\section{Results with Corrugated Substrate}
\label{sec:sqr3}

In this section we present results obtained by using a more
realistic model of the graphite substrate.  Most of the 
results shown in this section were obtained with a simulation
cell with the dimensions $25.56\AA \times 22.14 \AA$.  The commensurate
density corresponds to 36 helium atoms.  Periodic boundary
conditions were used in the plane of the substrate.  Finite-size effects
were examined by repeating some calculations using a $34.08 \AA \times
29.51 \AA$ simulation cell, for which the commensurate density 
corresponds to 64 particles.

\subsection{At And Below Commensurate Density}

%%%%%%%%%%%%%%
\begin{figure}[ht]
%\centerline{\psfig{figure=distrib636n36.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{distrib636n36.ps}}
\caption{Distribution plot at the commensurate density, 0.0636 atom/$\AA^2$,
for  T=2.99 K and T=4.0 K.  The filled circles indicate graphite
adsorption sites.}
\label{fig:dist636}
\end{figure}
%%%%%%%%%%%%%%

The principal conclusion we obtain from these calculations is that the low
density, low temperature phase of the first layer consists of commensurate solid
clusters, rather than a liquid phase.  In  an earlier publication\cite{pierce99b} 
we observed the commensurate phase and investigated its melting
with both static structure and specific heat calculations.
The calculated $\sqrt 3 \times \sqrt 3$ commensurate solid phase 
and its melt are shown in Fig.\ \ref{fig:dist636}.  At 3 K,
the film has solidified into the commensurate structure.  
The solid forms a sublattice that contains
one-third of the adsorption sites.  The remaining two-thirds of the sites
form two equivalent sublattices that are unoccupied.
Raising the temperature to 4 K causes 
melting.  At this temperature the helium atoms no longer inhabit a single
sublattice of substrate adsorption sites.  All adsorption sites will be
occupied with equal probability if the simulation is run for a sufficiently
long time.

%%%%%%%%%%%%%%
\begin{figure}[ht]
%\centerline{\psfig{figure=distrib566.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{distrib566.ps}}
\caption{Probability distributions for 0.0566 atom/$\AA^2$ at
T=1.0 (top) and 2.5 K (bottom).
The filled circles give the locations of graphite potential minima.
}
\label{fig:distrib566}
\end{figure}
%%%%%%%%%%%%%%

%%%%%%%%%%%%%%
\begin{figure}[ht]
\epsfxsize=\figwidth\centerline{\epsffile{distrib530.ps}}
\caption{Distribution plot at 0.0530 atom/$\AA^2$, T=2.5 K 
and T=3.0 K}
\label{fig:dist530}
\end{figure}
%%%%%%%%%%%%%%

%%%%%%%%%%%%%%
\begin{figure}[ht]
\epsfxsize=\figwidth\centerline{\epsffile{distrib424.ps}}
\caption{Distribution plot at 0.0424 atom/$\AA^2$, T=1.0 K 
and T=2.0 K}
\label{fig:dist424}
\end{figure}
%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\caption{Estimates of melting temperatures from the temperature
dependence of the static structure peaks.}
\vspace{.2in}
\begin{tabular}{|c|c|} \hline
Coverage (atom/$\AA^2$) & T$_{melt}$ \\ \hline
0.0424 & 2.0 \\
0.0530 & 2.5 \\
0.0566 & 3.0 \\
0.0636 & 3.33 \\ \hline
\end{tabular}
\label{tab:melt}
\end{center}
\end{table}
%%%%%%%%%%%%%%

Next, we present direct evidence that the low density (below the cimmensurate solid 
density of 0.0634 atom/$\AA^2$) first layer consists of a solid cluster surrounded 
by a low density vapor at low temperatures. No liquid phase forms and so there 
is no possibility for first layer superfluidity.  These findings are in contrast 
to the most recent proposal for this region\cite{greywall91}.

The presence of a solid with vacancies and phase separation can be visualized
with contour plots of the probability distributions, shown 
in Fig.\ \ref{fig:distrib566} for the indicated coverage and
temperatures.  At the lower temperature, 1.0 K, the vacancies have condensed
into a single bubble region, as can be seen in the top plot of 
Fig.\ \ref{fig:distrib566}.  Note that periodic boundary conditions are
being used, so the vacancy regions in the figure are 
connected.  The vacancies move very slowly at this temperature; we
found equilibration times to be very long for the vacancies to
condense if they were initially 
spread throughout the lattice.  We thus calculated the
energy for this system twice:  first with the vacancies initially spread 
through the solid and then with the 
vacancies condensed.  The condensed energy was found to be lower.  
At higher temperatures, the vacancies acquire enough kinetic energy
to leave the phase separated state and diffuse into the solid.  
As a result, vacancies can
become isolated.  This is illustrated at 2.5 K in the lower plot
of Fig.\ \ref{fig:distrib566}.  A series of probability distribution plots 
reveals that these vacancies move in the simulation, so the equilibration
problem encountered at 1.0 K is not present at this temperature.
We note that we still see evidence of phase separation in contour plots
at 2.0 K, while heat capacity peaks from experiment seem to indicate
a transition at 1.5 K.  


Figures \ref{fig:dist530} and \ref{fig:dist424} further confirm
that the low density region contains solid clusters at low
temperatures and that these clusters exhibit
the melting behavior discussed in the previous section.  We have calculated
distribution plots for densities as low as 0.0207 atom/$\AA^2$ and observe
solid clusters at all densities.

We have also attempted to place a vacancy in a solid cluster to see if
the cluster could support an isolated vacancy and at the same
time be in equilibrium with the low density vapor.
We found that at 0.0424 atom/$\AA^2$ and 1.0 K,
the vacancy was spontaneously expelled from the solid cluster
during thermalization.  We conclude that 
at low temperatures the solid clusters will not support isolated vacancies.
  

Further evidence for solidification into the commensurate
structure in the simulation comes from calculations of the 
static structure factor.  Typical results for static structure 
factors for coverages at and immediately below the
commensurate solid density have been reported in our earlier paper\cite{pierce98}.
In addition, the melting of the commensurate solid phase can be 
determined from the temperature dependence of the static structure 
peak height. Melting is signaled by a significant drop in the peak 
height and a large statistical fluctuation in a peak  value near 
the melting temperatures.  The melting temperature determined in this 
manner are given in Table \ref{tab:melt}. The density dependence 
of the melting temperature is consistent with the experimental 
phase diagram, although our melting temperatures are slightly higher than the 
experimental values.  Heat capacity measurements indicate that 
the commensurate solid melts at about 3 K, and the low density 
($< 0.045$ atom/$\AA^2$) melting peaks are at about 1.5 K.

%%%%%%%%%%%%%%
\begin{figure}[ht]
\epsfxsize=\figwidth\centerline{\epsffile{engtemp.ps}}
\caption{Temperature dependence of the energy per particle.  
The coverages are 0.0353 (circles), 0.0566 (triangles), 
and 0.0636 atom/$\AA^2$ (squares).}
\label{fig:engtemp}
\end{figure}
%%%%%%%%%%%%%%


%%%%%%%%%%%%%%
\begin{figure}[ht]
\epsfxsize=\figwidth\centerline{\epsffile{specheat.eps}}
\caption{Specific heat for 0.0353 (filled circles)
and 0.0636 atom/$\AA^2$ (squares).  The dashed line is a guide to
the eye.  The solid lines are the experimentally determined specific heat  at 0.0367 
and 0.0634 atom/$\AA^2$.
}
\label{fig:specheat}
\end{figure}
%%%%%%%%%%%%%%


Another way of estimating melting temperatures is from the 
temperature dependence of the energy. This is shown in 
Fig.\ \ref{fig:engtemp} for various densities.  
These curves possess inflection points that lead to specific
heat maxima when differentiated.  These peaks
indicate melting.  Figure \ref{fig:specheat} shows two sample 
calculations of the specific heat at the indicated densities.  
The peak height and location change dramatically with density.  Melting 
occurs at about 1.5 K and 3.5 K for 0.0353 (filled circes) and 0.0636 atom/$\AA^2$ (open squares)
respectively.  In Fig.\ref{fig:specheat} the calculated specific heat for 0.0353  and 
0.0636 atoms/$\AA^2$  is shown. For comparison we also present the
experimental measure specific heat (solid lines) for densities of 0.0367 and 0.0634 atms/$AA^2$.
The calculated specific heat has peaks  somewhat above the experimental values but
are consistent with the melting behavior exhibited in the contour plots and in the static structure
factor calculations.


\subsection{Maxwell Construction}

%%%%%%%%%%%%%%
\begin{figure}[ht]
%\centerline{\psfig{figure=shiftsqr3.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{energ.eps}}
\caption{Total energy versus coverage.  The dashed
line is a straight line connecting the lowest density accessible in our
lattice and the density that corresponds to 1/3 coverage.
}
\label{fig:engsqr3}
\end{figure}
%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\caption{Energy/particle versus coverage.  The first column gives the 
temperature of the calculation.  The number in parenthesis gives the
error in the last decimal place.}
\vspace{.2in}
\begin{tabular}{|c|c|c|} \hline
T(K) & $\sigma$($\AA^{-2}$) & E/N (K) \\ \hline
1.00 & 0.0353 & -143.73(8) \\
1.00 & 0.0424 & -144.03(9) \\
1.00 & 0.0530 & -144.31(7) \\
1.00 & 0.0566 & -144.56(7) \\ 
1.00 & 0.0636 & -145.12(8) \\
1.33 & 0.0707 & -142.71(8) \\ \hline
\end{tabular}
\label{tab:engsqr3}
\end{center}
\end{table}
%%%%%%%%%%%%%%

%%%%%%%%%%%%%%

We determine phase ranges by applying the Maxwell construction to the
total ground state energy.  For a system at constant volume with a 
varying number of particles, a region of phase separation will be
signaled by an unphysical upward curvature in the total free energy's
dependence on density.  The upper and lower bounding densities of this
region are connected by a common tangent line.  The total free energy
for all intermediate densities lies on or above this line, either because
creating a surface between the two phases costs a finite amount of 
energy or because the system remains unphysically homogeneous.  In the
thermodynamic limit the system will separate into the two phases
at the bounding densities.

The Maxwell construction at nonzero temperatures should be applied to the
total free energy.  This is not directly accessible from PIMC, however.
Instead, we use a limiting process to determine effectively ground
state energy values.  All energy calculations are performed at low
temperatures.  The temperature is then raised and the energy is 
recalculated.  If the two values are the same within error bars, we
conclude that we have obtained effectively zero temperature energy values.
This allows us to apply the Maxwell construction to the total energy,
since it is the same as the total free energy at zero temperature.

We now can apply this procedure to the total energy calculations of the
first layer solid.  Total energy calculations 
for a range of densities using a simulation cell 
designed to accommodate the commensurate solid are shown in
Fig.\ \ref{fig:engsqr3}.  The values for the energy per particle
are given in Table \ref{tab:engsqr3}. 
The cell dimensions
are $25.560 \AA \times 22.136 \AA$, and the number of particles varies
from 20 to 40.   
The dashed line in Fig.\ \ref{fig:engsqr3} is a straight line 
connecting the lowest density accessible in our
lattice and the density that corresponds to 1/3 coverage.
Notice that all intermediate energy values are
above this straight line.  From the Maxwell construction, this
indicates that the intermediate energy values are unstable and will
phase separate into the two stable phases (the vapor and the 
commensurate solid) that bound the unstable region.

The binding energy for a single particle on the substrate can be
easily calculated.  We find this to be
$E_B=-143.09 \pm 0.27$ .  This is comparable to
the estimated values \cite{cole81} 
$E_B=-141.75 \pm 1.50$ K from scattering \cite{derry79,carlos80}
and $E_B=-142.33 \pm 1.97$ K
from thermodynamic analysis \cite{elgin74}.
Our value for the binding energy
was calculated at 0.4 K and confirmed to be the ground
state value by the limiting procedure discussed 
above.  By subtracting this energy from
the energy per particle 
of the commensurate solid phase, we obtain the condensation
energy per particle for the two-dimensional solid,
$E_{2D}=-2.03 \pm 0.20$ K.  This is comparable to, but slightly higher
than, the two-dimensional 
energy (-1.06 K) for the commensurate phase 
found by the variational calculations of
\cite{bruch93} for the same interaction model.



\subsection{Above Commensurate Density}

Figures \ref{fig:dist742} and \ref{fig:dist994} illustrate the domain
wall solid and incommensurate triangular solid phases, respectively.  The
domain wall solid consists of patches of commensurate solid on different
sublattices.  Linear domain walls occur at the boundaries between 
these regions.  In the incommensurate solid, the atoms form a 
triangular solid that does not
have a periodic relationship with the underlying adsorption sites for
lengths scales less than the minimum dimension of the simulation cell.

%%%%%%%%%%%%%%
\begin{figure}[ht]
%\centerline{\psfig{figure=distrib742m200n1.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{distrib742m200n1.ps}}
\caption{Distribution plot of the domain wall solid at 0.0742 atom/$\AA^2$
(N=42 atoms) and T=1.0 K.  Filled circles indicate adsorption sites.
}
\label{fig:dist742}
\end{figure}
%%%%%%%%%%%%%%

%%%%%%%%%%%%%%
\begin{figure}[ht]
%\centerline{\psfig{figure=distrib994m100n1.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{distrib994m100n1.ps}}
\caption{Distribution plot of the incommensurate solid phase at 0.0994 
atom/$\AA^2$ (N=50 atoms) and T=2.0.  Filled circles indicate adsorption
sites.
}
\label{fig:dist994}
\end{figure}

\section{Tests of  Our Conclusions}

The strength of our conclusions is limited by the accuracy of the
interaction model that we use.  First, it is possible that the
substrate may substantially alter the helium-helium 
interaction\cite{mclachlan64,bruch82}.  Inclusion
of this effect, the so-called McLachlan interaction,
has been shown to change the ground state phase from the 
commensurate solid to a low density liquid in two-dimensional variational
calculations.\cite{bruch93} We have repeated the 
low temperature density scans
using the same mediated interactions employed in \cite{bruch93}. 
We found that while the energy per particle increases for
all coverages, the commensurate solid
remains the energetically favored phase.  Second, 
another potential problem is that
the helium-graphite interaction\cite{cole80} is too corrugated, thus favoring
solidification.  We have repeated the
low temperature calculations at the commensurate density and at
0.0424 atom/$\AA^2$ with the corrugation strength reduced by 25 \%. 
The commensurate phase remains energetically favored.  Calculations
were also performed with the corrugation reduced by 50 \%, but it was 
found that the commensurate solid would not form for temperatures as low
as 2 K, thus indicating the corrugations had been underestimated.  
We note finally that the melting behavior of the commensurate solid phase
was not sensitive to the inclusion of the McLachlan term.  

Finally, we wish to discuss the arguments of Greywall and Busch (GB)
against solid clusters and in favor of the superfluid phase.
Their primary objection to solid-vapor
coexistence is that this should be signaled by
linear heat capacity isotherms for the entire region from
zero coverage up to the commensurate density.  Their
published data shows that for temperatures from 0.2 K to
0.5 K, the isotherms are linear only between 0.025 and 0.060 atom/$\AA^2$.  
At 0.1 K, the upper endpoint is about 0.055 atom/$\AA^2$.  
As a possible explanation, we suggest that the departure from linearity below
0.025 atom/$\AA^2$  is caused by the presence of multiple finite-sized clusters.  At low densities,
solid clusters nucleate around surface defects.
Initially, there are many small metastable clusters with large 
perimeter-to-area ratios.  Increasing the density increases the size of the clusters until
the surface is covered by a few large solid clusters with negligible
boundary effects.  Thus, the heat capacity exhibits linear 
behavior only after the solid clusters are sufficiently large
so that the perimeter-to-area ratio is small.  This
presumably occurs for coverages above 0.025 atom/$\AA^2$.  
GB have used a similar explanation
in their arguments for solid-liquid and liquid-gas coexistence in 
regions that do not have linear isotherms.

GB's identification of coverages near 0.04 atom/$\AA^2$ as
liquid is based partly on simulation results for 2D helium on a flat
substrate, the most relevant calculations then
available.  As GB note, the large peak associated with 
the melting of the uniform commensurate solid phase first 
emerges above 0.04 atom/$\AA^2$.  2D helium is a liquid
near this density \cite{whitlock88}, suggesting that 
first layer coverages below 0.04 
may be liquid.  Unlike the purely 2D simulations, 
our calculations take the role of
substrate effects into account.  As we have shown, surface corrugations
push the density of the 
energy minimum up from about 0.04 on a flat substrate to 0.0636
atom/$\AA^2$ and produce solidification.  
GB also show that their low density heat capacity
results are in general agreement with a PIMC 
calculation for 2D superfluid helium \cite{ceperley89}, 
suggesting that there might be a superfluid
transition in the first layer.  We have
shown in Fig. \ref{fig:specheat} 
that these rounded heat capacities 
are produced by the melting of a solid cluster and are not associated with 
a superfluid transition.

In closing, we would like to remark that a promising direction for 
monolayer superfluidity lies with helium adsorbed on alkali substrates,
particularly lithium \cite{cheng89,boninsegni98}.
These substrates have much smaller corrugations and
a much weaker attraction, allowing the first layer helium film to be a 
liquid.  The phenomenon competing with superfluidity for these substrates
is pre-wetting, rather than solidification.  


\section{Summary}
Our first layer calculations were performed both with and without 
substrate corrugations.  When corrugations are neglected,
the first layer film resembles a purely two-dimensional film.  We determined
that the film consists of gas, liquid, and solid phases.  These are separated
by coexistence regions, and we determined the coverage ranges for all
phases at low temperatures by using the Maxwell construction.  The first
layer liquid has an equilibrium density of 0.0450 atoms/$\AA^2$.  Below this
density the system phase separates.  This region is divided into an
unstable region, where liquid droplets form, and a metastable region in
which the system over-expands instead of forming an interface.  These
two regions are separated by a spinodal point at 0.034 atoms/$\AA^2$.  At
higher densities the system enters a narrow region of liquid-solid coexistence
between 0.0675 and 0.0700 atoms/$\AA^2$.  Above these coverages the system
is in a triangular solid phase.  The beginning of layer promotion occurs
between 0.116 and 0.119 atoms/$\AA^2$.  All of our calculations are in
agreement with recent Green's function Monte Carlo results.\cite{whitlock98}

When corrugations are included in the first layer, the phase diagram is
substantially altered.  We find that a $\sqrt 3 \times \sqrt 3$ commensurate
solid occurs, in agreement with numerous experiments.  By examining 
the temperature dependence of the static structure function and the 
specific heat, we find that this solid melts at approximately 3.5 K, 
compared with the experimental melting temperature of 3 K.  
We further find that the commensurate solid phase is
energetically favored.  At densities below 
commensuration, the system phase
separates into commensurate solid clusters and a low density vapor.  No
liquid phase occurs at low temperatures.

\section{Acknowledgments}
This work was supported in part by the National Aeronautics and Space
Administration under grant number NAG3-1841.  Some of the calculations
were performed using the facilities of the Supercomputer Computations
Research Institute at Florida State University.  We also wish to
thank L. W. Bruch for useful comments on our work.

