below $h_{F}$.
\begin{figure}
  \includegraphics[width=.5\textwidth,height=40mm]{En8.eps}
\caption{Difference of energy between the lowest odd and even
eigenvalues of $H$ as a function of the transverse field for a
8-spin chain. Given the anisotropy amplitude $\gamma=0.6$, the
factorizing field is $h_F=0.8$. As predicted, we observe $N/2$
level crossing point, the last of them being $h_F$.}
\label{energie}
\end{figure}

While spontaneous symmetry breaking arises only for $N\rightarrow
\infty $, it can be seen from the previous analysis (see also Ref.
\cite{rossignoli}) that at the factorizing point $h_{F}$
degeneracy appears for any $N$. It is simple to show that
$E_{0}^{+}\left( h_{F}\right) =E_{0}^{-}\left( h_{F}\right) $
holds for any $N$, while the positions of all the other level
crossing points $h_{i}$ change with $N$. Then, at the special
field $h_{F}$, the Hamiltonian symmetry is broken independently of
the system size, and any linear superposition of the two symmetric
eigenstates $\left| \alpha ^{\pm }\right\rangle $ ($\left| \alpha
^{+}\right\rangle $ for the even eigenstate and $\left| \alpha
^{-}\right\rangle $ for the odd eigenstate), is a possible
eigenstate. Obviously, each one of the factorized states can be
expressed as a linear combination of the two\thinspace symmetric
eigenstates
\begin{equation}
\left| \Psi _{F}^{\pm }\right\rangle =u_{+}\left| \alpha
^{+}\right\rangle \pm u_{-}\left| \alpha ^{-}\right\rangle ,
\label{ftoa}
\end{equation}
with $u_{\pm }=\left[ \left( 1+\cos ^{N}2\alpha \right) /2\right] ^{1/2}$.
Notice that, for finite size systems, $\left| \Psi _{F}^{+}\right\rangle $
and $\left| \Psi _{F}^{-}\right\rangle $ are not orthogonal, while $\
\left\langle \Psi _{F}^{+}|\Psi _{F}^{-}\right\rangle =0$ in the
thermodynamic limit.

\section{XXX model\label{XXX}}

The homogeneous (ferromagnetic) Heisenberg model is defined by the
Hamiltonian
\begin{equation}
H_{XXX}=-J\sum_{l=0}^{N-1}\left( \sigma _{l}^{x}\sigma _{l+1}^{x}+\sigma
_{l}^{y}\sigma _{l+1}^{y}+\sigma _{l}^{z}\sigma _{l+1}^{z}\right) ,
\end{equation}
with the boundary condition $\sigma _{N}^{\epsilon }=\sigma
_{0}^{\epsilon }$. The model has been solved using the Bethe
ansatz \cite {bethe}. As far as the ground state properties are
concerned, a simple argument can be introduced to show that, for
any number of spins, any factorized
state $\left| \Phi \left( \theta ,\phi \right) \right\rangle =\otimes _{l
\left[ \cos \theta \left| \uparrow \right\rangle +\exp \left( i\phi \right)
\sin \theta \left| \downarrow \right\rangle \right] $ minimizes the energy.
Given the invariance of $H_{XXX}$ with respect to rotations of arbitrary
amplitude $\beta $ around any direction $\hat{n}$ $\mathcal{R}\left( \beta
\hat{n}\right) =\prod_{l}\exp \left[ i\beta \vec{\sigma}_{l}\cdot \hat{n
\right] $, due to $\left[ H_{XXX},\mathcal{R}\left(
\beta ,\hat{n}\right) \right] =0$, we shall restrict the attention on the particular state $\left| \Phi
\left( 0,0\right) \right\rangle =\left| \uparrow ,\uparrow ,\ldots ,\uparrow
\right\rangle $. It can be immediately seen that $\left| \Phi \left(
0,0\right) \right\rangle $ belongs to the ground state subspace of any of
two-body terms of $H_{XXX}$, and then its energy represents the minimum
achievable value. To make a link with the XY model, we could say that factorization point for the Heisenberg model corresponds to $h=0$.

In the absence of spontaneous symmetry breaking, i.e. for finite systems,
and in the absence of external fields, the ground state belongs to an $\left(
N+1\right) $-dimensional manifold, and can be expanded in the
over-complete set of factorized states
\begin{equation}
\left| \Phi \right\rangle =\frac{1}{\mathcal{N}^{\prime }}\int d\theta \int
d\phi f\left( \theta ,\phi \right) \left| \Phi \left( \theta ,\phi \right)
\right\rangle ,
\end{equation}
where $f\left( \theta ,\phi \right) $ is a weight function.The inner product
between states pointing in different directions reads
\begin{equation}
\left\langle \Phi \left( \theta ^{\prime },\phi ^{\prime }\right) |\Phi
\left( \theta ,\phi \right) \right\rangle =\left[ \cos \theta \cos \theta
^{\prime }+e^{i\left( \phi -\phi ^{\prime }\right) }\sin \theta \sin \theta
^{\prime }\right] ^{N}.  \label{inner}
\end{equation}
Then, only in the thermodynamic limit we have a set of orthogonal states.
Given the continuous $SU(2)$ symmetry of the model, spontaneous symmetry
breaking implies that the system will select one direction out of all the possible
choices in the $\left( \theta ,\phi \right) $ space.

As we are interested in studying the problem of vanishing of
peculiar quantum properties, we shall choose initial states with
given symmetry properties, which, in the finite size limit, do
exhibit those properties.

\section{Transition to the Schr\"{o}dinger cat regime\label{2sp}}

According to the superposition principle, every linear combination of
quantum states is allowed. On the other hand, it is well known that
superposition cannot be observed on the macroscopic scale because of
superselection, the most convincing argument being the Schr\"{o}dinger cat
paradox. Then, on this scale, all but a small set of states belonging the
total Hilbert space are actually forbidden. This process, which leads to a
diagonal form of the density operator in a preferred basis, implies the
vanishing of the most peculiar of quantum properties: state interference.

In order to study the vanishing of state interference we analyze two
different quantities: two-spin entanglement and the fluctuation properties
of the order parameter $M_{x}=\left( \sum_{l}\sigma _{l}^{x}\right) /N$.
Given a density matrix $\rho $, fluctuations statistics
 is associated to the generating function
\begin{equation}
G_{\rho }\left( \lambda \right) =\mathrm{Tr}\left\{ \rho e^{\frac{i\lambda }
N}\sum_{l}\sigma _{l}^{x}}\right\} ,
\end{equation}
which is the Fourier transform of the probability distribution
function of $M_{x}$. When the system can be observed in $m$ states
$\Psi _{1},\Psi _{2},\ldots \Psi _{m}$, whit related generating
functions $G_{\Psi _{n}}\left( \lambda \right)$, quantum
superposition effects appear if
\begin{equation}
\Delta G=G_{\rho }\left( \lambda \right) -\frac{1}{m}\sum_{n=1}^{m}G_{\Psi
_{n}}\left( \lambda \right) \neq 0.
\end{equation}
We expect that in the symmetry broken regime $\lim_{N\rightarrow
\infty }\Delta G=0$.

Similar considerations, carried out about entanglement properties,
lead to establish that, when $N\rightarrow \infty$, only
factorized states can be observed.

Even if superselection can be assumed as a principle, the size dependence of
quantum interference effects will be related to the particular system
observed. In the following, we find different decaying behaviors for
the XY and the XXX models, which are caused by the difference in the symmetry
of the two systems. We will start in both models by considering symmetric
states (which are expected  not to survive in the  thermodynamic limit) and
we shall study two-spin entanglement as a function of $N$. In fact, the existence of
degenerate ground state manifolds for any $N$ allows to calculate coherence
properties as a function of the size of the system.

For qubit systems, like spins, two-body entanglement can be measured through
concurrence \cite{wootters}. As shown in Ref. \cite{verrucchi}, for states
which are invariant under the action of the parity operator, the concurrence
$\mathcal{C}_{ij}$ of two spins at sites $i$ and $j$ is related to the
quantum correlation functions by simple relations which will be used here.
As pointed out in Ref. \cite{rossignoli}, for the XY model at the factorizing
field the two-spin concurrence does not depend on the spin distance $|i-j|$. Similar
arguments can be used also for the XXX chain. Since we are dealing with
superpositions of ferromagnetic states, the entanglement will be
of ferromagnetic kind as well. In this case,
\begin{equation}
\mathcal{C}_{ij}=\frac{1}{2}\left| p_{1}-p_{2}\right| -p_{III},
\end{equation}
where $\left| p_{1}-p_{2}\right| $ is the average value of $\left| \uparrow
\uparrow \right\rangle \left\langle \downarrow \downarrow \right| +\left|
\downarrow \downarrow \right\rangle \left\langle \uparrow \uparrow \right|
, and $p_{III}$ is the average value of $\left| \uparrow \downarrow
\right\rangle \left\langle \uparrow \downarrow \right| $.

\subsection{XY model}

Let us first consider the order parameter fluctuations for the symmetric states $\left| \alpha ^{\pm }\right\rangle $. These states could be obtained by starting with $h\neq h_F$. In this case the exact ground state would have definite parity. For instance, for $h> h_F$, the ground state is even. By lowering the field until the value $h_F$ is reached, the system is driven in  $\left| \alpha ^{+ }\right\rangle $.The generating function
is
\begin{equation}
G\left( \lambda ,\alpha ^{\pm }\right) =\frac{1}{4u_{\pm }^{2}}\left[
G\left( \lambda ,\Psi _{F}^{+}\right) +G\left( \lambda ,\Psi _{F}^{-}\right)
+\tilde{G}\left( \lambda ,\Psi _{F}^{+},\Psi _{F}^{-}\right) +\tilde{G
\left( \lambda ,\Psi _{F}^{-},\Psi _{F}^{+}\right) \right],
\end{equation}
where
\begin{equation}
\tilde{G}\left( \lambda ,\Psi _{F}^{\pm },\Psi _{F}^{\mp }\right)
=\left\langle \Psi _{F}^{\pm }\right| e^{\frac{i\lambda
}{N}\sum_{l}\sigma _{l}^{x}}\left| \Psi _{F}^{\mp }\right\rangle.
\end{equation}
It is easy to show that
\begin{equation}
G\left( \lambda ,\Psi _{F}^{\pm }\right) =\left( \cos \frac{\lambda }{N
\pm i\sin \frac{\lambda }{N}\sin 2\theta \right) ^{N},
\end{equation}
\begin{equation}
\tilde{G}\left( \lambda ,\Psi _{F}^{\pm },\Psi _{F}^{\mp
}\right)=\left( \cos \frac{\lambda }{N}\cos 2\theta \right) ^{N}.
\end{equation}
Then, interference effects (manifested by the off diagonal
elements) disappear exponentially with $N$.

As a second characterization, we study the concurrence for the symmetric
states $\left| \alpha ^{\pm }\right\rangle $. This can be easily derived
using the expression of $\left| \alpha^{\pm }\right\rangle $ in terms of $\left| \Psi _{F}^{\pm }\right\rangle $.
The result (see also Ref. \cite{rossignoli}) is given by
\begin{equation}
\mathcal{C}_{ij}\left( \alpha ^{\pm }\right) =\left( \cos 2\alpha \right)
^{N-2}\frac{\sin ^{2}2\alpha }{1\pm \left( \cos 2\alpha \right) ^{N}},
\end{equation}
where the factor $\left( \cos 2\theta \right) ^{N-2}$ derives from the
non-orthogonality of $\left| \Psi _{F}^{+}\right\rangle $ and $\left| \Psi
_{F}^{-}\right\rangle $ and determines the speed of classicalization. In the
macroscopic limit, $\mathcal{C}_{ij}\left( \alpha ^{\pm }\right) $ vanishes
as $\left[ \left( 1-\gamma \right) /\left( 1+\gamma \right) \right] ^{N/2}$.
Then, for every finite value of the anisotropy $\gamma $, entanglement
decays exponentially with $N$. A small but finite anisotropy will enhance
entanglement. Actually, the $\gamma =0$ limit implies a non-analytic change

\subsection{XXX model}

In analogy with the previous case, we introduce a state which is
invariant under a given spin rotation. In particular, if we choose
the state $\left| \Phi _{e}\right\rangle$ invariant under
rotations about the $y$ axis  $\exp \left[ -i\theta \sum_{l}\sigma
_{l}^{y}\right] \left| \Phi _{e}\right\rangle =\left| \Phi
_{e}\right\rangle $, we have
\begin{equation}
\left| \Phi _{e}\right\rangle =\frac{1}{\mathcal{N}}\int_{0}^{2\pi
}d\theta \left| \Phi _{\theta }\right\rangle,   \label{equat}
\end{equation}
where $\left| \Phi _{\theta }\right\rangle =\otimes _{l}\left| \Phi
_{l}\left( \theta ,0\right) \right\rangle $, and where $\left| \Phi
_{l}\left( \theta ,0\right) \right\rangle =\cos \theta \left| \uparrow
_{l}\right\rangle +\sin \theta \left| \downarrow _{l}\right\rangle $.
Requiring the normalization of $\left| \Phi _{e}\right\rangle $ implies
\int d\theta ^{\prime }d\theta \left[ \cos \left( \theta -\theta ^{\prime
}\right) \right] ^{N}=\mathcal{N}^{2}$. It is easy to verify that $\left|
\Phi _{e}\right\rangle $ is also an eigenstate of the parity operator. In fact, the
integration over $\theta $ cancels, in the superposition, all terms with an odd number of down spins. Each $\left| \Phi _{\theta }\right\rangle$ would be the actual ground state in the presence of an external field directed along the direction $\theta $.

Let us analyze the order parameter. First, we calculate fluctuations for a
given element of the ground state degeneracy manifold, obtaining
\begin{equation}
G_{\Phi _{\theta }}\left( \lambda \right) =\frac{1}{\mathcal{N}^{2}}\left(
\cos \frac{\lambda }{N}+i\sin \frac{\lambda }{N}\sin 2\theta \right) ^{N}.
\end{equation}
For large $N$ we see that $G_{\Phi _{\theta }}\left( \lambda \right) \simeq
\exp \left( i\lambda \sin 2\theta \right) $. \ This is the typical
expression of the generating function of a non fluctuating quantity. Its
Fourier transform, which is the probability distribution function of the
order parameter is indeed, for any $\theta $, a Dirac's delta distribution
around $\sin 2\theta $.

Furthermore, in the superposition state we have
\begin{equation}
G_{\Phi _{e}}\left( \lambda \right) =\frac{1}{\mathcal{N}^{2}}\int d\theta
^{\prime }d\theta \left[ g\left( \lambda ,\theta ,\theta ^{\prime }\right)
\right] ^{N},
\end{equation}
where
\begin{equation}
g\left( \lambda ,\theta ,\theta ^{\prime }\right) =\cos \frac{\lambda }{N
\cos \left( \theta -\theta ^{\prime }\right) +i\sin \frac{\lambda }{N}\sin
\left( \theta +\theta ^{\prime }\right).
\end{equation}
As $N$ gets large, the vanishing of interference is observed. In the large $N$
limit, the steepest descent method gives
\begin{equation}
\Delta G_{\Phi _{e}}\left( \lambda \right) =\int d\theta G_{\Phi _{\theta
}}\left( \lambda \right) \left[ \exp \left( \frac{\lambda ^{2}\cos 2\theta }
2N}\right) -1\right],
\end{equation}
implying
\begin{equation}
\Delta G_{\Phi _{e}}\left( \lambda \right) \sim 1/N
\end{equation}
in the asymptotic regime.

As far as the two-spin concurrence is concerned, it is straightforward to find
\begin{equation}
\left| p_{1}-p_{2}\right| =\frac{1}{\mathcal{N}^{2}}\int d\theta
d\theta ^{\prime }\left[ \cos ^{2}\theta \sin ^{2}\theta ^{\prime
}+\cos ^{2}\theta ^{\prime }\sin ^{2}\theta \right] \left[ \cos
\left( \theta -\theta ^{\prime }\right) \right] ^{N-2},
\end{equation}
and
\begin{equation}
p_{III}=\int d\theta d\theta ^{\prime }\cos \theta \cos \theta
^{\prime }\sin \theta \sin \theta ^{\prime }\left[ \cos \left(
\theta -\theta ^{\prime }\right) \right] ^{N-2},
\end{equation}
yielding
\begin{equation}
\mathcal{C}_{ij}\left( \Phi _{e}\right) =\frac{1}{2}\frac{\int d\theta
d\theta ^{\prime }\left[ \tan \left( \theta -\theta ^{\prime }\right) \right]
^{2}\left[ \cos \left( \theta -\theta ^{\prime }\right) \right] ^{N}}{\int
d\theta d\theta ^{\prime }\left[ \cos \left( \theta -\theta ^{\prime
}\right) \right] ^{N}}.
\end{equation}
This result is very simple for $N$ even. In that case one gets
\begin{equation}
\mathcal{C}_{ij}^{N\text{\textrm{\ even}}}\left( \Phi _{e}\right) =\frac{1}
2\left( N-1\right) }.
\end{equation}
This result can be understood taking into account that, given Eq. (\ref
{inner}), the inner product between $\left| \Phi \left( \theta ,0\right)
\right\rangle $ and $\left| \Phi \left( \theta ^{\prime },0\right)
\right\rangle $ vanishes exponentially with $N$ . This allows to evaluate
integrals, in the large $N$ regime, by means of the steepest descent method.
It is clear that, when $N$ gets large, $\left[ \cos \left( \theta -\theta ^{\prime
}\right) \right] ^{N}$ is different from zero only for
\left( \theta -\theta ^{\prime }\right) $ $\simeq 0$. Expanding all
terms around this value, the concurrence is well approximated by the ratio
between two Gaussian integrals
\begin{equation}
\mathcal{C}_{ij}\left( \Phi _{e}\right) \simeq \frac{1}{2}\frac
\int_{-\infty }^{\infty }x^{2}\exp \left( -\frac{Nx^{2}}{2}\right) dx}
\int_{-\infty }^{\infty }\exp \left( -\frac{Nx^{2}}{2}\right) dx},
\end{equation}
which eventually gives $\mathcal{C}_{ij}\left( \Phi _{e}\right) \simeq 1/2N$
in the asymptotic regime.

\section{Discussion\label{disc}}

We tackled the problem of describing how quantum coherence effects
vanish as the system size becomes macroscopic. Even if this
phenomenon is expected to appear in generic systems, two different
symmetry-broken model have been considered when exact and analytic
treatment are possible. In the first one (the XY model in a
transverse field), because of the discrete symmetry, the ground
state spans a two-dimensional manifold. As for the Heisenberg
model, the dimension of the manifold grows with $N$, eventually
reaching a dense structure. In Fig. \ref{figura} we plot the
behavior of the concurrence $\mathcal{C}_{ij}$ for the two models.
In the case of the XY chain, we used also different values of the
anisotropy
\begin{figure
  \includegraphics[width=.5\textwidth,height=40mm]{fig2.eps}
\caption{Two-spin concurrence as a function of
   the system size $N$ for the XXX model (dots) and for the XY model (continuous lines).
    In this latter case, we considered two different anisotropy parameters: the red line represents $\gamma=0.5$, and the blue line
      corresponds to $\gamma=0.8$.}
\label{figura}
\end{figure}
.
A necessary step to determine local quantities, like two-spin
entanglement or magnetization, is the introduction of the reduced
density matrix. Given the peculiar structure of the factorized
states we have considered here, calculating the reduced density
matrix requires the computation of inner products between states
defined on $\left( N-m\right) $ (where $m$ is a finite number)
spin subspaces) aligned along different directions . Both for the
XY and for the isotropic model these quantities vanish
exponentially in the large $N$ limit, destroying in such a way the
quantum interference between different components. However, since
the XXX model has a continuous symmetry, the ground state manifold
is continuous as well, and all matrix elements are integrated. The
integration implies a reduction in the decoherence rate, which
turns out to be linear in $1/N$.

In this paper we considered two exactly solvable models, and studied how superselection tends to destroy their quantum properties. A typical tool used to study problems whose solution is not known is the mean-field approximation, that, in fact, consists in the introduction of ``product states'' with the same aspect of those described in this paper. For example, in the BCS theory of superconductivity the solution introduced is factorized in the space of the modes $k$. Since in the finite-size case the symmetry is expected to be conserved, while the mean-field states are widely unsymmetrical, the linear superposition of degenerate states is a way to restore it. Once the thermodynamic limit is performed, all the considerations made in this paper apply. Then, we can conclude that our results apply not only to the models explicitly studied, but they could used, in the limit of validity of the mean-field theory, in all systems belonging to the classes of symmetry discussed.

\acknowledgments
  The authors wish to acknowledge S. Paganelli for useful discussions. GLG acknowledges the “Juan de la Cierva” fellowship of the Spanish Ministry of Science and Innovation.