text:
\section{}

\section{Introduction and Theoretical Techniques}

Crystals are regular periodic physical systems with the atomic
constituents organized in periodic arrays, where the periodicity is a characteristic
of the crystal in its equilibrium configuration and a manifestation of long-range
positional order.  However, the effect of thermal fluctuations must be taken into
consideration at finite temperatures where thermally excited lattice vibrations
may degrade or destroy crystalline order.  An early theoretical treatment developed by
Lindemann~\cite{uno} examined the effect of lattice vibrations in a framework which
neglected the correlations of atomic motions, but which nonetheless provides a reasonable
description (on an order of magnitude basis) for the melting of three dimensional
solids.

A salient component of the Lindemann analysis is the Lindemann criterion
where the melting of a solid is considered to have taken place when mean square deviations
from equilibrium exceed a tenth of a lattice constant.  X-ray diffraction data, which may provide a
direct measure of long-range order in a crystal lattice (and hence a means to determine
temperatures where crystalline order is lost) finds reasonable agreement~\cite{unopuntocinco,unopuntosiete} with the Lindemann
criterion. The accord is manifest in the finding
that Bragg peaks corresponding to broken translational symmetry vanish
when mean square fluctuations from equilibrium (also determined from an analysis of
X-ray diffraction data) are in the vicinity of a tenth of a lattice constant, as
specified in the Lindemann result.

A factor of significance for the effectiveness of the Lindemann criterion is the tendency for
atoms in three dimensional crystals to have a large number of neighbors [e.g. a dozen
nearest neighbors in the face of face centered cubic (fcc) lattices].  Hence, mean
field treatments in the spirit of Weiss molecular mean field theory are more likely
to provide a reasonable theoretical description
since statistical fluctuations tend to suppressed somewhat by averaging when a
relatively large number of
neighbors are present.

On the other hand, it should
be understood that apart from the number of nearest neighbors,
dimensionality is a very important parameter which may affect the thermodynamic
characteristics and integrity of a crystal lattice to a large degree.
Nano-scale engineering often takes place in systems of low dimensionality such
as carbon-nanotubes where the length may exceed the width by several orders of
magnitude; nanotubes tend to be regarded as one dimensional systems.
Graphene sheets, covalently bonded single layer honeycomb lattices of carbon atoms,
may be considered genuine monolayers, and hence possess strongly two
dimensional character.  The thermodynamic stability of a system is strongly
dependent on its dimensionality with statistical
fluctuations becoming more important for two dimensional systems, and very
important for essentially 1D structures such as nanotubes.

An important theoretical result known as the Mermin-Wagner theorem~\cite{dos,tres}
predicts that as the bulk limit is approached, thermal fluctuations destroy
long-range crystalline order in the context of 1D lattices.  However, this result does not preclude the
stabilization of positional order for $T > 0$ if the interaction between
atomic members is long-ranged (e.g. decaying as a power law in the separation between
positions in the crystal lattice).  In this work, it is our program to examine
conditions which preserve long range order in low dimensional systems at finite
temperatures.

In one dimension, the deleterious effect of thermal fluctuations is
felt most severely, and ultimately only short-range order exists if the
interatomic interaction is finite in range.  In three
dimensions, long-range positional order is intact for finite temperatures
below the melting temperature $T_{m}$.  Two dimensional solids are often regarded as
an intermediate case where thermal fluctuations are strong enough to destroy
long-range order as the size of the system is increased, but only in a very
gradual manner.  Although crystalline order in 2D systems does not survive
in the thermodynamic limit if the interaction is confined to nearest neighbors or is
otherwise finite in range, a long-range coupling with power law decay may
stabilize long-range positional order.  In fact, even for one dimensional
solids, we find a critical decay exponent $\alpha_{c}^{\mathrm{1D}} = 1.615 \pm 0.005$ below which crystalline
order remains stable for $T > 0$, whereas long-range order is only gradually lost
in the bulk for power law decays where $\alpha > \alpha_{c}$.
Similarly, the corresponding exponent in 2D is $\alpha_{c}^{\mathrm{2D}} = 3.15 \pm 0.025$,
where long range crystalline order is preserved for $\alpha < \alpha_{c}^{\mathrm{2D}}$, whereas
thermal fluctuations destroy positional order if $\alpha > \alpha_{c}^{\mathrm{2D}}$.  Within the bounds of
numerical error, we obtain the
same value for the threshold exponent $\alpha_{c}^{\mathrm{2D}}$ for distinct
lattice geometries including square lattices, triangular lattices, and honeycomb lattices.

We examine various types of coupling schemes, including very short-ranged interactions
where atoms interact with only a few nearest neighbors and
perhaps also next-nearest neighbors.  We also consider extended schemes where there is a finite
coupling to all neighbors, but where the interaction is still short-ranged,
with a rapid decay profile, such as that of an inter-atomic potential  with an
exponential dependence $V \propto e^{-\gamma r}$  where $\gamma^{-1}$ is the finite
length scale corresponding to the coupling scheme.
Finally, we also consider a long-ranged algebraically decaying coupling of the
form $V(r) = r^{-\alpha}$ where $\alpha$ may assume different values, though for
the energy per atom to be finite in the bulk limit, the exponent must exceed
threshold values which depend on dimensionality of the lattice.
For single dimensional systems, one must have $\alpha > \alpha_{L}^{1 \mathrm{D}} = 1$ and
$\alpha > \alpha_{L}^{2 \mathrm{D}} = 2$ for two dimensional crystals.

We report on a calculation of the atomic root mean square deviation $\delta_{\mathrm{RMS}}$ about
positions of equilibrium in 1D and 2D crystals. In section I, we discuss theoretical methods
used to calculate the partition function and hence calculate salient thermodynamic quantities by decoupling the vibrational states
used to gauge the integrity of long-range crystalline order
such as $\delta_{\mathrm{RMS}}$.  In Section II, we examine one dimensional systems, finding
positional order to be destroyed except in a long-range coupling scheme, (i.e. a  power law dependence where the
decay exponent $\alpha$ must lie between $\alpha_{L}^{1 \mathrm{D}} = 1$ and an upper bound exponent
$\alpha_{c}^{1 \mathrm{D}} = 1.615 \pm 0.005$).  In section III, we perform a similar analysis for two dimensional
square lattices, where we generalize to an extended scheme, and find a gradual destruction
of crystalline order with increasing system size
$L$ for short-ranged couplings.  However, we find that a power law decay profile where
$\alpha < \alpha_{c}^{\mathrm{2D}} = 3.15 \pm 0.025$ is sufficient to maintain
positional order at finite temperatures.  In addition to the square lattice, in Section IV we also examine triangular and
honeycomb lattices, finding the ability of a long-range interaction between atoms to
preserve crystalline order is not affected by the specific type of lattice geometry under
consideration, and the threshold exponents in all three cases are identical within the bounds of error
in the calculations. Finally, in section V, we consider motion transverse to the plane of the crystal
lattice for locally stiff dual layer systems where even if interaction between particles is taken
to be long-ranged,
we find the perpendicular motions rapidly compromise long-range order as the size of the
system is increased.

A salient component of our treatment is the explicit accounting for atomic motions in
discrete systems.  The harmonic approximation, which neglects anharmonic terms in the potential
set up by geometric effects has been tested directly in the context of Monte Carlo
simulations and found to be accurate to within one part in $10^{3}$ for the systems
considered~\cite{cuatro}.

Since our interest is in equilibrium thermodynamic characteristics of the system, we begin with the
lattice potential
\begin{align}
V = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{n_{i}} V(r_{ij}),
\end{align}
where $n_{i}$ is the number of neighbors corresponding to each atom in the system indexed with
the label $i$, and $N$ is the total number of particles contained in the crystal lattice.
The factor of $1/2$ in the lattice potential expression is present to compensate for double
counting of the energy associated with individual ``bonds'' between atomic pairs $i$ and $j$.

For small deviations from equilibrium positions, a ``harmonic approximation'' is possible, and one
finds instead
\begin{align}
V = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{n_{i}} \frac{K_{ij}}{2} (l_{ij} - l_{ij}^{0})^{2},
\end{align}
the first nonzero term of a Taylor expansion of $V(r_{ij})$ where $l_{ij} \approx l_{ij}^{0}$,
and $K_{ij}$ is the second derivative of the potential $V(r_{ij})$ at $r_{ij} = l_{ij}^{0}$.
In the results we report on here, we restrict attention to temperatures below those which would cause
melting in the bulk (determined by the Lindemann criterion), where the harmonic approximation would tend to fare well.
A primary issue of interest is whether there is any finite temperature
range where long-range positional order is intact, and our calculations are in the context of temperatures not of the magnitude that
would disrupt the bonding topology and create dislocations, but thermal regimes considerably below the
temperature range which might begin to rupture bonds between neighbors.
For covalent solids such as two dimensional sheets of graphene and carbon nanotubes where energies stored
in covalent bonds are far in excess of $k_{\mathrm{B}} T$ at 300K, even room temperature may be considered
a ``low'' temperature in the sense of being considerably below temperature scales where thermal
fluctuations would perturb the
local bonding scheme in a significant way.

In one dimension, the bonds are collinear, and the potential will remain quadratic as the energies of all bonds
between atoms are summed.  However, in two dimensional geometries, restoring forces to oppose displacements from
equilibrium will be exerted in different directions along distinct bond axes between an interacting
pair of atoms.  As a consequence, it will be
necessary to make an additional harmonic approximation in order to obtain a quadratic Hamiltonian and
subsequently exploit translational invariance for the regular lattices we examine.

In general, a bond length $l_{ij}$ between atoms $i$ and $j$ will appear as
\begin{align}
l_{ij} = \sqrt{ \begin{array}{c} (\Delta_{ij}^{0x} + \delta_{i}^{x} - \delta_{j}^{x} )^{2} +
( \Delta_{ij}^{0y} + \delta_{i}^{y} - \delta_{j}^{y})^{2} \\ +
(\Delta_{ij}^{0z} + \delta_{i}^{z} - \delta_{j}^{z})^{2} \end{array}}
\end{align}
where $\Delta_{ij}^{0x} \equiv (x_{i}^{0} - x_{j}^{0})$, $\Delta_{ij}^{0y} \equiv (y_{i}^{0} - y_{j}^{0})$,
and $\Delta_{ij}^{0z} \equiv (z_{i}^{0} - z_{j}^{0})$.  Thus, the potential energy stored in the bonds
depends only on the difference of coordinates such as, e.g., $\Delta_{ij}^{0x}$ for the equilibrium $x$
coordinate differences and $(\delta_{i}^{x} - \delta_{j}^{x})$ for differences constructed from
the corresponding shifts from equilibrium.  If the
latter are sufficiently small in relation to the former, it is appropriate to expand about
$\Delta_{ij}^{0x}$, $\Delta_{ij}^{0y}$, $\Delta_{ij}^{0z}$, and to quadratic order one will have
$(l_{ij} - l_{ij}^{0})^{2} \approx \left [ \hat{\Delta}_{ij} \cdot (\vec{\delta}_{i} - \vec{\delta}_{j} ) \right] ^{2}$ where
$\hat{\Delta}_{ij}$ is a unit vector formed by subtracting the position vectors $\mathbf{x}^{0}_{i}$ and
$\mathbf{x}^{0}_{j}$
corresponding to the atom $i$ and its neighbor $j$, such that
$\hat{\Delta}_{ij} = (\mathbf{x}_{i}^{0} - \mathbf{x}_{j}^{0})/ \| \mathbf{x}_{i}^{0} - \mathbf{x}_{j}^{0} \|$.
Hence, the atomic potential may be written to quadratic order in $\delta_{i}$ and $\delta_{j}$, and one has in
particular
\begin{align}
V_{\mathrm{Har}} = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{n_{i}} \frac{K_{ij}}{2} \left[ \hat{\Delta_{ij}}
\cdot (\vec{\delta}_{i} - \vec{\delta}_{j} ) \right]^{2}
\end{align}
It will be necessary to solve an eigenvalue problem to decouple the vibrational
modes.  However, with Fourier analysis, the problem may be reduced to the task of diagonalizing a
$2 \times 2$ matrix, a $4 \times 4$ matrix in the case of
a lattice with a honeycomb geometry, or
at most a $6 \times 6$ matrix for the case of the locally stiff dual-layer system,
even in cases where the coupling scheme is extended to encompass many neighbors for each atomic
member.  We will consider systems in one dimension where we show that long-range crystalline order may be stabilized
in the case of slowly decaying power law potentials, but not for localized exponentially decaying coupling schemes.
We also examine two dimensional lattice geometries, and find similar phenomena; again, a long-range power law decay is needed
to preserve crystalline order at finite temperatures.

Finally, we are careful to restrict motion to collinear displacements in the context of 1D systems
and intraplanar motion for the two dimensional crystals.  The lattices are very easily disturbed by
transverse displacements, and we find that relaxing the collinear and coplanar restrictions in
1D and 2D yields mean square fluctuations which diverge rapidly with increasing system size
$N$.  As we find with explicit calculation in section V, this rapid (i.e. at a linear) growth in $N$ occurs even if
the interaction between atoms is a long-ranged power law decay in locally stiff dual layer crystal geometries.

We use the results for the eigenvalues for the vibrational modes to calculate thermodynamic properties related to
crystalline order such as the thermally averaged mean square fluctuations about equilibrium per site, $\delta_{\mathrm{RMS}}$.
As noted elsewhere~\cite{cuatro} and summarized here, the mean square displacements about equilibrium may be calculated in terms of
the eigenvalues for the vibrational states.

In terms of the vibrational modes, the lattice energy may be written as
\begin{align}
{\mathcal H}^{\mathrm{Har}} = \frac{1}{2} \sum_{\alpha = 1}^{3M} \lambda_{\alpha} c_{\alpha}^{2}
\end{align}
with $M$ the total number of particles contained in the lattice.
The connection between the vibrational states and the mean square fluctuations is
\begin{align}
\langle \delta_{\mathrm{RMS}} \rangle^{2} = \frac{1}{M} \sum_{i = 1}^{M} \langle (\delta_{i}^{x})^{2} + (\delta_{i}^{y})^{2} + (\delta_{i}^{z})^{2} \rangle
\end{align}
With the eigenvectors indexed with the label $\alpha$ and using, e.g., $\delta_{i}^{x} = \sum_{\alpha = 1}^{3M} c_{\alpha} v_{\alpha}^{ix}$,
to express the displacements in terms of the eigenvectors, one finds for a specific system configuration
\begin{align}
\delta^{2} = \frac{1}{M} \sum_{i=1}^{M} \sum_{\alpha=1}^{3M} \sum_{\alpha^{'} = 1}^{3M} \left[ c_{\alpha} c_{\alpha^{'}} \left( v_{\alpha}^{ix}
v_{\alpha^{'}}^{ix} + v_{\alpha}^{iy} v_{\alpha^{'}}^{iy} + v_{\alpha}^{iz} v_{\alpha^{'}}^{iz} \right ) \right ]
\end{align}
In the thermal average, the factor $c_{\alpha} c_{\alpha^{'}}$ will be as often positive as negative, and hence only in the case $\alpha = \alpha^{'}$
will there be a net contribution to the thermal average $\langle \delta_{\mathrm{RMS}} \rangle^{2}$.  If the vectors are taken to be
normalized, one finds
\begin{align}
\langle \delta_{\mathrm{RMS}} \rangle ^{2} = \frac{1}{M} \sum_{\alpha = 1}^{3M} \langle c_{\alpha}^{2} \rangle
\end{align}
With the lattice energy expressed in this way, the partition function becomes of a product of Gaussian integrals,
\begin{align}
Z = \prod_{\alpha = 1}^{3M} \int_{-\infty}^{\infty} e^{- \beta \lambda_{\alpha} c_{\alpha}^{2}/2} d c_{\alpha} =
\prod_{\alpha = 1}^{3M} \left( \frac{2 \pi \tau}{\lambda_{\alpha}} \right)^{1/2},
\end{align}
where $\tau \equiv k_{\mathrm{B}} T$.  Finally, a thermal derivative of $Z$ leads to
\begin{align}
\langle \delta_{\mathrm{RMS}} \rangle^{2} = \frac{d}{d \tau} \textrm{Ln} (Z) = \sum_{\alpha = 1}^{3M} \lambda_{\alpha}^{-1} \tau
\end{align}
Hence, for the mean square deviation, we have
\begin{align}
\langle \delta_{\mathrm{RMS}} \rangle = \tau^{1/2} \sqrt{ \sum_{\alpha = 1}^{3M}} \equiv \tau^{1/2} \delta_{\mathrm{RMS}}^{n}
\end{align}
The term in the radical is not temperature dependent, but is instead determined by characteristics of the
lattice geometry and the bonding scheme between atomic members.  In this work, we calculate $\delta_{\mathrm{RMS}}^{n}$,
a mean square RMS deviation normalized with respect to temperature.
Zero eigenvalues are artifacts of the periodic boundary conditions, correspond to global translations of the
lattice, and are excluded from the sum.

\section{Systems in One Dimension}
In the 1D systems we consider, only longitudinal displacements are examined; similarly, for the two dimensional
geometries, lattice vibrations are considered to be confined to the two dimensional plane with no transverse
motions considered.  Periodic boundary conditions are implemented in both the one and two dimensional
cases.
An important characteristic of systems in one dimension is the
fact that all bonds are collinear, and hence there is no purely geometric source of anharmonic effects.
The lattice potential energy will have the form
\begin{align}
V = \sum_{l=1}^{n} \sum_{m = 1}^{n} K_{m} (\delta_{l} - \delta_{l+m})^{2}
\label{eq:Eq5}
\end{align}
Since only longitudinal motions are considered, the label ``$x$'' that would normally appear as a
subscript on the ``$\delta$'' symbols is suppressed.  The sum recorded in Eq.~\ref{eq:Eq5} is configured to avoid the
redundant summation over bonds, and the counting ``1/2'' factor is not required.
To maximize the number of neighbors coupled to any particular atom while avoiding multiple couplings to the
same atom via the periodicity condition, we set $n = (N-1)/2$ and we always consider an odd
number of atomic members.

It is convenient to operate in terms of Fourier components, where we have $\delta_{l} = \sum_{k} e^{ikl} \delta_{k}$; on
substitution, the expression for the lattice energy has the form
\begin{align}
E = 2 \sum_{k_{x}} \left[ \sum_{m = 1}^{n} K_{m} (1 - \cos mk_{x} ) \right] |\delta_{k_{x}}|^{2},
\end{align}
which has been diagonalized with the use of Fourier components $\delta_{k}$.  The eigenvalues are given by
$\lambda_{k_{x}} = 2 {\displaystyle \sum_{m=1}^{n}} K_{m} (1 - \cos km )$, and the normalized thermally induced shift $\delta_{\mathrm{RMS}}^{n}$ has
the form
\begin{align}
\delta_{\mathrm{RMS}}^{n} = \left( \sum_{k_{x}} \lambda_{k_{x}}^{-1} \right)^{1/2}
\end{align}
We first examine a localized potential which in the 1D context
 would certainly be expected to yield a divergent mean square fluctuation $\delta_{\mathrm{RMS}}^{n}$
with increasing system size $N$.  As a companion result to gain complementary insight, we also
calculate the density of states
corresponding to the system.  One merit of obtaining the vibrational density of states is the fact that it may be
computed in the thermodynamic limit without encountering divergences
with the aid of Monte Carlo sampling.  On the other hand, the divergence or convergence of
the mean square deviations will be signaled by specific signatures in the low eigenvalue regime of the density of
states without the need for an extrapolation to the thermodynamic limit.

In the case where interactions are confined to nearest neighbors, the
eigenvalues $\lambda_{k_{x}}$ have the simple form $2K_{0} (1 - \cos k_{x} x)$.  The corresponding
thermally averaged $\delta_{\mathrm{RMS}}^{n}$ values in the case of finite systems
are shown in the graph in Fig.~\ref{fig:Fig1}, and there is a steady  rise with
in the RMS fluctuations with increasing $n$.
The expansion of the mean square deviations from equilibrium is sub-linear, but it may be shown that the
increase continues indefinitely (i.e. diverges in the thermodynamic limit) by graphing instead
$(\delta_{\mathrm{RMS}}^{n})^{2}$, as in the inset of Fig.~\ref{fig:Fig1}.
The dependence on $N$ quickly reverts to an asymptotically linear increase with
$N$, and to a good approximation $\delta_{\mathrm{RMS}}^{n} = N^{1/2}$ for moderate to large systems.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig1.eps}
\caption{\label{fig:Fig1} Mean square deviation curves are graphed
versus system size $N$ in inset (a).  The vertical axis of the graph in inset (b) is $(\delta_{\mathrm{RMS}})^{2}$ to
emphasize the linear behavior of the square of the mean square fluctuations for
a broader range of system sizes.}
\end{figure}

We next extend the coupling scheme to many neighbors where the coupling decays at an exponential rate,
as might be found at least on a qualitative level for a covalently bonded system where the
rapidly decaying overlap of the orbitals of atomic neighbors (and hence the
magnitude of the exchange coupling) has an asymptotically exponential
decay as the separation between the pair of atoms becomes sufficiently large.  The lattice energy will have the form
\begin{align}
V_{\mathrm{exp}}^{\gamma} = \sum_{l = 1}^{N} \sum_{m = 1}^{n} K_{0} e^{-\gamma m}  ( \delta_{l} - \delta_{l+m})^{2}
\end{align}
Hence in terms of Fourier components, the total lattice potential becomes
\begin{align}
V_{\mathrm{exp}}^{\gamma} =  \sum_{k} \left( 2 K_{0} \sum_{m=1}^{n} e^{-\gamma m} [1 - \cos mk ] \right)
\end{align}
Again, operating in terms of Fourier components decouples the modes, and the appropriate
eigenvalues $\lambda_{k}$ are given by
\begin{align}
\lambda_{k} = 2 \sum_{m=1}^{n} e^{-\gamma m} [1 - \cos mk ]
\end{align}
where the prefactor $K_{0}$ has been suppressed.
In addition, the label ``$x$'' on the wave vector has also been suppressed
for the sake of convenience.
Although the coupling scheme is extended to many neighbors, the potential is in
an important sense still a local interaction due to its rapid decay, where
the appropriate length scale is the inverse decay rate $\gamma^{-1}$.
By appealing to the formula for a geometric sum, $r + r^{2} + \ldots + r^{n} = (r-r^{n+1})/(1-r)$ (where $r$ is
taken to be complex and $\lvert  r \rvert  < 1$) and using the
fact that $\cos m k = \tfrac{1}{2} (e^{imk} + e^{-imk})$, one may obtain an
explicit expression for $\lambda_{k}$ which does not require the intermediate summation.
Applying the geometric series formula for a finite series yields
\begin{align}
&\lambda_{k} = 2 e^{-\gamma} \left( \frac{1 - e^{-\gamma n}}{1 - e^{-\gamma}} \right ) -& \\ \nonumber
&e^{-\gamma} \left( e^{ik}\left[ \frac{1  - e^{n(-\gamma + ik)}}{1 - e^{-\gamma + ik}} \right] +
e^{-ik}\left[ \frac{1 - e^{n(-\gamma - ik)}}{1 - e^{-\gamma - ik}} \right] \right)&
\end{align}
Combining the last two fractional terms gives
\begin{align}
\lambda_{k} = 2 \! \left( \! \! \frac{1 - e^{-\gamma n}}{e^{\gamma} - 1} \! \right )\!  - \! \left( \! \frac{ \begin{array}{c}
\cos k  + e^{-\gamma (n+1)} \cos nk\\
 -e^{-\gamma}  - e^{-\gamma n} \cos (n+1)k] \end{array} }{\cosh \gamma - \cos k} \! \! \right ) \! \! ,
\end{align}
a tidier and computationally convenient expression
to use in calculating the RMS displacements $\delta_{\mathrm{RMS}}^{n}$.

As $n$ becomes large, terms proportional to $e^{-\gamma n}$ quickly become suppressed by the
rapid exponential decay.  Hence, for $n \gg \gamma^{-1}$, one will obtain
\begin{align}
\lambda_{k} = \frac{2}{e^{\gamma} - 1} + \frac{e^{-\gamma} - \cos(k)}{\cosh (\gamma) - \cos (k)}
\label{eq:Eq10}
\end{align}

The results are shown in Fig.~\ref{fig:Fig3} for the scaling of $\delta_{\mathrm{RMS}}^{n}$ with respect to the size $N$
of the system.  To keep the results for different values of $\gamma$
on the same footing, we use the prefactor $K_{0}$ as a normalization of the
coupling with $K_{0} \equiv e^{\gamma}$.  A similar procedure is also used in calculations involving exponentially
decaying extended couplings in 2D.  In this manner, the convergence to the results for the case where only
nearest neighbors are involved in the coupling scheme is easier to see.

In the main part of the graph, the
square of the RMS deviation is graphed with respect to system size $N$ for a broad range of system sizes.
The curves corresponding to the different decay constants are asymptotically linear in the system size
although the slopes decrease with decreasing $\gamma$ as the coupling becomes longer in range.
The inset of the graph shows a closer view of $\delta_{\mathrm{RMS}}^{n}$.  Each of the curves rises steadily for
sufficiently large $N$, notwithstanding non-monotonicities for small to moderate $N$ in the case
$\gamma = 0.25$ where the decay of the interaction is relatively slow.
In latter case, there is competition between thermal fluctuations and an increase in lattice
rigidity which occurs as the linear crystal grows, providing atoms with more neighbors.  Eventually, however,
$N$ exceeds the length scale $\gamma^{-1}$ of the coupling between atoms, and the balance shifts in
favor of thermal fluctuations.  The latter increase in importance with increasing $N$ and thus eventually destroy long-range
crystalline order.

We also examine the eigenvalue density of states for different decay rates $\gamma$.
With the range of the potential being set by $\gamma^{-1}$, larger values of $\gamma$ would correspond to a more
rapid decay and a shorter range of the interaction between neighbors.  In calculating the density of states, we use a Monte Carlo sampling process where
the values of $k$ are not quantized, permitting one to genuinely achieve the bulk limit for the purpose of
obtaining the density of states.  To obtain a smooth curve a large number of
eigenvalues (i.e. $2.5 \times 10^{8}$ for the
histograms corresponding to the exponentially decaying coupling scheme) are sampled.
The formula given in the continuum limit in Eq.~\ref{eq:Eq10} is the appropriate expression to use for
$\lambda_{k}$ in the Monte Carlo sampling process.

The normalized density of states for a range of decay constants $\gamma$ appears in Fig.~\ref{fig:Fig3}.
Panel (a) is a standard plot with the density of states on the vertical axis, while
to facilitate the viewing of the DOS curves, the logarithm of the DOS curves in shown in panel (b).
Even for relatively long-range cases such as $\gamma = 0.25$, the density of states retains the ``U''-shaped
profile of the nearest neighbor case.  The latter corresponds effectively to $\gamma = \infty$, and is
shown in red in the graphs.  For convenience in comparing results, we again choose $K_{0} = e^{\gamma}$ in calculating the
DOS curves.  The convergence to the $\gamma = \infty$ case with increasing $\gamma$ is
evident for the case $\gamma = 2.0$, where close agreement with the DOS calculated for the nearest neighbor case
is evident in panel (b) of Fig.~\ref{fig:Fig1}. In the latter, the ordinate is chosen to be $\log_{10} (\textrm{DOS})$ to help show the
structure of the density of states curves more clearly.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig2.eps}
\caption{\label{fig:Fig2} (Color Online) Mean square deviation curves are shown for a range of
$\gamma$ values.  The vertical axis of the main graph is $(\delta_{\mathrm{RMS}})^{2}$ to
emphasize the linear behavior of the square of the mean square fluctuations, while the inset is a
graph of $\delta_{\mathrm{RMS}}^{n}$ with respect to $N$ for a smaller range of system sizes.
$2.5 \times 10^{8}$ eigenvalues were sampled to calculate the DOS curves.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig3.eps}
\caption{\label{fig:Fig3} (Color Online) Eigenvalue Density of States (DOS) for $\gamma$ values
ranging from $\gamma = 0.25$ to $\gamma = 2.0$ to, effectively, $\gamma = \infty$ with the
DOS on the ordinate for panel (b).
Panel (b) shows the density of states as well, but the
ordinate is $\log_{10} (\textrm{DOS})$.}
\end{figure}

Finally, we examine a genuinely long-range inter-atomic potential with a power law decay profile.
The lattice potential energy will have the form
\begin{align}
E = \sum_{l=1}^{N} \sum_{m=1}^{n} K_{0} m^{-\alpha} (\delta_{l+m} - \delta_{l})^{2}
\end{align}
where $\alpha$ is the decay exponent of the power law interaction ($\alpha > 1$), and
again $n = (N-1)/2$.
In terms of Fourier components, one will have
\begin{align}
E = \sum_{k} \sum_{m=1}^{n} 2 K_{0} m^{-\alpha} (1 - \cos mk) \lvert \delta_{k} \rvert^{2}
\label{eq:Eq15}
\end{align}
Hence, the modes are now decoupled with eigenvalues given by $\lambda_{k} = {\displaystyle \sum_{m=1}^{n}}
m^{-\alpha} (1 - \cos mk)$; we evaluate this expression directly in order to obtain $\delta_{\mathrm{RMS}}^{n}$
and the DOS profile appropriate to particular exponent $\alpha$ in the bulk limit.
Again, we calculate $\delta_{\mathrm{RMS}}^{n}$ and generate plots with respect to system
size.  To test for divergence or convergence in the bulk limit, it is useful also to prepare log-log
plots (we use base ten logarithms in all cases), and the results appear in the inset of Figure~\ref{fig:Fig4}.
We examine systems ranging in size from $N = 3$ to on the order of a few hundred thousand atomic members.
A crucial question is whether there is a threshold value $\alpha_{c}$ above $\alpha = 1$ (where the
lattice energy may diverge with increasing system size) below which long range crystalline
order is stable with respect to thermal fluctuations in one dimensional lattices.

To identify $\alpha_{c}^{\mathrm{1D}}$, we calculate the normalized mean square fluctuations $\delta_{\mathrm{RMS}}^{n}$ with respect to
system size $n$, producing log-log graphs.  The highest value of $\alpha$ where the mean square
deviations converge is identified as $\alpha_{c}^{\mathrm{1D}}$, the upper limit for the decay exponent in the extended
power law decay scheme where long-range crystalline order is still supported at finite temperatures.

The mean square
deviations, useful thermodynamic quantities with which to diagnose the presence or absence of
long-range crystalline order, are shown in Fig.~\ref{fig:Fig4} and Fig.~\ref{fig:Fig5}
(with the abscissa shown as a base ten logarithm over five decades of system
sizes $N$).  In Fig.~\ref{fig:Fig4}, $\delta_{\mathrm{RMS}}^{n}$ curves are shown
for a relatively wide range of $\alpha$ values.  Over the broad range of systems on
the horizontal axis, five orders of magnitude, the mean square displacements rise
monotonically for $\alpha = 2.0$ and $\alpha = 1.75$, while $\delta_{\mathrm{RMS}}^{n}$
decreases steadily for $\alpha = 1.5$ and $\alpha = 1.25$.  The curves suggest a
decay exponent $\alpha_{c}$ in the vicinity of $\alpha = 1.6$ as a boundary between
crystals where long-range order is unstable at finite temperatures, and one dimensional
solid where crystalline order is retained for $T > 0$.
The inset is the corresponding log-log graph of the mean square fluctuations
plotted for the same $\alpha$ values as in the main graph, which is a semi-logarithmic plot.

In Fig.~\ref{fig:Fig5}, RMS deviation curves are shown for a tighter span of power law
decay exponents
(ranging from $\alpha = 1.60$ to $\alpha = 1.65$) to identify with greater accuracy the
numerical value of $\alpha_{c}$.  To facilitate the determination of the
exponent separating crystals with long-range order and those disrupted by thermal
fluctuations, we place dark circles over the maxima of the
$\delta_{\mathrm{RMS}}^{n}$ curves.  For $\alpha > 1.625$, the maxima are located at the
edge of the plot, consistent with a steady increase (and likely
divergence in the bulk limit) of the RMS curves.
On the other hand, for $\alpha < 1.6125$, the thermally averaged RMS deviations
are non-monotonic, reaching a
maximum for finite values of $N$ and then declining, presumably toward a stable
bulk value.
We identify the boundary as $\alpha_{c} = 1.615 \pm 0.005$.
It should be emphasized that while long-range order is not supported for decay exponents
in excess of $\alpha_{c}$, the divergence of $\delta_{\mathrm{RMS}}^{n}$ with increasing
system size is nonetheless quite slow, sublinear in $\log_{10}(N)$, whereas a strictly logarithmically
diverging mean square deviation would instead rise at a more rapid linear rate.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig4.eps}
\caption{\label{fig:Fig4} (Color Online) $\delta_{\mathrm{RMS}}^{n}$ is plotted on a
semi-logarithmic (all logs are base ten)
scale in the main draft for selected values of $\alpha$ for
one dimensional systems.  The inset is
a log-log graph of the mean square deviations for the same values of $\alpha$ as in the main
graph.}
\end{figure}

\begin{figure}
\includegraphics[width=.4\textwidth]{dallsfig5.eps}
\caption{\label{fig:Fig5} (Color Online) $\delta_{\mathrm{RMS}}^{n}$ is plotted on a
semi-logarithmic scale for a relatively tight range of $\alpha$ values.  The large dark
circles indicate maxima in the mean square deviation curves; it is concluded that
$\alpha_{c} = 1.615 \pm 0.005$.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig6.eps}
\caption{\label{fig:Fig6} (Color Online) Eigenvalue Density of States (DOS) for the one
dimensional lattice with a power law decay where the decay exponent $\alpha$ varies from
$\alpha = 1.5$ in panel (a) to $\alpha = 2.5$ in panel (d).}
\end{figure}

To obtain information complementary to the RMS fluctuations, we again calculate the eigenvalue
density of states.
We also use Monte Carlo sampling where wave numbers are chosen at random,
with uniform probability, to calculate the vibrational density of states with the results shown in Fig.~\ref{fig:Fig6}.
The double sum in Eq.~\ref{eq:Eq15} requires careful consideration, in that one must be certain that enough terms
have been included in the inner sum that a convergent result is obtained.  To be certain convergence has
been achieved,
we prepare eigenvalue histograms for successive doublings of the number of terms contained in the inner sum indexed by
$m$.  The number of terms which must be included in order to attain suitable convergence
increases with decreasing $\alpha$ for crystal lattices where the coupling is more slowly
decaying.  In general, however, the oscillatory cosine term in Eq. does act to somewhat hasten
convergence and hence limit the number of terms which need to be summed.

\section{Two Dimensional Crystals}

For the case of a two dimensional system, the analysis is in many respects parallel to that applied
for the one dimensional lattices.  However, the additional dimension makes available richer choices
for the lattice geometry.  We examine various coupling schemes for three types of lattices; the
square lattice, the triangular lattice, and the honeycomb lattice.  In Fig.~\ref{fig:Fig7}  panel (a) represents
the square lattice, the triangular lattice is depicted in panel (b), and the honeycomb lattice
appears in panel (c). A peculiarity of the honeycomb lattice is the presence of inequivalent sites,
and this characteristic is highlighted in panel (c) of Fig.~\ref{fig:Fig7} where different colors are used in labeling
the sites.  Although the geometries we examine have different characteristics, the
essential qualitative characteristics and the most salient physics are found to share much in common.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig7.eps}
\caption{\label{fig:Fig7} (Color Online) Three distinct lattice geometries are shown.
Panel (a) is a portion of a square lattice, panel (b) represents the triangular lattice, and
panel (c) shows the honeycomb lattice with inequivalent sites represented with distinct colors.}
\end{figure}

We first consider the square lattice, and
we initially take into account only interactions between nearest neighbors where at the present level of
approximation the lattice lacks rigidity.   The lattice energy is
\begin{align}
E = \frac{K}{2} \sum_{i,j=0}^{N-1} [ (\delta_{i+1}^{x} - \delta_{i}^{x})^{2} +
(\delta_{ij+1}^{y} - \delta_{ij}^{y})^{2}]
\end{align}
We express the displacements in terms of Fourier components with, e.g., $\delta_{ij}^{x}
= {\displaystyle \sum_{\mathbf{k}}} \delta_{\mathbf{k}}^{x} e^{I(k_{x} i + k_{y} j)}$, with $I$ being the
imaginary unit.  In terms of $\delta_{\mathbf{k}}^{x}$ and $\delta_{\mathbf{k}}^{y}$, the
energy has the form
\begin{align}
E = \frac{k}{2} \sum_{\mathbf{k}} [ (1 - \cos k_{x} ) \lvert \delta_{\mathbf{k}}^{x} \rvert^{2} +
(1 - \cos k_{y} ) \lvert \delta_{\mathbf{k}}^{y} \rvert^{2} ];
\label{eq:Eqfloppy}
\end{align}
in this manner the degrees of freedom are decoupled.  Inspection of Eq.~\ref{eq:Eqfloppy}
reveals that the eigenvalues are $2N$ fold degenerate and identical to the eigenvalues obtained
for the case of the one dimensional crystal where only interactions between nearest neighbors
were considered.
Since the eigenvalues are the same as those in the 1D case with interactions only between
nearest neighbors, crystalline order is readily disrupted by thermal fluctuations.
Hence, $(\delta_{\mathrm{RMS}}^{n})^{2}$ will scale with $N$ just as was the case for the 1D counterpart.

If one takes into account coupling to next-nearest neighbors as well, then the lattice energy in
real space is
\begin{align}
E = \tfrac{K}{2} \! \sum_{i,j=0}^{N-1} \left( \! \! \begin{array}{c} \left[ \hat{x} \cdot
(\vec{\delta}_{i+1j} - \vec{\delta}_{ij}) \right]^{2} + \left [ \hat{y} \cdot
( \vec{\delta}_{ij+1} - \vec{\delta}_{ij} ) \right]^{2} \\
+ \left[ \tfrac{1}{\sqrt{2}} ( \hat{x} + \hat{y} ) \cdot \left( \vec{\delta}_{i+1j+1}
- \vec{\delta}_{ij} \right) \right ]^{2} \\ + \left[ \frac{1}{\sqrt{2}} ( \hat{x} - \hat{y})
\cdot \left( \vec{\delta}_{i+1j-1} - \vec{\delta}_{ij} \right) \right]^{2} \end{array} \! \!\right)
\end{align}
Operating in terms of Fourier components, one diagonalizes the matrix
\begin{align}
\left[ \begin{array}{cc} \left( \begin{array}{c} 2 - \cos k_{x} \\ - \cos k_{x} \cos k_{y} \end{array}
\right) & \sin k_{x} \sin k_{y} \\
\sin k_{x} \sin k_{y} & \left( \begin{array}{c} 2 - \cos k_{y} \\ - \cos k_{x} \cos k_{y} \end{array} \right) \end{array} \right]
\end{align}
The eigenvalues are given by
\begin{align}
&\lambda_{\pm} = (4 - \cos k_{x} \cos k_{y} - 2 \cos k_{x} \cos k_{y} &  \\
&\pm \sqrt{(\cos k_{x} - \cos k_{y} )^{2} + 4 \sin^{2} k_{x} \sin^{2} k_{y}}&
\end{align}

The results for the mean
square deviations $\delta_{\mathrm{RMS}}^{n}$ appear in Fig.~\ref{fig:Fig8}.
One may also calculate the vibrational DOS, and the results appear in panel (a) of
Fig.~\ref{fig:Fig8}.  The introduction of next-nearest neighbor interactions is very effective in
reducing the deleterious effect of thermal fluctuations on long-range
crystalline order, though there is still a weak divergence in $\delta_{\mathrm{RMS}}^{n}$ in the
bulk limit.  The square $( \delta_{\mathrm{RMS}}^{n})^{2}$ quickly assumes an asymptotically
linear form with respect to $\log_{10}N$.
The much slower increase of the RMS deviations with $N$ is
reflected in the DOS profile, where instead of
exhibiting a sharp cusp in the low eigenvalue regime, the
DOS curve terminates smoothly.  However, the fact that the DOS tends to a finite value as the eigenvalue vanishes is still
enough to cause a divergence in the mean square displacements from equilibrium.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig8.eps}
\caption{\label{fig:Fig8} (Color Online) The mean square displacements
for the square lattice with nearest and next-nearest neighbor couplings. The main graph shows $(\delta_{\mathrm{RMS}}^{n})^{2}$,
with the corresponding vibrational density of states graphed in inset (a) and the raw RMS displacements $\delta_{\mathrm{RMS}}^{n}$ displayed
in inset (b).}
\end{figure}

We next examine a general case where there are interactions with many neighbors.
In real space, the energy stored in the lattice has the form
\begin{align}
E = \frac{1}{2} \! \sum_{i,j=-n}^{N-1} \sum_{l,m=-n}^{n} \! \! \! \! \tfrac{1}{2} K(r_{lm}) \! \left( \! \hat{\Delta}_{lm} \! \cdot \!
[ \vec{\delta}_{i+lj+m} - \vec{\delta}_{ij} ] \! \right )^{2}
\end{align}
where the inner ``$\tfrac{1}{2}$'' factor compensates for multiple counting of bond energies and the choice
$n = (N-1)/2$ allows each atomic member to interact with all of the atoms contained in
crystal while avoiding multiple interactions with the same particle.
Since $\vec{\Delta}_{lm} = [l,m]$, the appropriate unit vector directed between particles
given the labels ``$i+l,j+m$'' and ``$ij$'' is $\vec{\hat{\Delta}}_{lm} = [l,m]/\sqrt{l^{2} + m^{2}}$.
Again, we may decouple the vibrational modes by expressing the coordinate shifts in
terms of Fourier components.  The lattice potential energy may then be written as
\begin{align}
E =  \frac{1}{2} \sum_{i,j=0}^{N-1} \sum_{l,m=-n}^{n} \tfrac{K(r_{lm})}{2r_{lm}^{2}} \left[ \begin{array}{c} l (\delta_{i+lj+m}^{x} -
\delta_{ij}^{x} )  + \\ m ( \delta_{i+lj+m}^{y} - \delta_{ij}^{y} ) \end{array}
 \right]^{2}
\end{align}
with the ``1/2'' factor present to compensate for redundant bond counting.
The range radius $r_{lm}$ is defined
with $r_{lm} \equiv \sqrt{l^{2} + m^{2}}$, with the full vector given by $\mathbf{r}_{lm} =
l \hat{x} + m \hat{y}$.
In terms of Fourier components, one will have
\begin{align}
&E = \frac{1}{2} \sum_{k_{x},k_{y}} \sum_{l,m=-n}^{n} \frac{K(r_{lm})}{r_{lm}^{2}} [1 - \cos (\mathbf{k} \cdot \mathbf{r}_{lm})]&  \\ \nonumber
&\times \left( \begin{array}{c}
l^{2} \lvert \delta_{\mathbf{k}}^{x} \rvert^{2}  + m^{2} \lvert \delta_{\mathbf{k}}^{y} \rvert^{2} \\
+ ml [\delta_{\mathbf{k}}^{x} \delta_{\mathbf{k}}^{y*} + \delta_{\mathbf{k}}^{x*} \delta_{\mathbf{k}}^{y}]
\end{array} \right )&
\end{align}
Hence, in order to to decouple the vibrational modes, one must diagonalize the
matrix
\begin{align}
\tfrac{1}{2} \sum_{l,m=-n}^{n} K(r_{lm}) r_{lm}^{-2} [1 - \cos (\mathbf{k} \cdot \mathbf{r}_{lm})]
\left[ \begin{array}{cc} l^{2} & ml \\ ml & m^{2} \end{array} \right ],
\end{align}
which may also be written as
\begin{align}
\tfrac{1}{2} \! \! \! \! \! \sum_{l,m=-n}^{n} \! \! \! \! K(r_{lm}) [1 - \cos (\mathbf{k} \! \cdot \! \mathbf{r}_{lm})] \! \! \left [ \! \!
\begin{array}{cc} \hat{\Delta}_{lm}^{x} \hat{\Delta}_{lm}^{x} & \hat{\Delta}_{lm}^{x} \hat{\Delta}_{lm}^{y} \\
\hat{\Delta}_{lm}^{y} \hat{\Delta}_{lm}^{x} & \hat{\Delta}_{lm}^{y} \hat{\Delta}_{lm}^{y}
\end{array} \! \right ] \! \! ,
\end{align}
a representation which will prove more compact for more complicated systems such as the honeycomb lattice crystals with
more than one layer in the direction transverse to the crystal plane, examined in Section V.

We first turn to the case of an exponentially decaying coupling scheme, and we calculate
the $\delta_{\mathrm{RMS}}^{n}$ curves with respect to the size $N$ of the system.
The results for the thermally averaged means square displacements
are shown in Fig.~\ref{fig:Fig9} for a range of different decay constants $\gamma$.
The computational burden of calculating the auxiliary sum will grow with $N$, but one
aspect of the exponential decay that is of assistance in the calculations is the fact that the sum may be safely
truncated when the distance between interacting atoms becomes several times
greater than the range of the short-ranged coupling (i.e. terms beyond
beyond $m$ and $l$ pairs such that $\sqrt{l^{2} + m^{2}} \gg \gamma^{-1}$) need not be included.
In particular, we obtain results which are very well converged if we discard terms beyond 20 decay lengths $\gamma^{-1}$.
Ultimately, thermally induced deviations from equilibrium destroy long-range order, and
the RMS deviations diverge slowly [$(\delta_{\mathrm{RMS}}^{n})^{2}$ again scales linearly
with $\log_{10}(N)$], but the rate of divergence decreases with decreasing $\gamma$.  In
particular, as the range $\gamma^{-1}$ of the inter-atomic coupling is increased,
the slope of the graph of $(\delta_{\mathrm{RMS}}^{n})^{2}$ with respect to
$\log_{10}(N)$ decreases, although the RMS deviations eventually still diverge in
the thermodynamic limit.

The DOS is also calculated, with results appearing in Fig.~\ref{fig:Fig10} for a range of $\gamma$ values.
We use Monte Carlo sampling to choose $k_{x}$ and $k_{y}$ from a continuum range, and thereby
operate in the thermodynamic limit for the purpose of calculating the DOS curves.
At least $10^{6}$ eigenvalues are sampled in generating the DOS curves.
The low eigenvalue region of the DOS graph is very similar to the corresponding regime of the
density of states where only interactions with nearest and next-nearest neighbors are
included.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig9.eps}
\caption{\label{fig:Fig9} (Color Online) Mean square deviations for an
exponentially decaying interaction.  The main graph shows $(\delta_{\mathrm{RMS}}^{n})^{2}$, while
the raw mean square deviations are plotted in the inset of the Figure.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig10.eps}
\caption{\label{fig:Fig10} (Color Online) Vibrational density of states for an
exponentially decaying coupling.  The DOS curves are plotted for assorted values of the
decay parameter $\gamma$.}
\end{figure}

Next, we examine the much slower power law decays $K_{ij} = K r_{ij}^{-\alpha}$ in
the inter-atomic separation $r_{ij}$ where the
exponent $\alpha$ controls the rate of the decay, long-ranged in
the sense that there is not a length scale to
set the range of the coupling.  Again, we first calculate the mean square displacements
with respect to the size of the system, and then we examine the density of states for
the eigenvalues.  The $\delta_{\mathrm{RMS}}^{n}$ results are shown in Fig.~\ref{fig:Fig11}.

Computational subtleties similar to those encountered for the case of the one dimensional
solid in with a long-ranged coupling scheme must be carefully navigated since the
higher dimensionality ($d = 2$) will cause the computational burden to grow even more rapidly (nominally as
$L^{4}$ if interactions with all neighbors are included) with system size.  Again, we use Monte Carlo sampling to select wavevectors and accumulate
eigenvalues to build up the vibrational DOS.  The auxiliary sum
over dummy indices $l$ and $m$ giving the eigenvalue
would in principle contain an infinite number of terms (in the bulk limit, each atom in the
crystal would have an infinite number of neighbors), but we truncate the sum at a
finite range. The presence of sinusoidal terms in the
sum, as in the corresponding 1D case, provides an oscillatory element and will hasten the convergence of the sum, thereby reducing the
computational burden.  We check convergence with respect to the
truncation range by calculating the DOS with successive doublings of the truncation length $N_{\Delta}$
until the DOS profile ceases to change with additional doublings of
the system size.  The results for the vibrational DOS appear in Fig.~\ref{fig:Fig12}.
One notes that the convergence with respect to the truncation radius is least
rapid for $\alpha = 2.5$. However, the graphs are relatively well converged for $\alpha = \alpha_{c}^{\mathrm{2D}}$ and
higher values of the decay exponent.  When $\alpha$ is in the vicinity of $\alpha_{c}^{\mathrm{2D}}$, there is
very little support in the low eigenvalue regime.  On the other hand, with increasing $\alpha$ the
interaction decays more rapidly, and ultimately the histogram amplitude in the zero eigenvalue limit rises to a finite value,
contributing to a divergence in the mean square fluctuations with increasing system size.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig11.eps}
\caption{\label{fig:Fig11} (Color Online) Mean square deviations for an interaction
decaying as a power law for decay exponents $\alpha$ near the threshold $\alpha_{c}^{\mathrm{2D}}$.  The
inset shows a broader view, while the main graph is a closer view of the transition from
converging to diverging $\delta_{\mathrm{RMS}}^{n}$ curves.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig12.eps}
\caption{\label{fig:Fig12} (Color Online) The vibrational density of states for the two
dimensional square lattice with a power law decay interaction.  DOS curves for various values of the
decay parameter $\alpha$ are shown, with traces for different values of the truncation
length $N_{\Delta}$ on the same plot.}
\end{figure}

\section{Alternate Two Dimensional Geometries}

The treatment in the case of the triangular and honeycomb lattices is very similar to the
approach used in the case of the square lattice.
Interestingly,
the triangular lattice is rigid with only a nearest
neighbor interaction in the context of the harmonic
approximation, and the mere inclusion of nearest neighbors is enough to set up quasi-long range
order where thermally induced fluctuations about equilibrium diverge very slowly [i.e.
$(\delta_{\mathrm{RMS}}^{n})^{2}$ increases as the logarithm of the system size just as
in short-ranged extended interactions for the square lattice].
For the triangular lattice, the lattice energy in real space has the form
\begin{align}
E = \tfrac{K}{2} \sum_{i,j=0}^{N-1} \left( \! \! \begin{array}{c} \left[ \hat{x} \cdot \left (
\vec{\delta}_{i+1j} - \vec{\delta}_{ij} \right ) \right]^{2} + \\
\left[ (\tfrac{1}{2} \hat{x} + \tfrac{\sqrt{3}}{2} \hat{y} ) \cdot \left(
\vec{\delta}_{ij+1} - \vec{\delta}_{ij} \right) \right]^{2} + \\ \left[ (\tfrac{1}{2} \hat{x} -
\tfrac{\sqrt{3}}{2} \hat{y} ) \cdot \left( \vec{\delta}_{i+1j-1} - \vec{\delta}_{ij} \right )\right]^{2} \end{array} \! \! \right)
\end{align}
and the eigenvalues for the decoupled vibrational modes are obtained by diagonalizing the $2 \times 2$ matrix
\begin{align}
\left[ \! \! \! \! \begin{array}{cc} \left( \! \! \begin{array}{c} 3  - 2 \cos k_{x} \! - \!  \tfrac{1}{2} \cos k_{y} \\ -
\tfrac{1}{2} \cos [k_{y} \! - \! k_{x} ] \end{array} \! \! \! \right) & \tfrac{\sqrt{3}}{2} ( \cos [k_{y} \! - \! k_{x} ] - \cos k_{y} ) \\
\tfrac{\sqrt{3}}{2} (\cos [k_{y} \! - \! k_{x} ]  \cos k_{y}) & \left( \! \! \begin{array}{c} 3 - \tfrac{3}{2} \cos k_{y} \\
- \tfrac{3}{2} \cos [k_{y} \! - \! k_{x} ]  \end{array} \! \! \right) \end{array} \! \! \! \right] \! \! ,
\end{align}
yielding
\begin{align}
&\lambda_{\mathbf{k}}^{\pm} = [3 - \cos k_{x} - \cos k_{y} - \cos (k_{y} \! - \! k_{x} ) ]& \\ \nonumber
&\pm \sqrt{\begin{array}{c} \cos^{2} k_{x} + \cos^{2} k_{y} + \cos^{2} (k_{y} \! - \! k_{x}) - \cos k_{x} \cos k_{y} \\
- \cos k_{y} \cos (k_{y} \! - \! k_{x} )  - \cos k_{x} \cos (k_{y} \! - \! k_{x}) \end{array}}&
\end{align}

The results for the mean square fluctuations as well as the vibrational DOS are given in Fig.~\ref{fig:Fig13}.
\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig13.eps}
\caption{\label{fig:Fig13} (Color Online) The mean square deviation for the triangular lattice with nearest neighbor
couplings.  The main graph is a semi-logarithmic plot of $\delta_{\mathrm{RMS}}$, while inset (a) shows the vibrational
density of states.  A graph of the raw $\delta_{\mathrm{RMS}}^{n}$ appears in panel (b).}
\end{figure}

We generalize the nearest-neighbor case to an extended scheme where
each atomic member may interact with many neighbors.
In real space, the lattice potential energy may be written as
\begin{align}
E = \frac{1}{2} \sum_{i,j=0}^{N} \sum_{l,m=-n}^{n} \tfrac{1}{2}
K(r_{lm}) \vec{\hat{\Delta}}_{lm} \! \cdot \! (\vec{\delta}_{i+l,j+m} - \vec{\delta}_{ij}),
\end{align}
where the components of the unit vector $\vec{\hat{\Delta}}_{lm}$ are $\hat{\Delta}_{lm}^{x} = (l+m/2)/r_{lm}$ and
$\hat{\Delta}_{lm}^{y} = \sqrt{3}m/2r_{lm}$, where $r_{lm} = (l^{2} + m^{2} + lm)^{1/2}$ is the distance separating interacting pairs
in the triangular lattice geometry.
After expressing the displacements in terms of Fourier components, one calculates the eigenvalues of the $2 \times 2$
matrix
\begin{align}
\frac{1}{2} \sum_{l,m=-n}^{n} g_{lm} \left[ \! \! \begin{array}{cc} \hat{\Delta}^{x}_{lm} \hat{\Delta}^{x}_{lm} &
\hat{\Delta}^{x}_{lm} \hat{\Delta}^{y}_{lm} \\
\hat{\Delta}^{y}_{lm} \hat{\Delta}^{x}_{lm} & \hat{\Delta}^{y}_{lm} \hat{\Delta}^{y}_{lm} \end{array} \! \! \right ],
\end{align}
where $g_{lm} = [1 - \cos(k_{x}l + k_{y}m)] K(r_{lm})$.
As in the case of the square lattice, we consider for the triangular lattices an
exponentially decaying coupling scheme, and the results are shown in Fig.~\ref{fig:Fig14}
for over four decades of system sizes.
Again, the quantity $(\delta_{\mathrm{RMS}}^{n})^{2}$ increases linearly as $\beta \log_{10} N$ with the slope $\beta$
decreasing with decreasing decay rate $\gamma$ (and hence increasing range of the coupling).

We also prepare graphs of the vibrational density of states, shown in the four graphs in Fig.~\ref{fig:Fig15}
for a range of values of the decay constant $\gamma$.
The wide separation between the RMS curves corresponding to $\gamma = 2.0$ and $\gamma = 1.0$, $\gamma = 0.75$, and
$\gamma = 0.5$ is mirrored in the DOS curves where for the smaller decay rates the eigenvalue histogram curve
intersects the ordinate with very low amplitudes.  On the other hand, for the more rapid decay where $\gamma = 2.0$,
the amplitude in the regime of low eigenvalues is much higher, and the DOS graph resembles that of the nearest neighbor case
to a much greater degree than DOS profiles corresponding to lower decay rates of the exponential coupling.

\begin{figure}
\includegraphics[width=.45\textwidth]{dallsfig14.eps}
\caption{\label{fig:Fig14} (Color Online) Normalized Mean square curves and eigenvalue density of
states for the triangular lattice with an exponential coupling scheme.  The main plot shows $(\delta_{\mathrm{RMS}}^{n})^{2}$ with
respect to the base ten logarithm of the system size $N$, while inset (a) is the corresponding
graph for $\delta_{\mathrm{RMS}}^{n}$.  Inset (b) shows the density states curve for the triangular lattice
for $2.5 \times 10^{8}$ eigenvalues sampled in the bulk limit via Monte Carlo}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig15.eps}
\caption{\label{fig:Fig15} (Color Online) Density of states curves for $\gamma = 2.0$,
$\gamma = 1.0$, $\gamma = 0.75$, and $\gamma = 0.50$ in panels (a), (b), (c), and (d) respectively
 for the triangular lattice with an exponentially decaying coupling scheme.}
\end{figure}

As for the square lattice geometry, we examine a long-ranged power law interaction in the context of
triangular lattices.  The mean square deviations from equilibrium are graphed in Fig.~\ref{fig:Fig16} with the inset of the plot
showing a closer view of the $(\delta_{\textrm{RMS}}^{n})^{2}$ curves.  A salient question is if lattices with
geometries which differ from those of the square lattice will exhibit long-range crystalline order for the
same range of decay exponents $\alpha$ as in the context of the
square lattice. We find up to the bounds of error $\alpha_{c}^{\mathrm{2D}} = 3.15 \pm 0.025$ calculated for the triangular lattice to be identical
to the threshold exponent for the square lattice.

We show the corresponding eigenvalue histograms for the power law decay
for the decay exponents $\alpha = 5.0$, $\alpha = 4.0$, $\alpha = 3.5$, and $\alpha = 3.125$ in panels
(a), (b), (c), and (d) respectively of Fig.~\ref{fig:Fig17}  While the eigenvalue histograms plotted in the panel (a) and panel (b)
correspond to decay exponents significantly higher than $\alpha_{c}^{\mathrm{2D}}$, the DOS curve in panel (c),
is plotted for a decay exponent only slightly above the threshold value, and the histogram in panel (d) of Fig. corresponds
to a value of $\alpha$ just below (though very nearly equal) to $\alpha_{c}^{\mathrm{2D}}$.
Whereas the DOS curves in panel (a) and (b) clearly tend to a finite value as the eigenvalue approaches zero, the
amplitude for the slower decay $\alpha = 3.125$ and tends to zero in the the limit that the
eigenvalue is very small; for the case $\alpha = 3.5$, the amplitude reaches a finite
but very small value in the zero eigenvalue limit.
 A DOS amplitude tending to zero in the low eigenvalue limit, as seen for $\alpha = 3.125$ is
consistent with the preservation of long-range
crystalline order indicated in the convergence of the mean square deviations graphed in Fig.~\ref{fig:Fig16}.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig16.eps}
\caption{\label{fig:Fig16} (Color Online) The square of the normalized RMS deviations for
various values of the decay exponent $\alpha$, with the inset showing a closer view of the
$(\delta_{\mathrm{RMS}}^{n})^{2}$ curves.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig17.eps}
\caption{\label{fig:Fig17} (Color Online) Eigenvalue histogram curves plotted for $\alpha = 5.0$,
$\alpha = 4.0$, $\alpha = 3.5$, and $\alpha = 3.125$ respectively. The red trace corresponds to
a relatively short truncation radius where $N_{\Delta} = 25$, and the truncation is less drastic for
the blue DOS profiles where $N_{\Delta} = 50$.}
\end{figure}

We examine the honeycomb lattice, which differs from the square in triangular lattices in that it possesses
two inequivalent sites (labeled ``A'' and ``B'' for convenience).
We again appeal to translational invariance, operating in terms of Fourier components, to
decouple the vibrational modes for the honeycomb lattice.
The relationship of sites of type ``A'' and ``B'' to nearest neighbors is
illustrated in Fig.~\ref{fig:Fig20}.
\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig18.eps}
\caption{\label{fig:Fig18} (Color Online) Labeling and indexing scheme for the honeycomb lattice
for inequivalent sites labeled ``A'' and ``B'', and their immediate vicinity.}
\end{figure}

Following this labeling convention, the lattice energy in real space has the form
\begin{align}
E \! = \! \tfrac{K}{2} \! \sum_{i,j=0}^{N-1}  \left \{ \! \! \begin{array}{c}
\left[ \left( \tfrac{\sqrt{3}}{2}\hat{x} + \tfrac{1}{2} \hat{y} \right) \! \cdot \!
\left( \vec{\delta}_{ij}^{\mathrm{A}} - \vec{\delta}_{ij+1}^{\mathrm{B}} \right) \right ]^{2} + \\
 \left[ \left( -\tfrac{\sqrt{3}}{2} \hat{x} + \tfrac{1}{2} \hat{y} \right) \! \cdot \!
\left( \vec{\delta}_{ij}^{\mathrm{A}} - \vec{\delta}_{i-1j+1}^{\mathrm{B}} \right) \right]^{2}
\\ + \left[ -\hat{y} \! \cdot \! \left ( \vec{\delta}_{ij}^{\mathrm{A}} -
\vec{\delta}_{ij}^{\mathrm{B}} \right) \right]^{2}
\end{array} \! \! \! \right \}
\end{align}
where it is sufficient to sum over the three bonds surrounding the atoms labeled ``A''
with no factor of $\frac{1}{2}$ needed to compensate for double counting.
In Fourier space, the energy stored in the lattice has the form
\begin{align}
E = \tfrac{K}{2} \! \sum_{\mathbf{k}} \! \left \{ \! \! \! \begin{array}{c}
\tfrac{3}{2} \left( \lvert \delta_{\mathbf{k}}^{\mathrm{A} x} \rvert^{2} +
\lvert \delta_{\mathbf{k}}^{\mathrm{A} y}
\rvert^{2} + \lvert \delta_{\mathbf{k}}^{\mathrm{B} x} \rvert^{2} +
\lvert \delta_{\mathbf{k}}^{By} \rvert^{2} \right) \\
-\tfrac{1}{4} ( 1 + e^{-ik_{x}} ) e^{ik_{y}} \! \! \left (3 \delta_{\mathbf{k}}^{\mathrm{B} x}
\delta_{\mathbf{k}}^{\mathrm{A} x*} + \delta_{\mathbf{k}}^{\mathrm{B} y} \delta_{\mathbf{k}}^{\mathrm{A} x*}
\right) \\
-\tfrac{1}{4} (1 + e^{ik_{x}} ) e^{-ik_{y}} \! \! \left( 3 \delta_{\mathbf{k}}^{\mathrm{B} x*}
\delta_{\mathbf{k}}^{\mathrm{A} y} + \delta_{\mathbf{k}}^{\mathrm{B} y*} \delta_{\mathbf{k}}^{\mathrm{A} x}
\right)   \\
+ \tfrac{\sqrt{3}}{4} (e^{-ik_{x}} - 1) e^{ik_{y}} \! \! \left( \delta_{\mathbf{k}}^{\mathrm{B} x}
\delta_{\mathbf{k}}^{\mathrm{A} y*} + \delta_{\mathbf{k}}^{\mathrm{B} y} \delta_{\mathbf{k}}^{\mathbf{A} x*}
\right) \\
+ \tfrac{\sqrt{3}}{4} (e^{ik_{x}} - 1) e^{-ik_{y}} \! \! \left( \delta_{\mathbf{k}}^{\mathrm{B} x*}
\delta_{\mathbf{k}}^{\mathrm{A} y} + \delta_{\mathbf{k}}^{\mathrm{B} y*}
\delta_{\mathbf{k}}^{\mathrm{A} x} \right) \\ - \delta_{\mathbf{k}}^{\mathrm{A}y} \delta_{\mathbf{k}}^{\mathrm{B}y*} -
\delta_{\mathbf{k}}^{\mathrm{A} y *} \delta_{\mathbf{k}}^{\mathrm{B} y}
\end{array} \! \! \! \! \! \right \}
\end{align}
In addition to Fourier decomposition, the diagonalization of a $4 \times 4$ matrix
will be necessary to completely decouple the vibrational modes appropriate to the honeycomb lattice
with the nearest neighbor coupling scheme; the matrix in question is
\begin{align}
\left[ \begin{array}{cccc} c_{\mathrm{A}x \mathrm{A}x} & c_{\mathrm{A}x \mathrm{A}y} &
c_{\mathrm{A} x \mathrm{B} x} & c_{\mathrm{A} x \mathrm{B} y}  \\
c_{\mathrm{A} y \mathrm{A} x} & c_{\mathrm{A} y \mathrm{A} y} &
c_{\mathrm{A} y \mathrm{B} x} & c_{\mathrm{A} y \mathrm{B} y} \\
c_{\mathrm{B} x \mathrm{A} x} & c_{\mathrm{B} x \mathrm{A} y} &
c_{\mathrm{B} x \mathrm{B} x} & c_{\mathrm{B} x \mathrm{B} y} \\
c_{\mathrm{B} y \mathrm{A} x} & c_{\mathrm{B} y \mathrm{A} y} &
c_{\mathrm{B} y \mathrm{B} x} & c_{\mathrm{B} y \mathrm{B} y}
\end{array} \right]  = \left[ \begin{array}{cc} \hat{A} & \hat{B} \\ \hat{B}^{\dagger} & \hat{A}
\end{array} \right]
\end{align}
where $\hat{A}$ and $\hat{B}$, and $\hat{B}^{\dagger}$ is the
Hermitian conjugate of $\hat{B}$.  The  sub-matrices $\hat{A}$ and $\hat{B}$ are given by
\begin{align}
\hat{A} = \left [ \begin{array}{cc} \tfrac{3}{2} & 0 \\ 0 & \tfrac{3}{2} \end{array} \right ]
\end{align}
and
\begin{align}
\hat{B} = e^{ik_{y}} \! \left[ \! \! \begin{array}{cc} -\tfrac{3}{4} (1 + e^{-ik_{x}} ) &
\tfrac{\sqrt{3}}{4}  (e^{-ik_{x}} - 1) \\
\tfrac{\sqrt{3}}{4} (e^{-ik_{x}} - 1) & -e^{-ik_{y}} - \tfrac{1}{4}(1 + e^{-ik_{x}})  \end{array} \! \! \right]
\end{align}

However, to obtain a crystal which is locally stiff,
one must examine an extended scheme
where each atomic member interactions with many neighbors.  In real space, the lattice energy may
be expressed as
\begin{align}
&E \! \! =  \! \! \sum_{i,j = 0}^{N}& \\ \nonumber
&\sum_{l,m=-n}^{n} \! \! \left( \! \! \! \begin{array}{c} \tfrac{1}{2} \! K(r_{lm}^{t}) \! \left[ \!
(\hat{\Delta}_{lm}^{(t)x}  \hat{x} \! + \! \hat{\Delta}_{lm}^{(t)y} \hat{y} )  \! \cdot \!  (  \vec{\delta}_{i+lj+m}^{A} \! - \!
\vec{\delta}_{ij}^{A} ) \! \right]^{2} + \\ \tfrac{1}{2} K(r_{lm}^{t}) \! \!
\left[ \! (\hat{\Delta}_{lm}^{(t)x} \hat{x} \! + \! \hat{\Delta}_{lm}^{(t)y} \hat{y} )  \! \cdot \!  (  \vec{\delta}_{i+lj+m}^{B} \! - \!
\vec{\delta}_{ij}^{B} ) \! \right]^{2} + \\ \!  K(r_{lm}^{\mathrm{ab}}) \! \! \left[ \! (\hat{\Delta}_{lm}^{(\mathrm{ab})x} \hat{x} \! + \!
\hat{\Delta}_{lm}^{(\mathrm{ab})y} \hat{y}  )  \! \cdot  \! ( \vec{\delta}_{i+lj+m}^{B} \! - \!
\vec{\delta}_{ij}^{A} ) \! \right]^{2} \end{array} \! \! \! \! \right) &
\end{align}
where the first and second terms in the sum take into account interactions between atoms labeled
``A'' and ``B'', respectively; the identical form of ``A-A'' and ``B-B'' interaction terms is due
to the fact that the ``A'' and ``B'' species both define triangular lattices, as illustrated in Fig.
We take the lattice constant to be unity, and the components of the unit vector $\vec{\hat{\Delta}}^{t}_{lm}$ appropriate to the
triangular sublattices are  $\hat{\Delta}_{lm}^{x(t)} = \sqrt{3}(l + m/2)/r_{lm}^{t}$ and $\hat{\Delta}_{lm}^{y(t)} =(3m/2)/r_{lm}^{t}$, where
$r_{lm}^{t} = (3[l^{2} + m^{2} + lm])^{1/2}$.
On the other hand, the components of $\vec{\hat{\Delta}}^{\mathrm{ab}}_{lm}$ used in calculating interactions between
``A'' and ``B'' atoms are given by
$\hat{\Delta}_{lm}^{x(\mathrm{ab})} = \sqrt{3}(l + m/2)/r^{\mathrm{ab}}$ and
$\hat{\Delta}_{lm}^{y (\mathrm{ab})} = (\tfrac{3}{2} m - 1)/r^{\mathrm{ab}}_{lm}$, where
$r^{\mathrm{ab}}_{lm} = (3l^{2} + 3m^{2} +3lm -3m + 1)^{1/2}$

The exploitation of translational invariance by expressing the displacements in terms of Fourier
components reduces the decoupling of the vibrational modes to the diagonalization of the $4 \times 4$ matrix $\left[ \begin{array}{cc}
\hat{A} & \hat{B} \\ \hat{B}^{\dagger} & \hat{A} \end{array} \right]$,
where the sub-matrices are given by
\begin{align}
&\hat{A} = \sum_{l,m=-n}^{n}  g_{lm}
 \left[ \! \! \begin{array}{cc} \hat{\Delta}^{x (t)}_{lm}
\hat{\Delta}^{x (t)}_{lm} & \hat{\Delta}^{x (t)}_{lm} \hat{\Delta}^{y (t)}_{lm} \\
\hat{\Delta}^{x (t)}_{lm} \hat{\Delta}^{y (t)}_{lm} & \hat{\Delta}^{y (t)}_{lm} \hat{\Delta}^{y (t)}_{lm} \end{array} \! \! \right ]&
\\ \nonumber
&+ K(r_{lm}^{\mathrm{ab}}) \left[ \begin{array}{cc} \hat{\Delta}_{lm}^{x(\mathrm{ab})} \hat{\Delta}_{lm}^{x(\mathrm{ab})}
& \hat{\Delta}_{lm}^{x(\mathrm{ab})} \hat{\Delta}_{lm}^{y(\mathrm{ab})} \\
\hat{\Delta}_{lm}^{y(\mathrm{ab})} \hat{\Delta}_{lm}^{x(\mathrm{ab})}
& \hat{\Delta}_{lm}^{y(\mathrm{ab})} \hat{\Delta}_{lm}^{y(\mathrm{ab})} \end{array}\right]&
\end{align}
for $\hat{A}$ and
\begin{align}
\hat{B} \!  = \! \! \! \! \sum_{l,m=-n}^{n} \! \! e^{I(k_{x}l + k_{y}m)} \! K(r_{lm}^{\mathrm{ab}}) \! \!
\left[ \! \! \! \begin{array}{cc} \hat{\Delta}_{lm}^{x(\mathrm{ab})} \hat{\Delta}_{lm}^{x(\mathrm{ab})}
& \hat{\Delta}_{lm}^{x(\mathrm{ab})} \hat{\Delta}_{lm}^{y(\mathrm{ab})} \\
\hat{\Delta}_{lm}^{y(\mathrm{ab})} \hat{\Delta}_{lm}^{x(\mathrm{ab})}
& \hat{\Delta}_{lm}^{y(\mathrm{ab})} \hat{\Delta}_{lm}^{y(\mathrm{ab})}\end{array} \! \! \! \right]
\end{align}
where $g_{lm} \equiv (1 - \cos[k_{x} l + k_{y}m]) K(r_{lm}^{t})$, and
$g_{00} = 0$ to exclude self-interactions in the ``A-A'' and the ``B-B'' coupled pairs.

\begin{figure}
\includegraphics[width=.35\textwidth]{dallsfig19.eps}
\caption{\label{fig:Fig19} (Color Online) Honeycomb lattice geometry showing the
interpenetrating triangular lattices defined by the inequivalent sites labeled
``A'' and ``B''.}
\end{figure}

As we did for the square and triangular lattices, we examine a short-ranged exponential
interaction between atoms in the lattice with a length scale given by $\gamma^{-1}$.
On the other hand, as for the preceding two lattice geometries, we also consider a
long-ranged power law decay with $K(r) = K_{0} r^{-\alpha}$.  The $\delta_{\mathrm{RMS}}^{n}$
results for the mean square deviations for the exponential decay scheme are shown in Fig.~\ref{fig:Fig20} for a range of $\gamma$ values.
As in the cases of the square and triangular lattices in the extended schemes, increasing the range $\gamma^{-1}$
of the coupling slows (but does not halt) the rate of divergence of the mean square fluctuations.  The large separation
between the RMS curves corresponding to $\gamma = 2.0$ and the slower decays $\gamma = 1.0$, $\gamma = 0.75$, and $\gamma = 0.50$ is
consistent with changes in the density of states curves where the histogram amplitude in the low eigenvalue regime is
sharply diminished as $\gamma$ decreases from $\gamma = 2.0$ to $\gamma = 1.0$.

In the case of a power law decay, results for the mean square deviations are shown in the semi-logarithmic graphs in Fig.~\ref{fig:Fig23}
where the systems sizes considered span two decades,
with a closer view for a more restricted range of the decay exponent $\alpha$ in the main graph; the inset is a graph of RMS curves
for a broader set of $\alpha$ values.  As in the cases of the square and triangular lattices, the threshold exponent
$\alpha_{c}^{\mathrm{2D}}$ is determined by examining whether the RMS curves converge or diverge for very large system sizes.
In agreement with the square and triangular geometries, we find $\alpha_{c}^{\mathrm{2D}} = 3.15 \pm 0.025$ for the
critical decay exponent.

The eigenvalue histogram curves shown in Fig.~\ref{fig:Fig23} are consistent with the behavior of the mean square deviation
curves given in Fig.~\ref{fig:Fig22}.  For the more rapid decays $\alpha = 4.0$ and $\alpha = 4.5$, the amplitude of the
density of states is finite, which eventually contributes to a divergence in $\delta_{\mathrm{RMS}}^{n}$.
The divergence in the mean square fluctuations is much slower for $\alpha = 3.5$, a characteristic which is echoed in the
eigenvalue histogram in panel (c) of Fig.~\ref{fig:Fig23}, where in the zero eigenvalue limit the histogram amplitude is finite
but very small.  Finally, for $\alpha = 3.125$, just below $\alpha_{c}^{2D}$, the density of states curve tends to zero in the low
eigenvalue limit.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig20.eps}
\caption{\label{fig:Fig20} (Color Online) Mean square deviations for the honeycomb lattice for an
extended coupling scheme where the interaction decays exponentially.  The main graph is a semi-logarithmic plot of
$(\delta_{\mathrm{RMS}}^{n})^{2}$ for various values of the decay constant $\gamma$, while the inset is the corresponding
plot of $\delta_{\mathrm{rm}}^{n}$.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig21.eps}
\caption{\label{fig:Fig21} (Color Online) Density of states curves in the case of the
honeycomb lattice for $\gamma = 2.0$,
$\gamma = 1.0$, $\gamma = 0.75$, and $\gamma = 0.50$ in panels (a), (b), (c), and (d) respectively.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig22.eps}
\caption{\label{fig:Fig22} (Color Online) Mean square deviations for long-range interactions with a power law decay
exponent $\alpha$ for the honeycomb lattice geometry. The inset shows a relatively broad range of $\alpha$ values, whereas the
main graph is a closer view.}
\end{figure}

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig23.eps}
\caption{\label{fig:Fig23} (Color Online) Density of states curves in the case of the
honeycomb lattice with a power law coupling scheme for $\alpha = 4.5$,
$\alpha = 4.0$, $\alpha = 3.5$, and $\alpha = 3.125$ in panels (a), (b), (c), and (d) respectively.}
\end{figure}

\section{Transverse Displacements in an extended scheme}

In each of the preceding cases discussed in this work, thermally induced deviations of the
lattice sites have been confined either to motion withing the lattice plane in the context of two
dimensional systems, or collinear movements for the lattice in 1D.  However, we also consider
displacements perpendicular to the lattice for two dimensional systems.   To provide local stiffness
of covalent two dimensional lattices realized in nature where the finite
thickness would provide rigidity with respect to transverse displacements tending to push atoms above
or below the plane of the crystal, we examine a dual layer geometry where an extended scheme
provides a local stiffness.

In fact, to provide as much stability as possible, we consider a long-range coupling scheme in
the interaction between atoms within a layer as well as between layers decreases as a power law
(with the decay exponent designated $\alpha$) in the
separation between atomic species.

The potential energy stored in the lattice is similar in abstract form to the corresponding expression for the
honeycomb lattices, and is given by
\begin{align}
E \! = \! \! \sum_{i,j = 0}^{N} \sum_{l,m=-n}^{n} \! \! \left( \! \! \! \begin{array}{c} \tfrac{1}{2} K(r_{lm}^{s})\! \left[ \!
\vec{\hat{\Delta}}_{lm}^{s}  \! \cdot \! (\vec{\delta}_{i+lj+m}^{A} \! - \!
\vec{\delta}_{ij}^{A} ) \! \right]^{2} \\ + \tfrac{1}{2} K(r_{lm}^{s}) \!
\left[ \! \vec{\hat{\Delta}}_{lm}^{s} \! \cdot \! (\vec{\delta}_{i+lj+m}^{B} \! - \!
\vec{\delta}_{ij}^{B} ) \! \right]^{2} \\ + \! \! K(r_{lm}^{\mathrm{ab}}) \left[ \! \vec{\hat{\Delta}}_{lm}^{\mathrm{ab}}
 \! \cdot \! (\vec{\delta}_{i+lj+m}^{B} \! - \!
\vec{\delta}_{ij}^{A} ) \! \right]^{2} \end{array} \! \! \! \right)
\end{align}
in a real space representation, where the unit vectors corresponding the intraplanar couplings $\vec{\hat{\Delta}}_{lm}^{s}$ have
$x$ and $y$ components given by
\begin{align}
\hat{\Delta}_{lm}^{x(s)} = \frac{l}{r_{lm}^{s}};~~\hat{\Delta}_{lm}^{y(s)} = \frac{m}{r_{lm}^{s}},
\end{align}
where the separation between interacting sites within a plane is $r_{lm}^{s} = \sqrt{l^{2} + m^{2}}$
The components of the unit vector $\vec{\hat{\Delta}}_{lm}^{\mathrm{ab}}$
are identical to those of the planar case with the exception of a nonzero $z$ component $\hat{\Delta}_{lm}^{z(\mathrm{ab})} =
1/r_{lm}^{\mathrm{ab}}$ where the distance between sites in the two distinct lattice planes is $r_{lm}^{\mathrm{ab}} = \sqrt{m^{2} + l^{2} + 1}$.
 Operating in terms of Fourier components reduces the
decoupling of the vibrational modes to the diagonalization of a $6 \times 6$ matrix of the form
where $\hat{A}$ and $\hat{B}$ are $3 \times 3$ sub-matrices, with
$\hat{B}^{\dagger}$ the Hermitian conjugate of $\hat{B}$. The sub-matrices have the form
\begin{align}
\hat{A} = \! \! \! \sum_{l,m=-n}^{n} \! \! \left[ \! \! \! \begin{array}{ccc} (d_{lm}^{xx(\mathrm{ab})} + d_{lm}^{xx(s)}) & (d_{lm}^{xy(\mathrm{ab})} +
d_{lm}^{xy(s)}) & d_{lm}^{xz(\mathrm{ab})} \\
(d_{lm}^{yx(\mathrm{ab})} + d_{lm}^{yx(s)}) & (d_{lm}^{yy(\mathrm{ab})} + d_{lm}^{yy(s)}) & d_{lm}^{yz(\mathrm{ab})} \\
d_{lm}^{zx(\mathrm{ab}} & d_{lm}^{zy(\mathrm{ab}} & d_{lm}^{zz(\mathrm{ab})}
\end{array} \! \! \! \right ]
\end{align}
for $\hat{A}$, where for the sake of brevity we have used the notation, e.g.  $d_{lm}^{xy(\mathrm{ab})} \equiv K(r_{lm}^{\mathrm{ab}})
\hat{\Delta}_{lm}^{x(\mathrm{ab})}
\hat{\Delta}_{lm}^{y(\mathrm{ab})}$ and $d_{lm}^{xy(s)} \equiv K(r_{lm}^{s}) (1 - \cos [k_{x} l + k_{y} m])
\hat{\Delta}_{lm}^{x(s)} \hat{\Delta}_{lm}^{y(s)}$.
The complex sub-matrix $\hat{B}$ is given by
\begin{align}
\hat{B} = \! \! \sum_{l,m=-n}^{n} e^{i(k_{x}l + k_{y} m)}
\left[ \! \! \begin{array}{ccc} d_{lm}^{xx(\mathrm{ab})}  & d_{lm}^{xy(\mathrm{ab})} & d_{lm}^{xz(\mathrm{ab})} \\
d_{lm}^{yx(\mathrm{ab})}  & d_{lm}^{yy(\mathrm{ab})} & d_{lm}^{yz(\mathrm{ab})} \\
d_{lm}^{zx(\mathrm{ab})}  & d_{lm}^{zy(\mathrm{ab})} & d_{lm}^{zz(\mathrm{ab})}
 \end{array} \! \! \right ]
\end{align}

Results for the mean square deviation $\delta_{\mathrm{RMS}}^{n}$ are shown in Fig.~\ref{fig:Fig24} for a variety of
decay exponents ranging from a weak decay $\alpha = -2.5$ to a considerably more rapid decay with
the separation distance $\alpha = -6.0$.  A salient feature in each of the RMS curves
shown in the graph is an asymptotic linear growth
with the size $N$ of the system, though the slope of the linear diverge in system size decreases
with decreasing $\alpha$.  Broadly speaking, there are two regimes for each value of $\alpha$ in the
variation of the mean square deviation with system size.  For relatively small system sizes,
$\delta_{\mathrm{RMS}}$ changes quite slowly with increasing $N$.  However, ultimately the RMS
fluctuations begin to grow more rapidly and eventually   The size of the plateau where the mean square
fluctuations expand slowly is slightly broader for small values of the decay exponent $\alpha$, and
somewhat abbreviated for the more rapidly decaying coupling where $\alpha = -6$.  The latter is
closer to what one would find in covalently bonded (but non-polar) systems such as graphene where
Van der Waals interactions decreasing with the sixth power of the inter-atomic separation
constitute the
main source of long-range attraction between particles.  Hence, London interactions would not
prevent atomic displacements transverse to the lattice plane from destroying
crystalline order.

\begin{figure}
\includegraphics[width=.49\textwidth]{dallsfig24.eps}
\caption{\label{fig:Fig24} (Color Online) Mean square deviations for long-range interactions with a power law decay
exponent $\alpha$.}
\end{figure}

\section{Conclusions}
We have examined the effect of thermally induced lattice vibrations on long range order in
one and two dimensional crystals.
In the case of crystals in one dimension, long-range positional order is (as expected) disrupted by
thermal fluctuations with the $\delta_{\mathrm{RMS}}^{n}$ scaling as $N^{1/2}$.  For inherently
long-ranged interactions scaling as power laws in the distance between interacting atoms,
the divergence is much slower for $\alpha > \alpha_{c}^{\mathrm{1D}} = 1.615$, while crystalline order is
intact even at finite temperatures if $\alpha < \alpha_{c}^{\mathrm{1D}}$.

For two dimensional crystals, we find the same essential phenomena
with respect to thermodynamic stability of the crystal for square, triangular, and
honeycomb lattices.  For the latter two, thermal fluctuations destroy long-range crystalline
order at finite temperatures, but the divergence in $\delta_{\mathrm{RMS}}^{n}$ occurs very
slowly with increasing system size.  On the other hand, RMS deviations decay rapidly for
simple square lattices where only nearest neighbor couplings are active.  However, an extended
coupling scheme to both nearest and next-nearest neighbors considerably mitigates the effect
of thermal fluctuations on crystalline order, and the resulting slow divergence is
quantitatively similar to that seen in the triangular and honeycomb lattices where the coupling
is confined entirely to nearest-neighbors.

When we extend the coupling to many neighbors, but still implement a short-ranged coupling
(e.g. an exponential decay with the range set by the inverse decay rate $\gamma^{-1}$),
we find qualitatively the same results as for the square lattice with couplings both to
nearest and next-nearest neighbors, as well as the triangular and hexagonal lattices in
atomic members only interact with nearest neighbors with
$(\delta_{\mathrm{RMS}}^{n})^{2}$ scaling linearly with $\log_{10}(N)$.  However, the
slope of the linear dependence
becomes smaller as $\gamma$ is decreased, as a longer-ranged coupling is more effective at
suppressing the effects of thermal fluctuations.
As in the case of systems in 1D, a longer range coupling in the form of a power law
can maintain long-range crystalline order for $T > 0$ if the decay exponent does not
exceed $\alpha_{c}^{\mathrm{2D}} = 3.15$.

Allowing thermally induced fluctuations perpendicular to the lattice causes a rapid divergence
(i.e. linear) of the RMS deviations with the system size, even if the lattice geometry is dual-layered
in an extended scheme to provide local stiffness.
The growth of $\delta_{\mathrm{rms}}^{n}$ is asymptotically linear even if the coupling between sites is
long-ranged, decaying as a power law.  Hence, long-ranged London interactions would not be enough by themselves
to preserve positional order in a covalently bonded locally rigid two dimensional lattice.

\begin{acknowledgments}
Useful conversations with Yogesh Joglekar are gratefully acknowledged.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Hello! We are Emma and Shanice, two 2nd year Music Business and Arts Management students at Middlesex University. We are organising our first music event at The Slaughtered Lamb in Clerkenwell, as part of the charity Youth Music's, Give a Gig initiative. This will take place on Sunday, 24th March 2019.
Sunday Sessions Live, is a music event working with young musicians in order to showcase their talent at The Slaughtered Lamb, a popular pub and home to live music in buzzy Clerkenwell.
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We will be giving regular updates on our project via our Facebook page and we will also let you know how successful the event was and how much we raised for Youth Music the week after it takes place.
THANK YOU AND PLEASE HELP SPREAD THE WORD!
We would be very grateful for any donation that you are able to make to our crowdfunding campaign and/or your help sharing our crowdfunding page with anyone you feel may want to donate or attend the event itself. Your support means a lot to us (we really appreciate it!) and we are excited to be organising Sunday Sessions Live in aid of Youth Music.
A personalised thank you and shout out on our Facebook page!
A FREE Ticket to the Event! + A personalised thank you and shout out on our Facebook page!
A FREE ticket to the event, and a complimentary drink! + A personalised thank you and shout out on our Facebook page!


text:
How Much Does the White House Correspondents' Dinner Actually Raise for Scholarships?
https://money.com/white-house-correspondents-dinner-scholarship-fundraiser/
President Barack Obama, right, brings out actor Keegan-Michael Key from Key & Peele to play the part of "Luther, President Obama's anger translator" during his remarks at the White House Correspondents' Association dinner at the Washington Hilton last year.
Evan Vucci—AP
Comedian Larry Wilmore will take the stage at the Washington Hilton on Saturday night to roast President Obama in front of a powerful, wealthy crowd of celebrities, lobbyists, politicians, and media elite.
Lost in the levity may be a major reason the annual White House Correspondents' Dinner is supposedly held at all: to raise money for college scholarships.
As the event has grown in glitz and glamour, its role as a fundraiser to support aspiring journalists has faded.
The dinner, hosted by the White House Correspondents' Association (WHCA), spent nearly 60% of its revenue on scholarships in 2009, according to an audit of its financial records by the Washingtonian. That dropped to just 26% in 2013, and then fell again in 2014 to 21.5%, the Washingtonian found.
This year, the association will award about $77,500 to 18 students, the WHCA announced last week. Most awards are worth $5,000—not an insignificant sum for an individual college student. But that's small change compared to the incredible amount of money spent on and around what's been dubbed D.C.'s Nerd Prom.
There are 260 dinner tables (with 10 seats each) for sale at $3,000 per table, for an effective price of $300 a ticket. Guide Star reports the WHCA's total revenue from contributions in 2014 was $445,565. The Washington Post estimates the association's annual revenue at about $600,000, and says most of that money comes from the dinner.
And the dinner is now just a small part of the hoopla around Nerd Prom. There are dozens of events planned through Sunday (though not by the WHCA), from brunches to panels to after parties at undisclosed locations. C-SPAN will be on the Red Carpet. "Funny or Die" and Vanity Fair will host parties. Celebrities like Kendall Jenner, Will Smith, and Rachel McAdams will clink glasses with wealthy socialites, cabinet members, politicians, and even a few of the reporters for whom the event is named.
After being called out for its small-by-comparison scholarship activities, the association has defended itself by saying that it is not a scholarship organization. Instead, its primary mission is to advocate for openness and transparency at the White House by bargaining for access to events the press pool, for example. The WHCA also told the Washingtonian that it recently started putting some of its scholarship money aside in an endowment to fund future awards, which explains the decrease in current award dollars.
That very well may be, yet the association states on its homepage that proceeds of the dinner "go toward scholarships and awards that recognize aspiring and accomplished journalists."
No doubt hosting D.C.'s version of the Oscars isn't cheap. But if the total money awarded in scholarships is just a fraction of the revenue brought in, we have to wonder: Is the event truly a scholarship fundraiser? Or is it just another fancy dinner for the Washington elite, with college students getting the crumbs?

redpajama_set_name: RedPajamaCommonCrawl


Victoria Falls is the world's largest waterfall. It is bordered by the Mosi-O-Tunya National Park on the Zambian side and Victoria Falls National Park in Zimbabwe. The towns of Victoria Falls (ZIM) and Livingstone (ZAM) are quaint and historic. The 'Falls' are also very close to Chobe National Park, a perfect excursion into Botswana. Livingstone is also the home of the Royal Livingstone Express, a fine dining on rails experience that is not to be missed. Victoria Falls is the natural add on to a Zambian Safari or a perfect long weekend escape from South Africa.
Lovingly called 'VicFalls' the name evokes images of old style African safaris, hats, porters, gin and tonics and black and white etchings of the natural wonders of this continent…. Victoria Falls is also called 'the warm heart of Africa' – and when looking at a map of this continent, it does seem to be almost right in the middle.
Victoria Falls is a unique spot to visit. Surprisingly easy to reach, it offers the visitor stunning, dramatic landscapes, up close encounters with animals, a chance to experience some thrilling adventurous activities and has easy access to other destinations in Africa. As a choice for a safari trip, it is a perfect destination to include it all. We specialise in tailor made packages including Mosi-Oa-Tunya National Park Safaris, riverside lodges, Zambian culture and onward travel.
We offer the world famous Royal Livingstone Express, an evening safari through the bush and onto the bridge including a 5star dinner buffet, in the grand old style of Rhodes and his contemporaries.
The actual waterfalls of the Victoria Falls divide the nations of Zimbabwe and Zambia. Both have formed settlements on either side of Victoria Falls and her Zambezi river banks and both make a living out of tourism, wildlife, international visitors and the trade that passes the bridge between the towns and nations.
In Zambia, the settlement on the edge of the Zambezi river is called Livingstone – after Dr David Livingstone who first named these falls the Victoria Falls in 1855. Livingstone is a quaint colonial town where development seems to have stopped in 1964. It is also home to some excellent lodges on the Zambian side of the Zambezi river. From your lodge you can take river safaris up the Zambezi and see crocodiles and hippos from your steamer, or make a foray into Chobe National Park in Botswana or explore the Mosi-Oa-Tunya National Park.
Zambian Safaris in the south of the nation are not as high brow as some of their Tanzanian equivalents, but there is a real feeling of being in Africa here.
We specialise in tailor made Southern Zambian Safaris. You can combine these with a safari in South Luangwa National Park if needed. With a mix of attractions Zambia has so much to offer the visitor.
The Zambezi is the name of the river that runs roughly north to south and forms the border between Zambia and Zimbabwe. At the chasm in the earth called the Batoka Gorge, it drops over a rock curtain and forms the falls that are called The Victoria Falls. By then, this enormous river has traversed and touched upon the lands of Angola, Namibia and Botswana. After the Victoria Falls, it cuts through Mozambique before finally flowing into the ocean.
In Zambia, the river Zambezi runs over a plateau of basalt, with softer weaker sandstone flats. This is where 'cracks' have appeared over the course of thousands of years – the famous gorges that are found upstream of the now Victoria Falls. This is now called the 'First' Gorge. In reality, this spot where the falls thunder over the edge these days, is simply the latest enormous crack in the earth in a series of switchback gorges; all zigzagging through a flat scrubby landscape.
The name Mosi-Oa-Tunya then is the local Bantu name referring to the actual Victoria Falls but is now also the name of the National Park that stretches from the edge of the Falls (the first Gorge), to the town of Livingstone and encloses the banks of the Zambezi river for 20 kilometres upstream, on the Zambian side of the river. This park has 2 sections: the falls itself, and the wildlife part. Both are interlinked, and intrinsically depending on each other. The flow of the river towards the fall feeds and supplies hundreds of square miles of arid bushland with water, food, nutrients, harbouring migrating animal species and even supports a miniature climate system.
The name Victoria Falls covers a number of things – the bridge, the falls, the town – but let's start with the most eye catching human addition to this place of natural African beauty – the bridge over the Falls.
The bridge connecting what is now Zambia and Zimbabwe was constructed in England by the Cleveland Bridge & Engineering Company. It was then shipped in its entirety, to the port of Beira, Mocambique and then transported on the newly constructed railway to the Victoria Falls. It took just 14 months to construct and was completed in situ in 1905.
The bridge was the brainchild of Cecil Rhodes, (whose name was used to claim the colonies of North and South Rhodesia) and was a vital part of his grand, never-to-be-realised Cape to Cairo railway scheme. Rhodes is recorded as instructing the engineers to "build the bridge across the Zambezi where the trains, as they pass, will catch the spray of the Falls' – and it does indeed.
As with all natural phenomena and especially those around the equator – the seasons are a strong influence on life and nature. The rainy season occurs from late November to April whereas the rest of the year is usually dry. In those months the heat can become intense and downright brutal at the end, just before the rains start. The 'peak', also known as the flood season, usually starts from February to May with a peak in its volume around April. The water needs to run the full course of the river, before flooding at the lower banks and forming the Falls at the First Gorge.
During the flood season, the spray caused by the falls that is so famous, can reach as high as 400 meters (1300 feet) and can be seen as far away as 48 kilometres! However, when the thundering volume of water is so much, visitors can no longer see the foot of the falls, the façade along the gorges and sometimes, there is so much water that some of the paths and the bridges are closed for safety reasons. At such a time, the curtain of water would form a permanent mist, and at full moon, visitors can experience a moon-bow.
The Victoria Falls are the largest waterfalls in the world. This definition and label is based on the combination of the width (1708 metres wide) and height (108 metres high) of the Falls. A staggering average of 1.088 cubic metres per second (that means 1.088 tonnes per second) falls over the edge at Victoria Falls. Walking around, you can feel the ground shake with the power of this force and it is hard to imagine the sounds, rage and vibrancy of this display of nature.
With these numbers, Victoria Falls is twice as high as Niagara Falls in America. Only the Iguaza Falls in Argentina / Brazil are larger than this.
In sharp contrast to the power and majesty of the wet season and the flooding season, stands the dry season. Now, only 300 cubic metres of water per second flows over the edge. The banks of the river go dry, animals can wade into the water to drink without fear of being swept over the edge and boat cruises have to go further upstream to be able to move. The heat gets more intense, and the tropical rain forest that grows abundantly on the gorge cliffs, has retracted – the spray is no longer enough, and the plants and trees struggle on till the next wet season.
Visitors can now, at certain points, even wade through the river to Livingstone Island and swim in small pools on the edge of the highest falls of the world. While the water volume has gone down, in this season, it is possible to see the entire façade (the curtain) where the water would otherwise fall from – the full width of 1088 metres is visible, and the depth of the falls can be seen as the raging river runs below the bridge and footpaths. At this time, rafting tours are on offer, and wildlife boat cruises are a must to do!
Mosi-oa-Tunya National Park is 66 square kilometres. The Victoria Falls take up about 23 square kilometres and that leaves about 44 square kilometres dedicated to animal and wildlife protection – this is stretched out for 20 km upstream of the Zambezi river. On the other side of the river, in Zimbabwe, the banks and bushland are also part of National Parks, which means that animals can move freely between both the banks (west and east), and move with the flow of the water running north and south in search of food and shelter.
On the Zambian side, fences and the outskirts of Livingstone keep most of the animals inside the Mosi-oa-Tunya National Park. But animals are often seen roaming some of the lush lawns of the lodges immediately near the Falls when the water runs at a minimum and many people are lucky enough to see an impromptu elephant standing by the side of the road, eating a tree – having wandered a little too far.
The fauna immediately next to the Falls itself, is a typical tropical rainforest – the huge amount of water and year round moist air allows a unique flora to flourish here: this small area contains plants that are very rare for the area such as pod mahogany, ebony, ivory palm, wild date palm and a number of creepers and lianas.
Moving only a 200 mtrs away from this source of water, you will see the typical sub equator bushland appear: mopane woodland savannah predominates in this area, with patches of miombo and Rhodesian teak woodland, mixed with scrubland savannah.
The two parts of this tiny National Park contains abundant wildlife and offers a fantastic daytour to spend on viewing big game. The back drop of the river, the constant roar of the Falls during the wet season and the scent of wet air in a dry bushland all add to the excitement of this place.
Animals that can be spotted in the Mosi Oa Tunya Park include small herds of elephant, buffalo, giraffe, Grant's zebra, and a variety of antelope. Some resident prides of lions, African leopards and South African cheetahs are known – but only very occasionally seen. Vervet monkeys and baboons are common.
While on a cruise on the river or even while sitting on the lawn of a river lodge – pods of hippopotami and stealthy crocodiles can easily be spotted. The river above the Falls contains large populations of hippopotamus and crocodile and African bush elephants cross the river in the dry season at particular crossing points to avoid these hungry predators.
An interesting set of inhabitants of the Mosi Oa Tunya wildlife park, are the white rhinos that are living in the park. These are not native to this region, not even to Zambia – Zambia formerly only had black rhinos but they were hunted to extinction. The white rhino living in the park have been introduced from South Africa as Zambia has shown so much promise in protecting these wonderful creatures.
A special note on Avifauna: due to such an incredible diversity of ecosystems and available food in such a small area, the list of bird species that can be spotted is impressive- the gorges of the Falls are known to sustain a staggering 35 species of raptors alone. A few of these are the Taita falcon, black eagle, peregrine falcon and augur buzzard. These large birds of prey all breed in the Gorges itself. In the upstream region (so above the falls) groups of herons, fish eagles and numerous kinds of waterfowl are commonly seen. Kingfishers are a delight to spot and the whole family can be ticked off. Every morning and afternoon, the distinctive cry of the African Fish Eagle can be heard – this majestic bird and its call are one thing that will surely stay with you while visiting the largest falls in the world – The Victoria Falls in Zambia.


It was a mixed bag of cattle at the Finley sale, with not a lot of weight across the yard, although there was a small run of excellent supplementary fed yearlings and a small number of vealers. The balance of the yard was store conditioned cattle.
A few local producers were active on the lighter weighted stock, as feed lotters picked through the better bred lines of heavier yearlings. Prices lifted on all finished cattle to the trade and cow prices increased by 20c to 40c/kg on a fortnight ago. The best of the B muscled supplementary fed vealers made to 330c/kg.
The B and C muscled supplementary fed yearling steers sold from 295c to 330c/kg.
The feeder steer portion over 330kg lwt made from 210c to 257c/kg. The yearling heifers supplementary fed made to 328c/kg.
The 400-500kg steers reached 302c/kg. The 500-600kg steers made to 274c/kg. Only 96 cows were offered, with the C4 beef cows making to 212c/kg. The heavy Friesian cows made to 190c/kg.


text:
<?php

declare(strict_types=1);

namespace Ray\Compiler;

use Ray\Di\AbstractModule;
use Ray\Di\Scope;

class FakeCarModule extends AbstractModule
{
    protected function configure()
    {
        $this->bind(FakeCarInterface::class)->to(FakeCar::class); // dependent
        $this->bind(FakeEngineInterface::class)->to(FakeEngine::class); // constructor
        $this->bind(FakeHardtopInterface::class)->to(FakeHardtop::class); // optional setter
        $this->bind(FakeTyreInterface::class)->to(FakeTyre::class); // setter
        $this->bind(FakeMirrorInterface::class)->annotatedWith('right')->to(FakeMirrorRight::class)->in(Scope::SINGLETON); // named binding
        $this->bind(FakeMirrorInterface::class)->annotatedWith('left')->to(FakeMirrorLeft::class)->in(Scope::SINGLETON); // named binding
        $this->bind('')->annotatedWith('logo')->toInstance('momo');
        $this->bind(FakeHandleInterface::class)->toProvider(FakeHandleProvider::class);
        $this->bindInterceptor(
            $this->matcher->any(),
            $this->matcher->any(),
            [FakeInterceptor::class]
        );
        $this->bind(FakeCar::class);
        $this->bind(FakeRobot::class);
    }
}

redpajama_set_name: RedPajamaGithub