\section{Introduction}

The study of magnetic models has
generated considerable progresses in the understanding
of magnetic materials,
and lately, it has overcome the frontiers of magnetism,
being considered in many areas of knowledge.
Certainly, the Ising model represents one of the most
studied and important models of magnetism and
statistical mechanics~\cite{huang,reichl},
and it has been employed also to typify a wide variety of
physical systems, like lattice gases, binary alloys, and
proteins (with a particular interest in the problem of protein
folding).
Although real magnetic systems should be properly
described by means of Heisenberg spins (i.e.,
three-dimensional variables), many materials are
characterized by anisotropy fields that make these
spins prefer given directions in space, explaining
why simple models, characterized by
binary variables, became so important
for the area of magnetism.
Particularly, models defined in terms of Ising variables
have shown the ability for exhibiting a wide variety
of multicritical behavior
by introducing randomness, and/or competing
interactions, has attracted the attention of many
researchers~(see, e.g., Refs.~\cite{aharony,mattis,kaufman,nogueira98
nuno08a,nuno08b,salmon1,morais12}).

Certainly, the simplicity of Ising variables,
which are very suitable for both analytical and numerical
studies, has led to proposals of important models outside
the scope of magnetism, particularly in the
area of complex systems.
These models have been successful for describing
a wide variety of relevant
features in such systems, and have raised
interest in many fields, like
financial markets, optimization problems,
biological membranes, and social behavior.
In some cases, more than one Ising variable have been used,
especially by considering a coupling between them, as
proposed within the framework of choice
theories~\cite{fernandez}, or in plastic
crystals~\cite{plastic1,brandreview,folmer}.
In the former case, each set of Ising variables represents
a group of identical individuals, all of which can make two
independent binary choices.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=5.5cm]{figure1.eps}
\end{center}
\vspace{-1cm}
\caption{Illustrative pictures of the three phases as the temperature
increases, low-temperature (ordered) solid, intermediate
plastic crystal, and high-temperature (disordered) liquid phase.
In the plastic state the  centers of mass of the molecules form a
regular crystalline lattice but the molecules are
disordered with respect to the orientational degrees of freedom.}
\label{fig:fasesdecristais}
\end{figure}

The so-called plastic
crystals~\cite{plastic1,brandreview,folmer,michel85,michel87,%
galam87,galam89,salinas1,salinas2} appear as states
of some compounds considered to be simpler than those of canonical
glasses, but still presenting rather nontrivial
relaxation and equilibrium properties. Such a plastic
phase corresponds to an intermediate stable state, between a
high-temperature (disordered) liquid phase, and a low-temperature
(ordered) solid phase and both transitions,
namely, liquid-plastic and plastic-solid, are first order.
In this intermediate phase, the rotational disorder coexists
with a translationally ordered state, characterized by
the centers of mass of the molecules forming a regular crystalline
lattice with the molecules presenting disorder in their
orientational degrees of freedom, as shown
in Fig.~\ref{fig:fasesdecristais}.
Many materials undergo a liquid-plastic phase transition,
where the lower-temperature phase presents such a
partial orientational order, like the plastic-crystal
of Fig.~\ref{fig:fasesdecristais}.
The property of translational invariance makes the plastic crystals
much simpler to be studied from both analytical and numerical
methods, becoming very useful towards a proper
understanding of the glass transition~\cite{plastic1,brandreview,folmer}.
In some plastic-crystal models one introduces a coupling
between two Ising models, associating each of these
systems respectively, to the translational and rotational degrees of
freedom~\cite{galam87,galam89,salinas1,salinas2},
as a proposal for explaining satisfactorily
thermodynamic properties of the plastic phase.

Accordingly, spin variables $\{t_{i}\}$ and $\{r_{i}\}$ are introduced in
such a way to mimic translational
and rotational degrees of freedom of each molecule $i$, respectively.
The following Hamiltonian is
considered~\cite{galam87,galam89,salinas1,salinas2},

\begin{equation}
\label{eq:hamplastcrystals}
{\cal H} = - J_{t}\sum_{\langle ij \rangle}t_{i}t_{j}
- J_{r} \sum_{\langle ij \rangle}r_{i}r_{j}
- \sum_{i} (\alpha t_{i} + h_{i})r_{i}~,
\end{equation}

\vskip \baselineskip
\noindent
where $\sum_{\langle ij \rangle}$ represents a sum over
distinct pairs of nearest-neighbor spins.
In the first summation, the Ising variables $t_{i}=\pm 1$
may characterize two lattices A and B (or occupied and vacant sites).
One notices that the rotational
variables $r_{i}$ could be, in principle, continuous variables,
although the fact that the minimization of the coupling contribution
$\alpha t_{i}r_{i}$ is achieved
for $t_{i}r_{i} =1$ ($\alpha>0$), or
for $t_{i}r_{i} =-1$ ($\alpha<0$), suggests the simpler choice
of binary variables ($r_{i}=\pm 1$) to be appropriate,
based on the energy minimization requirement.

In the present model the variables $t_{i}$ and
$r_{i}$ represent very different characteristics of a
molecule. Particularly, the rotational variables $r_{i}$
are expected to change more freely than the translational ones;
for this reason, one introduces a random field acting only
on the rotational degrees of freedom.
In fact, the whole contribution $\sum_{i} (\alpha t_{i} + h_{i})r_{i}$
is known to play a fundamental role for the plastic phase of
ionic plastic crystals, like the alkalicyanides KCN, NaCN and RbCN.
In spite of its simplicity, the above Hamiltonian is able to capture
the most relevant features of the plastic-crystal phase, as well
as the associated phase transitions,
namely, liquid-plastic and plastic-solid ones~\cite{michel85,michel87,%
galam87,galam89,salinas1,salinas2,vives}.

A system described by a Hamiltonian slightly different
from the one of~\eq{eq:hamplastcrystals}, in which the
whole contribution
$\sum_{i} (\alpha t_{i} + h_{i})r_{i}$ was replaced by
$\sum_{i} \alpha_{i} t_{i}r_{i}$, i.e., with no random
field acting on variable $r_{i}$ separately, was considered
in Ref.~\cite{salinas2}. In such a work one finds a detailed
analysis of the phase diagrams and order-parameter behavior
of the corresponding model. However, to our knowledge,
previous investigations on the model defined
by~\eq{eq:hamplastcrystals} have not
considered thoroughly the effects of the random
field $h_{i}$, with a particular attention to the phase diagrams
for the case of a randomly distributed bimodal
one, $h_{i}=\pm h_{0}$;
this represents the main motivation
of the present work.
In the next section we define the model, determine its free-energy
density, and describe the
numerical procedure to be used.
In Section III we exhibit typical phase diagrams
and analyze the behavior of the corresponding order parameters,
for both zero and finite temperatures; the ability of the model
to exhibit a rich variety of phase diagrams, characterized
by multicritical behavior, is shown.
Finally, in Section IV we present our main conclusions.

\section{The Model and Free-Energy Density}

Based on the discussion of the previous section, herein
we consider a system composed by two interacting Ising models,
described by the Hamiltonian

\begin{equation}
\label{eq:hamiltonian1}
{\cal H}(\{h_{i}\}) = - J_{\sigma} \sum_{(ij)}\sigma_{i}\sigma_{j}
- J_{\tau} \sum_{(ij)}\tau_{i}\tau_{j} + D\sum_{i=1}^{N}\tau_{i}\sigma_{i}
-\sum_{i=1}^{N}h_{i}\tau_{i}~,
\end{equation}

\vskip \baselineskip
\noindent
where $\sum_{(ij)}$ represent sums over all distinct pairs of spins,
a limit for which the mean-field approximation becomes exact. Moreover,
$\tau_{i}= \pm 1$ and $\sigma_{i}= \pm 1$ ($i=1,2, \cdots , N$) depict
Ising variables,
$D$ stands for a real parameter, whereas both $J_{\sigma}$ and
$J_{\tau}$ are positive coupling constants, which will be
restricted herein to
the symmetric case, $J_{\sigma}=J_{\tau}=J>0$. Although this later
condition may seem as a rather artificial simplification of the
Hamiltonian in~\eq{eq:hamplastcrystals}, the application of a
random field $h_{i}$ acting separately on one set of variables, will
produce the expected distinct physical behavior associated with
$\{ \tau_{i} \}$ and $\{ \sigma_{i} \}$. The random fields
$\{ h_{i} \}$ will be considered as following
a symmetric bimodal probability distribution function,

\begin{equation}
\label{eq:hpdf}
 P(h_{i}) = \frac{1}{2} \, \delta(h_{i}-h_{0}) +\frac{1}{2} \, \delta(h_{i}+h_{0})~.
\end{equation}

\vskip \baselineskip
\noindent
The infinite-range character of the interactions allows one to write the above
Hamiltonian in the form

\begin{equation}
\label{eq:hamiltonian2}
{\cal H}(\{h_{i}\})= - \frac{J}{2N}{\left (\sum_{i=1}^{N}\sigma_{i} \right )}^{2}
- \frac{J}{2N}{\left (\sum_{i=1}^{N}\tau_{i} \right )}^{2}
+D\sum_{i=1}^{N}\tau_{i}\sigma_{i} -\sum_{i=1}^{N}h_{i}\tau_{i}~,
\end{equation}

\vskip \baselineskip
\noindent
from which one may calculate the partition function associated with
a particular configuration of the fields $\{ h_{i}\}$,

\begin{equation}
Z(\{h_{i}\}) =  {\rm Tr} \exp \left[- \beta {\cal H}(\{h_{i}\})  \right]~,
\end{equation}

\vskip \baselineskip
\noindent
where $\beta=1/(kT)$ and
${\rm Tr} \equiv {\rm Tr}_{\{ \tau_{i},\sigma_{i}=\pm 1 \}} $ indicates a sum over
all spin configurations. One can now make use of
the Hubbbard-Stratonovich transformation~\cite{dotsenkobook,nishimoribook}
to linearize the quadratic terms,

\begin{equation}
Z(\{h_{i}\}) =  \frac{1}{\pi} \int_{-\infty}^{\infty}dx dy \exp(-x^{2}-y^{2})
\prod_{i=1}^{N} {\rm Tr} \exp [ H_{i}(\tau,\sigma)]~,
\end{equation}

\vskip \baselineskip
\noindent
where $H_{i}(\tau,\sigma)$ depends on the random
fields $\{ h_{i}\}$,
as well as on the spin variables, being given by

\begin{equation}
H_{i}(\tau,\sigma) = \sqrt{\frac{2\beta J}{N}} \ x \tau + \sqrt{\frac{2\beta J}{N}} \ y \sigma
- \beta D \tau \sigma + \beta h_{i} \tau~.
\end{equation}

\vskip \baselineskip
\noindent
Performing the trace over the spins and defining new variables, related to
the respective order parameters,

\begin{equation}
\label{eq:mtausigma}
m_{\tau} = \sqrt{\frac{2kT}{JN}} \ x~; \qquad
m_{\sigma} = \sqrt{\frac{2kT}{JN}} \ y~,
\end{equation}

\vskip \baselineskip
\noindent
one obtains

\begin{equation}
Z(\{h_{i}\})= \frac{\beta J N}{2 \pi} \int_{-\infty}^{\infty} dm_{\tau} dm_{\sigma} \exp[N g_{i} (m_{\tau},m_{\sigma})]~,
\end{equation}

\vskip \baselineskip
\noindent
where

\begin{eqnarray}
g_{i}(m_{\tau},m_{\sigma}) &=& - \frac{1}{2} \beta J m_{\tau}^{2}
- \frac{1}{2} \beta J m_{\sigma}^{2} + \log \left \{
2e^{-\beta D} \cosh[\beta J(m_{\tau}+m_{\sigma}+h_{i}/J)]
\right. \nonumber \\ \nonumber \\
\label{eq:gimtausigma}
&+& \left. 2e^{\beta D} \cosh[\beta J(m_{\tau}-m_{\sigma}+h_{i}/J)] \right \}~.
\end{eqnarray}

\vskip \baselineskip
\noindent
Now, one takes the thermodynamic limit ($N \rightarrow \infty$), and uses the saddle-point
method to obtain

\begin{equation}
Z = \displaystyle \frac{\beta J N}{2 \pi} \int_{-\infty}^{\infty} dm_{\tau} dm_{\sigma}
\exp[-N \beta f(m_{\tau},m_{\sigma})]~,
\end{equation}

\vskip \baselineskip
\noindent
where the free-energy density functional $f(m_{\tau},m_{\sigma})$ results
from a quenched average of
$g_{i}(m_{\tau},m_{\sigma})$ in~\eq{eq:gimtausigma}, over the
bimodal probability distribution of~\eq{eq:hpdf},

\begin{equation}
\label{eq:freeenergy}
f(m_{\tau},m_{\sigma}) = \displaystyle  \frac{1}{2} J m_{\tau}^{2}
+ \frac{1}{2}  J m_{\sigma}^{2} - \frac{1}{2\beta}\log Q(h_{0})
- \frac{1}{2\beta}\log Q(-h_{0})~,
\end{equation}

\vskip \baselineskip
\noindent
with

\begin{equation}
Q(h_{0}) = 2e^{-\beta D} \cosh[\beta J(m_{\tau}+m_{\sigma} + h_{0}/J)]
+2e^{\beta D} \cosh[\beta J(m_{\tau}-m_{\sigma} + h_{0}/J)]~.
\end{equation}

\vskip \baselineskip
\noindent
The extremization of the free-energy density above with respect to the
parameters $m_{\tau}$ and $m_{\sigma}$ yields the following equations of state,

\begin{eqnarray}
\label{eq:mtau}
m_{\tau} &=& \frac{1}{2} \frac{R_{+}(h_{0})}{Q(h_{0})}
+ \frac{1}{2} \frac{R_{+}(-h_{0})}{Q(-h_{0})}~,
\\ \nonumber \\
\label{eq:msigma}
m_{\sigma} &=& \frac{1}{2} \frac{R_{-}(h_{0})}{Q(h_{0})}
+ \frac{1}{2} \frac{R_{-}(-h_{0})}{Q(-h_{0})}~,
\end{eqnarray}

\vskip \baselineskip
\noindent
where

\begin{equation}
R_{\pm}(h_{0}) = e^{-\beta D} \sinh[\beta J(m_{\tau}+m_{\sigma} + h_{0}/J)]
\pm e^{\beta D} \sinh[\beta J(m_{\tau}-m_{\sigma} +h_{0}/J)]~.
\end{equation}

\vskip \baselineskip
\noindent

In the following section we present numerical results for the
order parameters and phase diagrams of the model, at both
zero and finite temperatures.
All phase diagrams are represented
by rescaling conveniently the energy parameters of the system, namely,
$kT/J$, $h_{0}/J$ and $D/J$.
Therefore, for given values of these dimensionless parameters,
the equations of state [Eqs.(\ref{eq:mtau}) and~(\ref{eq:msigma})]
are solved numerically for $m_{\tau}$ and $m_{\sigma}$.
In order to avoid metastable states, all solutions obtained for
$m_{\tau} \in [-1,1]$ and $m_{\sigma} \in [-1,1]$ are
substituted in~\eq{eq:freeenergy},
to check for the minimization of the free-energy density.
The continuous (second order) critical frontiers are found by the set
of input values for which the order parameters fall continuously down to
zero, whereas the first-order frontiers were found through
Maxwell constructions.

Both ordered ($m_{\tau} \neq 0$ and $m_{\sigma} \neq 0$)
and partially-ordered ($m_{\tau}=0$ and $m_{\sigma} \neq 0$)
phases have appeared in our analysis, and will be labeled
accordingly.
The usual paramagnetic phase ({\bf P}),
given by $m_{\tau}=m_{\sigma}=0$, always occurs for sufficiently
high temperatures.
A wide variety of critical points appeared in our analysis
(herein we follow the classification due to Griffiths~\cite{griffiths}):
(i) a tricritical point signals the encounter of a continuous frontier
with a first-order line with no change of slope;
(ii) an ordered critical point corresponds to an isolated critical
point inside the ordered region, terminating a first-order line that
separates two distinct ordered phases;
(ii) a triple point, where three distinct phases coexist, signaling the
encounter of three first-order critical frontiers.
In the phase diagrams we shall use distinct symbols and
representations for the critical points and frontiers, as described below.

\begin{itemize}

\item Continuous (second order) critical frontier: continuous line;

\item First-order critical frontier: dotted line;

\item Tricritical point: located by a black circle;

\item Ordered critical point: located by a black asterisk;

\item Triple point: located by an empty triangle.

\end{itemize}

\section{Phase Diagrams and Behavior of Order Parameters}

\subsection{Zero-Temperature Analisis}

At $T=0$, one has to analyze the different spin orderings that
minimize the Hamiltonian of~\eq{eq:hamiltonian2}.
Due to the coupling between the two
sets of spins, the minimum-energy configurations will correspond to
$\{\tau_{i}\}$ and $\{\sigma_{i}\}$ antiparallel ($D>0$), or parallel ($D<0$).
Therefore, in the absence of random fields ($h_{0}=0$) one should have
$m_{\tau}=-m_{\sigma}$ ($D>0$), and
$m_{\tau}=m_{\sigma}$ ($D<0$),  where $m_{\sigma}=\pm1$.
However, when random fields act on the $\{\tau_{i}\}$ spins, there will
be a competition between these fields and the coupling parameter $D$,
leading to several phases, as represented in Fig.~\ref{fig:groundstate},
in the plane $h_{0}/J$ versus $D/J$. One finds three ordered
phases for sufficiently low values of $h_{0}/J$
and $|D|/J$, in addition to {\bf P} phases for $(|D|/J)>0.5$ and $(h_{0}/J)>1$.
All frontiers shown in Fig.~\ref{fig:groundstate} are first-order critical lines.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=5.5cm]{figure2.eps}
\end{center}
\caption{Phase diagram of the model defined by Hamiltonian
of~\eq{eq:hamiltonian2}, at zero temperature. All critical frontiers
represent first-order phase transitions; the empty triangles denote
triple points.}
\label{fig:groundstate}
\end{figure}

When $(h_{0}/J) \leq 1/2$ one finds ordered phases for all values of $D/J$,
with a vertical straight line at $D=0$ separating the
symmetric state ($D<0$), where $m_{\tau}=m_{\sigma}$, from the
antisymmetric one ($D>0$), characterized by $m_{\tau}=-m_{\sigma}$.
Two critical frontiers (symmetric under a reflection operation)
emerge from the triple point at
$(D/J)=0.0$ and $(h_{0}/J)=0.5$, given, respectively, by
$(h_{0}/J)=0.5 + (D/J)$ for $D>0$, and
$(h_{0}/J)=0.5 - (D/J)$ for $D<0$.
These critical frontiers terminate at $(h_{0}/J)=1.0$ and
separate the low random-field-valued ordered phases from
a partially-ordered
phase, given by $m_{\tau}=0$ and $m_{\sigma}= \pm 1$.
As shown in Fig.~\ref{fig:groundstate}, three triple points
appear, each of them signaling the encounter
of three first-order lines, characterized by a coexistence of three phases,
defined by distinct values of the magnetizations $m_{\tau}$ and
$m_{\sigma}$, as described below.

\begin{itemize}

\item $[(D/J)=-0.5$ and $(h_{0}/J)=1.0]$~:
$(m_{\tau},m_{\sigma})=\{ (0,0);(0,\pm 1); (\pm 1, \pm 1) \}$.

\item $[(D/J)=0.5$ and $(h_{0}/J)=1.0]$~:
$(m_{\tau},m_{\sigma})=\{ (0,0);(0,\pm 1); (\pm 1, \mp 1) \}$.

\item  $[(D/J)=0.0$ and $(h_{0}/J)=0.5]$~:
$(m_{\tau},m_{\sigma})=\{ (\pm 1,\pm 1);(\pm 1, \mp 1); (0, \pm 1) \}$.

\end{itemize}

Such a rich critical behavior shown for $T=0$ suggests that interesting
phase diagrams should occur when the temperature is taken
into account. From now on, we investigate the model
defined by the Hamiltonian of~\eq{eq:hamiltonian2} for finite temperatures.

\subsection{Finite-Temperature Analysis}

As shown above, the zero-temperature phase diagram presents
a reflection symmetry with respect
to $D=0$ (cf. Fig.~\ref{fig:groundstate}).
The only difference between the two sides of this phase
diagram concerns the magnetization solutions
characterizing the ordered phases for low random-field values,
where one has
$m_{\tau}=-m_{\sigma}$ ($D>0$), or
$m_{\tau}=m_{\sigma}$ ($D<0$).
These results come as a consequence
of the symmetry of the Hamiltonian of~\eq{eq:hamiltonian2},
which remains unchanged under the operations,
$D \rightarrow -D$, $\sigma_{i} \rightarrow -\sigma_{i}$ $(\forall i)$, or
$D \rightarrow -D$, $\tau_{i} \rightarrow -\tau_{i}$,
$h_{i} \rightarrow -h_{i}$ $(\forall i)$.
Hence, the finite-temperature phase diagrams should present similar
symmetries with respect to a change $D \rightarrow -D$. From now on,
for the sake of simplicity, we will restrict ourselves to the
case $(D/J) \geq 0$, for which the zero-temperature and
low-random-field magnetizations present
opposite signals, as shown in Fig.~\ref{fig:groundstate}, i.e.,
$m_{\tau}=-m_{\sigma}$~.

\begin{figure}[htp]
\begin{center}
\vspace{.5cm}
\includegraphics[height=5cm]{figure3a.eps}
\hspace{0.5cm}
\includegraphics[height=5cm]{figure3b.eps}
\end{center}
\vspace{-.5cm}
\caption{Phase diagrams of the model
defined by the Hamiltonian of~\eq{eq:hamiltonian2} in two
particular cases:
(a) The plane of dimensionless variables $kT/J$ versus $D/J$,
in the absence of random fields $(h_{0}=0)$;
(b) The plane of dimensionless variables $kT/J$
versus $h_{0}/J$, for $D=0$.
The full lines represent continuous phase transitions,
whereas the dotted line stands for a
first-order critical frontier.
For sufficiently high temperatures one finds a
paramagnetic phase ({\bf P}), whereas
the magnetizations $m_{\tau}$ and
$m_{\sigma}$ become nonzero
by lowering the temperature.
In case (b), two low-temperature phases appear, namely,
the ordered (lower values of $h_{0}$) and
the partially-ordered (higher values of $h_{0}$).
These two phases are separated by a continuous
critical frontier (higher temperatures), which turns into
a first-order critical line (lower temperatures) at a tricritical
point (black circle). The type of phase
diagram exhibited in (b) will be called herein of topology I.}
\label{fig:tdh00}
\end{figure}

In Fig.~\ref{fig:tdh00} we exhibit phase diagrams of the model
in two particular cases, namely, in the absence of fields $(h_{0}=0)$
[Fig.~\ref{fig:tdh00}(a)] and for zero coupling
$(D=0)$ [Fig.~\ref{fig:tdh00}(b)].
These figures provide useful
reference data in the numerical procedure
to be employed for constructing phase diagrams in
more general situations, e.g., in the plane
$kT/J$ versus $h_{0}/J$, for several values of $(D/J)>0$.

In Fig.~\ref{fig:tdh00}(a) we present the phase diagram
of the model in the plane of dimensionless variables $kT/J$ versus $D/J$,
in the absence of random fields ($h_{0}=0$),
where one sees the point $D=0$ that corresponds
to two noninteracting Ising models, leading to the well-known mean-field
critical temperature of the Ising model [$(kT_{c}/J)=1$].
Also in Fig.~\ref{fig:tdh00}(a),
the ordered solution $m_{\tau}=-m_{\sigma}$
minimizes the free energy at low temperatures
for any $D>0$; a second-order frontier separates this ordered phase
from the paramagnetic one that appears for sufficiently high temperatures.
For high values of $D/J$ one sees that this critical frontier approaches
asymptotically $(kT/J) = 2$.
Since the application of a random field results in
a decrease of the critical temperature, when compared with the one
of the case $h_{0}=0$~\cite{aharony,mattis,kaufman},
the result of Fig.~\ref{fig:tdh00}(a) shows that no
ordered phase should occur for $h_{0}>0$ and $(kT/J)>2$.

The phase diagram for $D=0$ is shown in
the plane of dimensionless variables $kT/J$
versus $h_{0}/J$ in Fig.~\ref{fig:tdh00}(b).
The {\bf P} phase occurs for $(kT/J)>1$, whereas
for $(kT/J)<1$ two phases appear, namely,
the ordered one (characterized by
$m_{\sigma} \neq 0$ and $m_{\tau} \neq 0$, with
$|m_{\sigma}| \geq |m_{\tau}|$),
as well as the partially-ordered phase
($m_{\sigma} \neq 0$ and $m_{\tau} = 0$).
Since the two Ising models are uncorrelated for $D=0$
and the random fields act only on the $\{\tau_{i}\}$
variables, one finds that the critical behavior associated
with variables $\{\sigma_{i}\}$ and $\{\tau_{i}\}$ occur
independently:
(i) The variables $\{\sigma_{i}\}$ order at $(kT/J)=1$, for
all values of $h_{0}$;
(ii) The critical frontier shown in
Fig.~\ref{fig:tdh00}(b), separating the two
low-temperature phases, is characteristic of an
Ising ferromagnet in the presence of a bimodal
random field~\cite{aharony}. The black circle
denotes a tricritical point, where the higher-temperature
continuous frontier meets the lower-temperature
first-order critical line. The type of phase
diagram exhibited in Fig.~\ref{fig:tdh00}(b)
will be referred herein as topology I.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=7.0cm,clip,angle=-90]{figure4a.eps}
\hspace{0.1cm}
\includegraphics[height=7.0cm,clip,angle=-90]{figure4b.eps} \\
\vspace{0.5cm} \hspace{-0.5cm}
\includegraphics[height=4.5cm,clip]{figure4c.eps}
\hspace{1.0cm}
\includegraphics[height=4.5cm,clip]{figure4d.eps}
\end{center}
\vspace{-.2cm}
\caption{Phase diagram and order parameters in the case
$(D/J)=0.1$.
(a) Phase diagram in the plane of dimensionless variables $kT/J$
versus $h_{0}/J$. At low temperatures, a first-order
critical frontier that terminates in an
ordered critical point (black asterisk) separates
the ordered phase (lower values of $h_{0}/J$) from
the partially-ordered phase (higher values of $h_{0}/J$);
this type of phase
diagram will be referred herein as topology II.
The order parameters $m_{\tau}$ and $m_{\sigma}$
are represented versus the dimensionless temperature
$kT/J$ for typical values of $h_{0}/J$:
(b) As one goes through the ordered
phase (low temperatures) to the {\bf P} phase;
(c) As one goes through the first-order critical
frontier, which separates the two ordered phases,
up to the {\bf P} phase;
(d) As one goes through the partially-ordered phase
(slightly to the right of the first-order critical frontier) up
to the {\bf P} phase. Equivalent solutions exist by
inverting the signs of $m_{\tau}$ and $m_{\sigma}$.}
\label{fig:d01}
\end{figure}

The effects of a small interaction [$(D/J)=0.1$]
between the variables $\{\sigma_{i}\}$ and
$\{\tau_{i}\}$ are presented in Fig.~\ref{fig:d01}, where
one sees that the topology I [Fig.~\ref{fig:tdh00}(b)] goes
through substantial changes, as shown
in Fig.~\ref{fig:d01}(a) (to be called herein as topology II).
As expected from the behavior presented
in Fig.~\ref{fig:tdh00}(a), one notices that
the border of the {\bf P} phase (a continuous frontier)
is shifted to higher temperatures.
However, the most significant difference between
topologies I and II consists in
the low-temperature frontier
separating the ordered and partially-ordered phases.
Particularly, the continuous frontier, as well
as the tricritical point shown
in Fig.~\ref{fig:tdh00}(b), give place to an
ordered critical point~\cite{griffiths}, at which
the low-temperature first-order critical
frontier terminates.
Such a topology has been found also in some
random magnetic systems, like the Ising and Blume-Capel
models, subject to random fields and/or
dilution~\cite{kaufman,salmon1,salmon2,benyoussef,
carneiro,kaufmankanner}.
In the present model, we verified that topology II holds
for any $0<(D/J)<1/2$, with
the first-order frontier starting at zero temperature and
$(h_{0}/J)=(D/J)+1/2$, which in Fig.~\ref{fig:d01}(a)
corresponds to $(h_{0}/J)=0.6$. Such a first-order line
essentially affects the parameter $m_{\tau}$, as will be
discussed next.

In Figs.~\ref{fig:d01}(b)--(d) the order parameters
$m_{\tau}$ and $m_{\sigma}$ are exhibited versus
$kT/J$ for conveniently chosen values of $h_{0}/J$,
corresponding to distinct physical situations of the
phase diagram for $(D/J)=0.1$.
A curious behavior is
presented by the magnetization
$m_{\tau}$ by varying $h_{0}/J$, and more
particularly, around the first-order critical line.
For $(h_{0}/J)=0.59$ [Fig.~\ref{fig:d01}(c)],
one starts at low temperatures
essentially to the left of the critical frontier and by increasing
$kT/J$ one crosses this critical frontier at $(kT/J)=0.499$,
very close to the ordered critical point.
At this crossing point,
$|m_{\tau}|$ presents an abrupt decrease, i.e.,
a discontinuity, corresponding
to a change to the partially-ordered phase; on
the other hand, the magnetization $m_{\sigma}$
remains unaffected when going through this critical frontier.
For higher temperatures,
$|m_{\tau}|$ becomes very small, but still finite,
turning up zero only at the {\bf P} boundary; in fact,
the whole region around the ordered critical point
is characterized by a finite small value of $|m_{\tau}|$.
Another unusual effect is presented in
Fig.~\ref{fig:d01}(d), for which $(h_{0}/J)=0.65$, i.e.,
slightly to the right of the first-order critical frontier:
the order parameter $m_{\tau}$ is zero
for low temperatures, but becomes nonzero by increasing the
temperature, as one becomes closer to the critical ordered
point. This rather curious phenomenon is directly related to
the correlation between the variables $\{\sigma_{i}\}$ and
$\{\tau_{i}\}$: since for $(kT/J) \approx 0.5$ the magnetization
$m_{\sigma}$ is still very close to its maximum value,
a small value for $|m_{\tau}|$ is induced, so that both
order parameters go to zero together only at the {\bf P} frontier.

Behind the results presented in Figs.~\ref{fig:d01}(a)--(d)
one finds a very interesting feature, namely, the
possibility of going continuously from the ordered phase to the
partially-ordered phase by circumventing the ordered critical point.
This is analogous to what happens in many substances, e.g., water,
where one goes continuously (with no latent heat)
from the liquid to the gas
phase by circumventing a critical end point~\cite{huang,reichl}.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=6.5cm,angle=-90]{figure5a.eps}
\hspace{0.2cm}
\includegraphics[height=6.5cm,angle=-90]{figure5b.eps}
\end{center}
\vspace{-.5cm}
\caption{The first-order critical line in Fig.~\ref{fig:d01}(a),
corresponding to $(D/J)=0.1$, is amplified, and
the dimensionless free-energy density $f/J$ of~\eq{eq:freeenergy}
(shown in the insets) is analyzed
at two distinct points along this frontier:
(a) A low-temperature point located at $[(h_{0}/J)=0.599,(kT/J)=0.010]$, showing the
coexistence of the ordered ($|m_{\tau}|=1$) and partially-ordered ($m_{\tau}=0$)
solutions;
(b) A higher-temperature point located at $[(h_{0}/J)=0.594,(kT/J)=0.387]$,
showing the coexistence of solutions with $|m_{\tau}|>0$, namely,
$|m_{\tau}|=0.868$ and $|m_{\tau}|=0.1$.
In both cases (a) and (b) the free energy presents four minima,
associated with distinct pairs of solutions
$(m_{\tau},m_{\sigma})$: the full lines show the two minima
with positive $m_{\sigma}$, whereas the dashed lines correspond
to the two minima with negative $m_{\sigma}$.}
\label{fig:freeenergyd01}
\end{figure}

In Fig.~\ref{fig:freeenergyd01} the free-energy density of~\eq{eq:freeenergy}
is analyzed at two different points along the first-order critical frontier of
Fig.~\ref{fig:d01}(a), namely, a low-temperature
one [Fig.~\ref{fig:freeenergyd01}(a)], and a point at a higher
temperature [Fig.~\ref{fig:freeenergyd01}(b)].
In both cases the free energy presents four minima
associated with distinct pairs of solutions
$(m_{\tau},m_{\sigma})$. The point at $(kT/J)=0.010$ presents
$(m_{\tau},m_{\sigma})=\{(-1,1);(0,1); (0,-1);(1,-1)\}$, whereas the
point at $(kT/J)=0.387$ presents
$(m_{\tau},m_{\sigma})=\{(-0.868, 0.991); (-0.100,0.991); (0.100, -0.991);
(0.868, -0.991)\}$.
The lower-temperature point represents a coexistence of the two phases
shown in the case $D=0$ [cf. Fig.~\ref{fig:tdh00}(b)], namely, the
ordered ($|m_{\tau}|=1$) and partially-ordered ($m_{\tau}=0$) phases.
However, the higher-temperature point typifies the phenomenon
discussed in Fig.~\ref{fig:d01}, where distinct solutions with
$|m_{\tau}|>0$ coexist, leading to a jump in this
order parameter as one crosses the critical frontier,
like illustrated in Fig.~\ref{fig:d01}(c) for the point
$[(h_{0}/J)=0.59,(kT/J)=0.499]$. Although the
magnetization $m_{\tau}$ presents a very
curious behavior in topology II [cf., e.g.,
Figs.~\ref{fig:d01}(b)--(d)],
$m_{\sigma}$ remains essentially
unchanged by the presence of the first-order
critical frontier of
Fig.~\ref{fig:d01}(a), as shown also in
Fig.~\ref{fig:freeenergyd01}.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=7cm,angle=-90]{figure6a.eps}
\hspace{0.2cm}
\includegraphics[height=7cm,angle=-90]{figure6b.eps}
\end{center}
\vspace{-.5cm}
\caption{Phase diagrams in the plane of dimensionless variables $kT/J$
versus $h_{0}/J$ for two different values of $D/J$:
(a) $(D/J)=0.5$, to be referred as topology III;
(b) $(D/J)=0.7$, to be referred as topology IV.}
\label{fig:phasediagd0507}
\end{figure}

In Fig.~\ref{fig:phasediagd0507} we present two other possible phase
diagrams, namely, the cases $(D/J)=0.5$ [Fig.~\ref{fig:phasediagd0507}(a),
called herein topology III] and
$(D/J)=0.7$ [Fig.~\ref{fig:phasediagd0507}(b), called herein topology IV].
Whereas topology III represents
a special situation that applies only for $(D/J)=0.5$, exhibiting the
richest critical behavior of the present model, topology IV holds
for any $(D/J)>0.5$.
In Fig.~\ref{fig:phasediagd0507}(a) one observes the appearance of
several multicritical points, denoted by the black circle (tricritical
point), black asterisk (ordered critical point), and
empty triangles (triple points):
(i) The tricritical point, which signals the
encounter of the higher-temperature continuous phase transition
with the lower-temperature first-order phase transition,
found in the $D=0$ phase diagram [cf. Fig.~\ref{fig:tdh00}(b)],
have curiously disappeared for $0<(D/J)<0.5$,
and emerged again for $(D/J)=0.5$;
(ii) The ordered critical point exists for any $0 < (D/J) \leq 0.5$
[as shown in Fig.~\ref{fig:d01}(a)];
(iii) Two triple points, one at a finite temperature, whereas the
other one occurs at zero temperature. It should be mentioned
that such a zero-temperature triple point corresponds
precisely to the one of Fig.~\ref{fig:groundstate}, at
$(D/J)=0.5$ and $(h_{0}/J)=1.0$.
The value $(D/J)=0.5$ is very special and will be considered as
a threshold for both multicritical behavior and correlations
between the two systems. We have observed that for
$(D/J) \gtrsim 0.5$, the critical points shown in
Fig.~\ref{fig:phasediagd0507}(a) disappear, except for the
tricritical point that survives for
$(D/J)>0.5$ [as shown in Fig.~\ref{fig:phasediagd0507}(b)].
Changes similar to those occurring
herein between topologies II and III, as well as
topologies III and IV,
were found also in some
magnetic systems, like the Ising and Blume-Capel
models, subject to random fields and/or
dilution~\cite{kaufman,salmon1,salmon2,benyoussef,
carneiro,kaufmankanner}.
Particularly, the splitting of the
low-temperature first-order critical frontier into
two higher-temperature first-order lines that terminate
in the ordered and tricritical points,
respectively [as exhibited in Fig.~\ref{fig:phasediagd0507}(a)],
is consistent with results found in
the Blume-Capel model under
a bimodal random magnetic, by
varying the intensity of the crystal
field~\cite{kaufmankanner}.

Another important feature of topology III concerns the
lack of any type of
magnetic order at finite temperatures for $(h_{0}/J)>1.1$,
in contrast to the phase diagrams for
$0 \leq (D/J) < 0.5$, for which there is $m_{\sigma} \neq 0$
for all $h_{0}/J$
[see, e.g., Figs.~\ref{fig:tdh00}(b) and~\ref{fig:d01}(a)].
This effect shows that $(D/J)=0.5$ represents a threshold value
for the coupling between the variables $\{\sigma_{i}\}$ and
$\{\tau_{i}\}$, so that for $(D/J) \geq 0.5$ the
correlations among these variables become significant.
As a consequence of these correlations, the fact
of no magnetic
order on the $\tau$-system ($m_{\tau} =0$)
drives the the magnetization of the
$\sigma$-system to zero as well, for $(h_{0}/J)>1.1$.
It is important to notice that the $T=0$ phase diagram
of Fig.~\ref{fig:groundstate}
presents a first-order critical line for $(D/J)=0.5$ and
$(h_{0}/J)>1.0$, at which
$m_{\tau} =0$, whereas in the $\sigma$-system both
$m_{\sigma}=0$ and  $|m_{\sigma}|=1$ minimize the Hamiltonian.
By analyzing numerically the free-energy density
of~\eq{eq:freeenergy} at low temperatures and $(h_{0}/J)>1.0$,
we have verified that for any infinitesimal value of
$kT/J$ destroys such a coexistence of solutions, leading to
a minimum free energy at
$m_{\tau}=m_{\sigma}=0 \ (\forall \, T>0)$. Consequently,
one finds that the low-temperature region in the interval
$1.0 \leq (h_{0}/J) \leq 1.1$ becomes part of the {\bf P} phase.
Hence, the phase diagram in
Fig.~\ref{fig:phasediagd0507}(a) presents
a reentrance phenomena for
$1.0 \leq (h_{0}/J) \leq 1.1$. In this region, by lowering
the temperature gradually, one goes from a {\bf P} phase
to the ordered phase
($m_{\tau} \neq 0$ ; $m_{\sigma} \neq 0$), and then back
to the {\bf P} phase. This effect appears frequently
in both theoretical and experimental investigations of
disordered magnets~\cite{dotsenkobook,nishimoribook}.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=7cm,clip,angle=-90]{figure7a.eps}
\hspace{0.5cm} \vspace{0.7cm}
\includegraphics[height=7cm,clip,angle=-90]{figure7b.eps} \\
\vspace{0cm} \hspace{-0.8cm}
\includegraphics[height=4.5cm,clip]{figure7c.eps}
\hspace{1.2cm}
\includegraphics[height=4.5cm,clip]{figure7d.eps}
\end{center}
\vspace{0.2cm}
\caption{(a) The region of multicritical points of the phase diagram for
$(D/J)=0.5$ [Fig.~\ref{fig:phasediagd0507}(a)] is amplified and three
thermodynamic paths are chosen for analyzing the magnetizations
$m_{\tau}$ and $m_{\sigma}$.
(b) Order parameters along thermodynamic path (1):
$(h_{0}/J)=0.97$ and increasing temperatures.
(c) Order parameters along thermodynamic path (2):
$(h_{0}/J)=1.05$ and increasing temperatures.
(d) Order parameters along thermodynamic path (3):
$(kT/J)=0.42$ and varying the field
strength in the interval $0.9 \leq (h_{0}/J) \leq 1.15$.
Equivalent solutions exist by inverting the signs of
$m_{\tau}$ and $m_{\sigma}$.}
\label{fig:magpaths123}
\end{figure}

In Fig.~\ref{fig:magpaths123} we analyze the behavior of the
$m_{\tau}$ and $m_{\sigma}$ for topology III,
in the region of multicritical
points of the phase diagram for
$(D/J)=0.5$, along three typical thermodynamic paths, as
shown in Fig.~\ref{fig:magpaths123}(a).
In Fig.~\ref{fig:magpaths123}(b) we exhibit the behavior of
$m_{\tau}$ and $m_{\sigma}$ along path (1), where one
sees that both parameters go through a jump by
crossing the first-order critical line [$(kT/J)=0.445$],
expressing a coexistence of different types
of solutions for $m_{\tau}$ and $m_{\sigma}$ at this
point. One notices a larger jump in $m_{\tau}$, so that
to the right of the ordered critical point
one finds a behavior similar to the one verified in topology II,
where $|m_{\tau}|$ becomes very small, whereas
$m_{\sigma}$ still presents significant values.
Then, by further increasing the temperature, these parameters
tend smoothly to zero at the continuous critical frontier
separating the ordered and {\bf P} phases.
In Fig.~\ref{fig:magpaths123}(c) we show the magnetizations
$m_{\tau}$ and $m_{\sigma}$ along path (2),
within the region of the phase diagram
where the reentrance phenomenon occurs; along this path,
one increases the temperature, going from the {\bf P} phase
to the ordered phase and then to the {\bf P} phase again.
Both parameters are zero for low enough temperatures,
jumping to nonzero values at $(kT/J)=0.396$, as one
crosses the first-order critical line. After such jumps,
by increasing the temperature, these parameters
tend smoothly to zero at the border of the
{\bf P} phase. The behavior shown in
Fig.~\ref{fig:magpaths123}(c) confirm the reentrance
effect, discussed previously.
Finally, in Fig.~\ref{fig:magpaths123}(d) we exhibit
the order parameters along thermodynamic path (3),
for which the temperature is fixed at $(kT/J)=0.42$, with
the field varying in the range
$0.9 \leq (h_{0}/J) \leq 1.15$. One sees that both
magnetizations $m_{\tau}$ and $m_{\sigma}$ display
jumps as one crosses each of the two first-order lines,
evidencing a
coexistence of different ordered states at the lower-temperature
jump, as well as a coexistence of the ordered and {\bf P} states
at the higher-temperature jump.

The behavior presented by the order parameters in
Figs.~\ref{fig:magpaths123}(b)--(d) shows clearly
the fact that $(D/J)=0.5$ represents a threshold value
for the coupling between the variables $\{\sigma_{i}\}$ and
$\{\tau_{i}\}$. In all these cases, one sees that jumps
in the magnetization $m_{\sigma}$ are correlated with
corresponding jumps in $m_{\tau}$.
These results should be contrasted with those for the
cases $(D/J)<0.5$, as illustrated
in Fig.~\ref{fig:d01}(c), where a discontinuity
in $m_{\tau}$ does not affect the smooth behavior presented
by $m_{\sigma}$.

\begin{figure}[htp]
\begin{center}
\includegraphics[height=7cm,angle=-90]{figure8a.eps}
\hspace{0.2cm}
\includegraphics[height=7cm,angle=-90]{figure8b.eps}
\end{center}
\vspace{-.5cm}
\caption{The order parameters $m_{\tau}$ and $m_{\sigma}$
are represented versus the dimensionless temperature
$kT/J$ for $(D/J)=8.0$ and two typical values of $h_{0}/J$:
(a) Slightly to the left of the tricritical point;
(b) Slightly to the right of the tricritical point.
The associated phase diagram corresponds
to topology IV [cf. Fig.~\ref{fig:phasediagd0507}(b)].
Equivalent solutions exist by inverting the signs of
$m_{\tau}$ and $m_{\sigma}$.}
\label{fig:magd80}
\end{figure}

The phase diagram shown in Fig.~\ref{fig:phasediagd0507}(b),
which corresponds to topology IV, is valid for any
for any $(D/J)>0.5$. Particularly,
the critical point where the low-temperature
first-order critical
frontier touches the zero-temperature axis
is kept at $(h_{0}/J)=1$, for all $(D/J)>0.5$,
in agreement with Fig.~\ref{fig:groundstate}.
We have verified
only quantitative changes in such a phase diagram
by increasing
the coupling between the variables $\{\sigma_{i}\}$ and
$\{\tau_{i}\}$. Essentially, the whole continuous critical
frontier moves towards
higher temperatures, leading to an increase in the values
of the critical temperature
for $(h_{0}/J)=0$, as well as in the temperature
associated with the tricritical point, whereas the abscissa
of this point remains typically unchanged. Moreover,
in what concerns the order parameters,
the difference between $|m_{\tau}|$
and $m_{\sigma}$ decreases, in such a way
that for $(D/J) \rightarrow \infty$, one obtains
$m_{\tau}=-m_{\sigma}$.
This later effect is illustrated in
Fig.~\ref{fig:magd80}, where we represent the
order parameters $m_{\tau}$ and $m_{\sigma}$
versus temperature, for a sufficiently large value of
$D/J$, namely, $(D/J)=8.0$, in
two typical choices of $h_{0}/J$, close
to the tricritical point.
In Fig.~\ref{fig:magd80}(a) $m_{\tau}$ and $m_{\sigma}$
are analyzed slightly to the left of the tricritical
point, exhibiting the usual continuous behavior,
whereas in Fig.~\ref{fig:magd80}(b) they
are considered slightly to the right of the tricritical
point, presenting jumps as one crosses the
first-order critical frontier. However,
the most important conclusion from Fig.~\ref{fig:magd80}
concerns the fact that in both cases one has essentially
$m_{\tau}=-m_{\sigma}$, showing that  the random
field applied solely to the $\tau$-system influences the
$\sigma$-system in a similar way, due to the
high value of $D/J$ considered.
We have verified that for $(D/J)=8.0$
the two systems become so strongly
correlated, such that
$m_{\tau}=-m_{\sigma}$ holds along
the whole phase diagram,
within our numerical accuracy.

\section{Conclusions}

We have analyzed the effects of a coupling $D$
between two Ising models, defined in terms of variables
$\{\tau_{i}\}$ and $\{\sigma_{i}\}$.
The model was considered in the limit of infinite-range
interactions, where all spins in each system
interact by means of an exchange coupling $J>0$, typical
of ferromagnetic interactions.
Motivated by a qualitative description of
systems like plastic crystals,
the variables $\{\tau_{i}\}$ and $\{\sigma_{i}\}$ would
represent rotational and translational degrees
of freedom, respectively. Since the rotational
degrees of freedom are expected to change more
freely than the translational ones,
a random field acting only on the variables
$\{\tau_{i}\}$ was considered.
For this purpose, a bimodal random field,
$h_{i} = \pm h_{0}$, with equal probabilities,
was defined on the $\tau$-system.
The model was investigated through its free energy
and its two order parameters, namely,
$m_{\tau}$ and $m_{\sigma}$.

We have shown that such a system presents a very rich
critical behavior, depending on the particular choices
of $D/J$ and $h_{0}/J$.
Particularly, at zero temperature, the phase diagram in the plane
$h_{0}/J$ versus $D/J$ exhibits ordered, partially-ordered,
and disordered phases. This phase diagram is symmetric
around $(D/J)=0$, so that for sufficiently low values of
$h_{0}/J$ one finds ordered phases characterized by
$m_{\sigma}=m_{\tau}=\pm 1$ ($D<0$) and
$m_{\sigma}=-m_{\tau}=\pm 1$ ($D>0$).
We have verified that $|D/J|=1/2$
plays an important role in the present model, such
that at zero temperature one has the disordered
phase ($m_{\sigma}=m_{\tau}=0$)
for  $|D/J|>1/2$ and $(h_{0}/J)>1$.
Moreover, the partially-ordered phase,
where $m_{\sigma}=\pm 1$ and $m_{\tau}=0$,
occurs for $(h_{0}/J)>1/2+|D/J|$ and $|D/J|<1/2$.
In this phase diagram all phase transitions are
of the first-order type, and three triple points were found.
In the case of plastic crystals,
the sequence of transitions from the disordered
to the partially-ordered, and then to the
ordered phases, would correspond to the
sequence of transitions from the liquid to
the plastic crystal, and then to ordered crystal
phases.

Due to the symmetry around $D=0$, the
finite-temperature phase diagrams were considered
only for $D>0$, for which the ordered
phase was identified by $m_{\sigma}>0$ and
$m_{\tau}<0$, whereas the partially-ordered phase
by $m_{\sigma}>0$ and
$m_{\tau}=0$ (equivalent solutions also exist by
inverting the signs of these order parameters).
Several phase diagrams in the
plane $kT/J$ versus $h_{0}/J$ were studied,
by varying gradually $D/J$. We have found
four qualitatively different types of phase diagrams,
denominated as topologies I [$(D/J)=0$], II [$0<(D/J)<1/2$],
III [$(D/J)=1/2$], and IV [$(D/J)>1/2$]. Such a
classification reflects the fact that $(D/J)=1/2$
represents a threshold value
for the coupling between the variables $\{\sigma_{i}\}$ and
$\{\tau_{i}\}$, so that for $(D/J) \geq 1/2$ the
correlations among these variables become significant,
as verified through the behavior of the order parameters
$m_{\tau}$ and $m_{\sigma}$.
From all these cases, only topology IV
typifies a well-known phase diagram,
characterized by a tricritical point, where the
higher-temperature continuous frontier meets
the lower-temperature first-order critical line.
This phase diagram is qualitatively similar to
the one found for the
Ising ferromagnet in the presence of a bimodal
random field~\cite{aharony}, and it does not
present the herein physically relevant
partially-ordered phase.
For $(D/J) \geq 1/2$, even though the random field
is applied only in the $\tau$-system, the correlations
lead the $\sigma$-system to follow a qualitatively
similar behavior.

The phase diagrams referred as topologies I and II
exhibit all three phases. In the later case we have found
a first-order critical line terminating at an ordered
critical point, leading to the potential physical realization
of going continuously from the ordered phase to the
partially-ordered phase by circumventing this
critical point.
In these two topologies, the sequence of transitions
from the disordered
to the partially-ordered, and then to the
ordered phase, represents the physical
situation that occurs in plastic crystals.
For conveniently chosen thermodynamic paths,
i.e., varying temperature and random field appropriately,
one may go from the liquid phase
($m_{\sigma}=m_{\tau}=0$), to a plastic-crystal phase
($m_{\sigma} \neq 0$; $m_{\tau}=0$), where the rotational degrees
of freedom are found in a disordered state, and then,
to an ordered crystal phase
($m_{\sigma} \neq 0$; $m_{\tau} \neq 0$).

 From the point of view of multicritical behavior,
topology III [$(D/J)=1/2$] corresponds to
the richest type of phase diagram, being
characterized by several critical lines and
multicritical points; one finds its most
complex criticality around $(h_{0}/J)=1$, signaling
a great competition among the different types of orderings.
Although the partially-ordered phase
does not appear in this particular case, one has also
the possibility of circumventing the ordered critical point,
such as to reach a region of the phase diagram
along which $|m_{\tau}|$ becomes very small,
resembling a partially-ordered phase.

Since the infinite-range interactions among
variables of each Ising system correspond to a limit
where mean-field approach becomes exact, an immediate
question concerns whether some of the results obtained above
represent an artifact of such limit.
Certainly, such a relevant point is directly related with the
existence of some of these features in the associated
short-range three-dimensional magnetic models. For example, the
tricritical point found in topologies III and IV is essentially
the same that appears within the mean-field approach of the
Ising model in the presence of a bimodal random field.
This later model has been extensively investigated on a cubic
lattice through different numerical approaches, where the
existence of this tricritical point is still very controversial.
On the other hand, a first-order critical frontier terminating
at an ordered critical point, and the fact that one can
go from one phase to another by
circumventing this point, represents a typical
physical situation that occurs in real
substances. The potential for exhibiting such a
relevant feature represents an important advantage
of the present model.

Finally, we emphasize that the rich critical behavior presented
in the phase diagrams corresponding to topologies II and III
suggest the range $0<(D/J) \leq 1/2$ as appropriate
for describing plastic crystals.
The potential of exhibiting successive transitions from the
ordered to the partially-ordered and then to the
disordered phase should be useful for a better
understanding of these systems.
Furthermore, the characteristic
of going continuously from the ordered phase
to the partially-ordered phase by circumventing an ordered
critical point represents a typical physical situation that
occurs in many substances,
and opens the possibility for
the present model to describe a wider range of materials.

\vskip 2\baselineskip

{\large\bf Acknowledgments}

\vskip \baselineskip
\noindent
 The partial financial supports from CNPq,
FAPEAM-Projeto-Universal-Amazonas,
and FAPERJ (Brazilian agencies) are acknowledged.

\vskip 2\baselineskip


\section{Introduction}
\label{intro}
It is now a well-established fact that according to our present theory of gravity, 85\%~of the matter content of our universe is missing. Observational evidence for this discrepancy ranges from macroscopic to microscopic scales, e.g. gravitational lensing in galaxy clusters, galactic rotation curves and fluctuations measured in the Cosmic Microwave Background. This has resulted in the hypothesised existence of a new type of matter called Dark Matter. Particle physics provides a well-motivated explanation for this hypothesis: The existence of (until now unobserved) massive weakly interacting particles (WIMPs). A favorite amongst the several WIMP candidates is the neutralino, the lightest particle predicted by Supersymmetry, itself a well-motivated extension of the Standard Model.

If Supersymmetry is indeed realised in Nature, Supersymmetric particles would have been copiously produced at the start of our Universe in the Big Bang. Initially these particles would have been in thermal equilibrium. After the temperature of the Universe dropped below the neutralino mass, the neutralino number density would have decreased exponentially. Eventually the expansion rate of the Universe would have overcome the neutralino annihilation rate, resulting in a neutralino density in our Universe today similar to the cosmic microwave background.

These relic neutralinos could then accumulate in massive celestial bodies in our Universe like our Sun through weak interactions with normal matter and gravity. Over time the neutralino density in the core of the object would increase considerably, thereby increasing the local neutralino annihilation probability. In the annihilation process new particles would be created, amongst which neutrinos. This neutrino flux could be detectable as a localised emission with earth-based neutrino telescopes like ANTARES.

This paper gives a brief overview of the prospects for the detection of neutrinos originating from neutralino annihilation in the Sun with the ANTARES neutrino telescope.

\begin{figure}[b]
\center{
\includegraphics[width=0.45\textwidth,angle=0]{NEA_60kHz0XOFF_off.png}
\caption{The ANTARES Neutrino Effective Area vs. $E_\nu$.}
\label{fig:1}
}
\end{figure}

\begin{figure*}[t]
\center{
\includegraphics[width=0.8\textwidth,angle=0]{psflux.png}
\caption{Predicted $\nu_\mu+\bar{\nu}_\mu$ flux from the Sun in mSUGRA parameter space.}
\label{fig:2}
}
\end{figure*}

\section{The ANTARES neutrino telescope}
\label{sec:1}
The ANTARES undersea neutrino telescope consists of a 3D~grid of 900~photomultiplier tubes arranged in 12~strings, at a depth of 2475~m in the Mediterranean Sea. Three quarters of the telescope have been deployed and half of the detector is already fully operational, making ANTARES the largest neutrino telescope on the northern hemisphere. The angular resolution of the telescope is of the order of one degree at low energy, relevant to dark matter searches, and reaches 0.3 degree at high energies ($>$~10~TeV).

The sensitivity of a neutrino detector is conventionally expressed as the Neutrino Effective Area, $A_{\rm eff}^{\nu}$. The $A_{\rm eff}^{\nu}$ is a function of neutrino energy $E_\nu$ and direction $\Omega$, and is defined as
\begin{equation}
A_{\rm eff}^{\nu}(E_\nu,\Omega) \;=\; V_{\rm eff}(E_\nu,\Omega)\;\sigma(E_\nu)\;\rho N_A\;P_E(E_\nu,\Omega)
\label{eq:1}
\end{equation}

\noindent where $\sigma(E_\nu)$ is the neutrino interaction cross section, $\rho\,N_A$ is the nucleon density in/near the detector,\linebreak $P_E(E_\nu,\Omega)$ is the neutrino transmission probability\linebreak through the Earth and $V_{\rm eff}(E_\nu,\Omega)$ represents the effective detector volume. This last quantity depends not only on the detector geometry and instrumentation, but is also on the efficiency of the trigger and reconstruction algorithms that are used.

The ANTARES $A_{\rm eff}^{\nu}$ for upgoing $\nu_\mu$ and $\bar{\nu}_\mu$'s, integrated over all directions as a function of the neutrino energy is shown in Fig.~\ref{fig:1}. The curves represent the $A_{\rm eff}^{\nu}$ after triggering only (``{\em Trigger level}'', in blue) and after reconstruction and selection (``{\em Detection level}'', in red). The increase of the $A_{\rm eff}^{\nu}$ with neutrino energy is mainly due to the fact that $\sigma(E_\nu)$ as well as the muon range are both proportional to the neutrino energy.

The detection rate $R(t)$ for a certain neutrino flux $\Phi(E_\nu,\Omega,t)$ is then defined as

\begin{equation}
R(t) \;=\; \iint\,A_{\rm eff}^{\nu}(E_\nu,\Omega)\;\frac{d\Phi(E_\nu,\Omega,t)}{dE_\nu\,d\Omega}\;dE_\nu\,d\Omega
\label{eq:2}
\end{equation}

\section{Neutralino annihilation in the Sun}
\label{sec:2}
We calculated the $\nu_\mu+\bar{\nu}_\mu$ flux resulting from neutralino annihilation in the centre of the Sun using the DarkSUSY simulation package \cite{DarkSUSY}. Furthermore, the effects of neutrino oscillations in matter and vacuum as well as absorption were taken into account. The top quark mass was set to 172.5~GeV and the NFW-model for the Dark Matter halo with a local halo density \mbox{$\rho_0 = 0.3$~GeV/cm$^3$} was used. Instead of the general Supersymmetric framework, we used the more constrained approach of minimal Supergravity (mSUGRA). In mSUGRA, models are characterized by four parameters and a sign: A common gaugino mass $m_{1/2}$, scalar mass $m_0$ and tri-linear scalar coupling $A_0$ at the GUT scale ($10^{16}$ GeV), the ratio of vacuum expectation values of the two Higgs fields $tan(\beta)$ and the sign of the Higgsino mixing parameter $\mu$. We considered only $\mu=+1$ models within the following parameter ranges: \mbox{$0<m_0<8000$~GeV,} \mbox{$0<m_{1/2}<2000$~GeV,}\linebreak \mbox{$-3m_0<A_0<3m_0$} and \mbox{$0<tan(\beta)<60$.} The SUSY parameters were subsequently calculated using the\linebreak ISASUGRA package \cite{Isasugra}.

\begin{figure}[b]
  \includegraphics[width=0.45\textwidth,angle=0]{neutrino_flux_relic_density.png}
\caption{Predicted $\nu_\mu+\bar{\nu}_\mu$ flux from the Sun vs. $m_\chi$.}
\label{fig:3}
\end{figure}

\begin{figure*}[t]
\center{
\includegraphics[width=0.8\textwidth,angle=0]{psexcl.png}
\caption{mSUGRA models 90\% CL excludable by ANTARES in mSUGRA parameter space.}
\label{fig:4}
}
\end{figure*}

\pagebreak Only a small subset of all mSUGRA models possess a relic neutralino density $\Omega_\chi h^2$ that is compatible with the Cold Dark Matter energy density $\Omega_{\rm CDM} h^2$ as measured by WMAP \cite{WMAP}. To investigate specifically those mSUGRA models, we sampled the mSUGRA parameter space using a random walk method based on the Metropolis algorithm where $\Omega_\chi h^2$ acted as a guidance parameter \cite{MarkovChain}.

The resulting $\nu_\mu+\bar{\nu}_\mu$ flux from the Sun per~km$^{2}$ per year, integrated above \mbox{$E_\nu=10$~GeV}, can be seen in the \mbox{$m_0$-$m_{1/2}$~plane} for different ranges of $tan(\beta)$ in Fig.~\ref{fig:2}. The white regions correspond to mSUGRA models without radiative electroweak symmetry breaking, models with $\Omega_\chi h^2>1$, models that are already experimentally excluded, or models where the neutralino is not the lightest superpartner. Models in the so-called ``Focus Point'' region\,\footnote{The region of mSUGRA parameter space around $(m_0,m_{1/2}) = (2000,400)$.} produce the highest neutrino flux: In this region the neutralino has a relatively large Higgsino component \cite{Nerzi}. This enhances the neutralino capture rate through $Z$-boson exchange as well as the neutralino annihilation through the\linebreak \mbox{$\chi\chi\rightarrow WW/ZZ$} channel, resulting in a large flux of relatively high energetic neutrinos.

The $\nu_\mu+\bar{\nu}_\mu$ flux can also be plotted against the neutralino mass $m_\chi$, as is shown in Fig.~\ref{fig:3}. In this plot, the mSUGRA models are subdivided into three categories according to how well their $\Omega_\chi h^2$ agrees with $\Omega_{\rm CDM} h^2$ as measured by WMAP\,\footnote{WMAP: $\Omega_{\rm CDM} h^2 = 0.1126_{-0.013}^{+0.008}$}: \mbox{$\Omega_\chi h^2-\Omega_{\rm CDM}h^2<2\sigma$} (black), \mbox{$0< \Omega_\chi h^2 < \Omega_{\rm CDM} h^2$} (blue) and \mbox{$\Omega_{\rm CDM} h^2 < \Omega_\chi h^2 < 1$} (magenta).

\begin{figure}[b]
  \includegraphics[width=0.45\textwidth,angle=0]{detection_rate_relic_density.png}
\caption{ANTARES detection rate per 3~years vs. $m_\chi$.}
\label{fig:5}
\end{figure}

\section{ANTARES prospects to detect neutralino annihilation in the Sun}
\label{sec:3}
The ANTARES detection rate (See Eq.~\ref{eq:2}) for the detection of neutralino annihilation in the Sun was calculated as follows: For each mSUGRA model considered in Sect.~\ref{sec:2}, the differential $\nu_\mu+\bar{\nu}_\mu$ flux from the Sun was convoluted with the Sun's zenith angle distribution as well as the ANTARES $A_{\rm eff}^{\nu}$ (see Eq.~\ref{eq:1} and Fig.~\ref{fig:1}). The resulting detection rate in ANTARES per 3~years is shown as a function of the neutralino mass in Fig.~\ref{fig:5}. The color coding in the plot corresponds to the one used in Fig.~\ref{fig:3}.

The ANTARES exclusion limit for the detection of neutralino annihilation in the Sun was calculated as follows: As can be seen from Fig.~\ref{fig:5}, the expected detection rates for all mSUGRA model considered in Sect.~\ref{sec:2} are small. Therefore the Feldman Cousins approach \cite{FeldmanCousins} was used to calculate 90\%~CL exclusion limits. The two sources of background were taken into account as follows: Since we know the Sun's position in the sky, the atmospheric neutrino background (Volkova parametrisation) was considered only in a 3~degree radius search cone around the Sun's position. After applying the event selection criteria used to determine the $A_{\rm eff}^{\nu}$ in Fig.~\ref{fig:1}, the misreconstructed atmospheric muon background was considered to be 10\% of the atmospheric neutrino background. mSUGRA models that are excludable at 90\%~CL by ANTARES in 3~years are shown in blue in Fig.~\ref{fig:6}, those that are non-excludable are shown in red. Bright colors indicate models which have \mbox{$\Omega_\chi h^2-\Omega_{\rm CDM}h^2<2\sigma$}. The fraction of ANTARES excludable models in mSUGRA parameter space is shown in Fig.~\ref{fig:4}.

\begin{figure}[t]
  \includegraphics[width=0.45\textwidth,angle=0]{detection_rate_exclusion.png}
\caption{mSUGRA models 90\% CL excludable by ANTARES per 3~years vs. $m_\chi$.}
\label{fig:6}
\end{figure}

\begin{figure}[b]
  \includegraphics[width=0.45\textwidth,angle=0]{crossection_exclusion_direct_limits.png}
\caption{Spin-independent $\chi p$~cross section vs. $m_\chi$.}
\label{fig:7}
\end{figure}

\begin{figure}[t]
  \includegraphics[width=0.44\textwidth,angle=0]{NEA_triggercomparison_off.png}
\caption{The ANTARES Neutrino Effective Area at the trigger level vs. $E_\nu$.}
\label{fig:8}
\end{figure}

\section{Comparison to direct detection}
To compare with direct detection experiments, the spin-independent $\chi p$~cross section versus the neutralino mass for all mSUGRA models considered in Sect.~\ref{sec:2} is shown in Fig.~\ref{fig:7}. The color coding in the plot corresponds to the one used in Fig.~\ref{fig:6}. The limits in this plot were taken from the Dark Matter Limit Plot Generator \cite{DirectDetection}. The spin-independent cross section is driven by CP-even Higgs boson exchange \cite{Nerzi}. Therefore, mSUGRA models in which the neutralino is of the mixed gaugino-Higgsino type will produce the largest cross sections. This implies a correlation between the models that are excludable by direct detection experiments and models excludable ANTARES, as can be seen from Fig.~\ref{fig:7}.

\section{Conclusion \& Outlook}
Nearly half of the ANTARES detector has been operational since January this year. The detector is foreseen to be completed in early 2008. The data show that the detector is working within the design specifications.

As can be seen from Fig.~\ref{fig:4}, mSUGRA models that are excludable by ANTARES at 90\%~CL are found in the Focus Point region. The same models should also be excludable by future direct detection experiments, as is shown in Fig.~\ref{fig:7}.

To improve the ANTARES sensitivity, a directional trigger algorithm has recently been implemented in the data acquisition system. In this algorithm, the known position of the potential neutrino source is used to lower the trigger condition. This increases the trigger efficiency, resulting in a larger $A_{\rm eff}^{\nu}$. In Fig.~\ref{fig:8}, the $A_{\rm eff}^{\nu}$ at the trigger level for the standard- and the directional trigger algorithm are shown in black (``{\em trigger3D}'') and red (``{\em triggerMX}'') respectively.


\section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}. The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection. In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}. In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}. The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage. The
magnitude of all weight elements in all stages are equal to unity.
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC.
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity. In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels. However in
practice, these phases are not known and should be estimated. In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights. The partial information is the
quarter of each channel phase in $(0,2\pi)$.

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced. In section \ref{S5} some simulation
examples illustrate the results of the proposed method. Finally the
paper is concluded in section \ref{S6}.

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$. The $m^{th}$ user
has its own code $p_m(.)$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$. It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$. At the
receiver we assume that perfect power control scheme is applied.
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission). According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$. Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user. It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$. Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector. The output of the last
stage is considered as the final estimate of transmitted bits. In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases.

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$. Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m}.
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n).
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T.
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition. From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n).
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$. To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0.
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$. In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm). In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$. In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$. At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$.

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$. The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$. Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$. From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}).
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))}.
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right.
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$. If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected. In this case only one
of the three members of $P^s$ has the same region as $\phi_m$. For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$. If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation. In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi.\]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges. What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$. It means that
$w^s_m(N)$ has not converged yet. In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e.g. if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$.
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4.\\
\end{array}\right.
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n).
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\}.
\end{equation}
Assuming $\phi_m\in R_i$, then
\begin{equation}
\label{qqpp} \hat{\phi}^0_m =\frac{(i-1)\pi+i\pi}{4}.
\end{equation}
Table \ref{tab4} shows the structure of the modified PLMS-PPIC
method. It is to be notified that
\begin{itemize}
\item Equation (\ref{qte5}) shows the conventional bit detection
method when the receiver only knows the quarter of channel phase in
$(0,2\pi)$. \item With $L=1$ (i.e. only one NLMS algorithm), the
modified PLMS-PPIC can be thought as a modified version of the
LMS-PPIC method.
\end{itemize}

In the following section some examples are given to illustrate the
effectiveness of the proposed method.

\section{Simulations}\label{S5}

In this section we have considered some simulation examples.
Examples \ref{ex2}-\ref{ex4} compare the conventional, the modified
LMS-PPIC and the modified PLMS-PPIC methods in three cases: balanced
channels, unbalanced channels and time varying channels. In all
examples, the receivers have only the quarter of each channel phase.
Example \ref{ex2} is given to compare the modified LMS-PPIC and the
PLMS-PPIC in the case of balanced channels.

\begin{example}{\it Balanced channels}:
\label{ex2}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex2})} \label{tabex5} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s = 2 &  $\hat{\phi}^s_m=\frac{3.24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3.18\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{3.24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3.18\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s = 2 &  $\hat{\phi}^s_m=\frac{2.85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.88\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{2.85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.88\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider the system model (\ref{e7}) in which $M$ users
synchronously send their bits to the receiver through their
channels. It is assumed that each user's information consists of
codes of length $N$. It is also assumd that the signal to noise
ratio (SNR) is 0dB. In this example there is no power-unbalanced or
channel loss is assumed. The step-size of the NLMS algorithm in
modified LMS-PPIC method is $\mu=0.1(1-\sqrt{\frac{M-1}{M}})$ and
the set of step-sizes of the parallel NLMS algorithms in modified
PLMS-PPIC method are
$\Theta=\{0.01,0.05,0.1,0.2,\cdots,1\}(1-\sqrt{\frac{M-1}{M}})$,
i.e. $\mu_1=0.01(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_4=0.2(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_{12}=(1-\sqrt{\frac{M-1}{M}})$. Figure~\ref{Figexp1NonCoh}
illustrates the bit error rate (BER) for the case of two stages and
for $N=64$ and $N=256$. Simulations also show that there is no
remarkable difference between results in two stage and three stage
scenarios. Table~\ref{tabex5} compares the average channel phase
estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and PLMS-PPIC, when the the number of users is
$M=15$.
\end{example}

Although LMS-PPIC and PLMS-PPIC, as well as their modified versions,
are structured based on the assumption of no near-far problem
(examples \ref{ex3} and \ref{ex4}), these methods and especially the
second one have remarkable performance in the cases of unbalanced
and/or time varying channels.

\begin{example}{\it Unbalanced channels}:
\label{ex3}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex3})} \label{tabex6} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s=2 &  $\hat{\phi}^s_m=\frac{2.45\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.36\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2.71\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.80\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s=2 &  $\hat{\phi}^s_m=\frac{3.09\pi}{8}$ & $\hat{\phi}^s_m=\frac{2.86\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2.93\pi}{8}$ & $\hat{\phi}^s_m=\frac{3.01\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider example \ref{ex2} with power unbalanced and/or channel loss
in transmission system, i.e. the true model at stage $s$ is
\begin{equation}
\label{ve7} r(n)=\sum\limits_{m=1}^{M}\beta_m
w^s_m\alpha^{(s-1)}_m c_m(n)+v(n),
\end{equation}
where $0<\beta_m\leq 1$ for all $1\leq m \leq M$. Both the LMS-PPIC
and the PLMS-PPIC methods assume the model (\ref{e7}), and their
estimations are based on observations $\{r(n),X^s(n)\}$, instead of
$\{r(n),\mathbf{G}X^s(n)\}$, where the channel gain matrix is
$\mathbf{G}=\mbox{diag}(\beta_1,\beta_2,\cdots,\beta_m)$. In this
case we repeat example \ref{ex2}. We randomly get each element of
$G$ from $[0,0.3]$. Figure~\ref{Figexp2NonCoh} illustrates the BER
versus the number of users. Table~\ref{tabex6} compares the channel
phase estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and modified PLMS-PPIC for $M=15$.
\end{example}

\begin{example}
\label{ex4} {\it Time varying channels}: Consider example \ref{ex2}
with time varying Rayleigh fading channels. In this case we assume
the maximum Doppler shift of $40$HZ, the three-tap
frequency-selective channel with delay vector of $\{2\times
10^{-6},2.5\times 10^{-6},3\times 10^{-6}\}$sec and gain vector of
$\{-5,-3,-10\}$dB. Figure~\ref{Figexp3NonCoh} shows the average BER
over all users versus $M$ and using two stages.
\end{example}

\section{Conclusion}\label{S6}

In this paper, parallel interference cancelation using adaptive
multistage structure and employing a set of NLMS algorithms with
different step-sizes is proposed, when just the quarter of the
channel phase of each user is known. In fact, the algorithm has been
proposed for coherent transmission with full information on channel
phases in \cite{cohpaper}. This paper is a modification on the
previously proposed algorithm. Simulation results show that the new
method has a remarkable performance for different scenarios
including Rayleigh fading channels even if the channel is
unbalanced.