\chapter{Energy functional for repulsive Fermi gas}
\label{appendix_functional}


\section{Energy functional}

The objective is to determine the energy functional, or equation of state, of the homogeneous Fermi gas at zero temperature by Fixed-Node Diffusion Monte Carlo (FN-DMC) method.
This technique has been popularly employed in several studies of the ground-state properties of resonantly interacting Fermi gases with balanced~\cite{astra2004,carlson2004} as well as imbalanced~\cite{carlson2005,lobo,pilati2008} populations of the two components.
Despite the fact that -- to circumvent the sign problem -- one has to introduce the fixed-node constraint, meaning that the ground-state wave function is forced to have the same nodal surface as a trial wave function, FN-DMC has proven to be extremely accurate. 
It provides a rigorous upper bound for the ground-state energy, which is exact if the nodes of the trial wave function coincide with those of the exact ground state. 
Predictions for the ground-state energies obtained with this technique have been benchmarked against experimental results for the low temperature equations of state of both normal and superfluid atomic gases on the attractive branch of Feshbach resonances~\cite{navon,nascimbenepilati}. 
In this appendix, we will extend the same technique to the repulsive branch \cite{dft_fermigas}.
\\ \\
\noindent Other than the fixed node approximation for interacting fermions, diffusion Monte Carlo is numerically exact with any suitable choice of initial trial wavefunction $\psi_T (\mathbf{r}^\uparrow_1 , \cdots , \mathbf{r}^\uparrow_{N_\uparrow}, \mathbf{r}^\downarrow_1 , \cdots , \mathbf{r}^\downarrow_{N_\downarrow})$ (or simply $\psi_T ( \{ {\bf r}_i^\uparrow \} , \{ {\bf r}_{i^\prime}^\downarrow \} )$ for $N_\uparrow$ spin-up and $N_\downarrow$ spin-down fermions \cite{dmc_solids}.
We adopt the Jastrow-Slater form \cite{pilati2010} 
%
\begin{equation}
\psi_T ( \{ {\bf r}_i^\uparrow \} , \{ {\bf r}_{i^\prime}^\downarrow \}  )  = \prod_{i,i^\prime} 
f(|{\bf r}_{i}^\uparrow - {\bf r}_{i^\prime}^\downarrow|) \, \det (  \phi_j^\uparrow ({\bf r}_i^\uparrow) ) \,  \det (  \phi_{j'}^\downarrow ({\bf r}_{i'}^\downarrow) )\, ;
\label{psiT}
\end{equation}
%
where the positive-definite Jastrow correlation term $f(r)$ between unlike spin fermions is obtained from the solution of the two-body scattering problem in free space 
%
\begin{equation}
-\frac{\hbar^2}{m} \frac{\partial^2}{\partial r^2} f(r) + V_{\mathrm{int}} (r) f(r) = \epsilon_\mathrm{int} f(r)  
\end{equation}
%
($0 < r < L$).
Here, $V_{\mathrm{int}} (r)$ is chosen to be the hardcore scattering potential that reproduces the s-wave scattering length $a_s$ at long range, and $f(r)$ is chosen correspondingly with the lowest positive eigenvalue $\epsilon_{\mathrm{int}}$ as well as its derivative $f'(r = L/2) = 0$.
$\phi_j^\uparrow ({\bf r}_i^\uparrow)$ and $\phi_{j'}^\downarrow ({\bf r}_{i'}^\downarrow)$ are plane wave orbitals.
%
\begin{figure}[h]
\begin{center}
\includegraphics[width=8.0cm]{./figures/fig-S1.eps}
\caption{(color on-line). Energy per volume as a function of the polarization for different values of the interaction parameter $k_Fa$. Squares are Monte Carlo data, solid (black) lines the global energy function equation~(\ref{global}), dashed and dot-dashed lines represent the low-$P$ and large-$P$ expansions, equations~(\ref{eossmallP}) and~(\ref{Hpolaron}), respectively.}
\label{figapp3}
\end{center}
\end{figure}
\noindent \\ 
\noindent The ground-state energy per volume of the repulsive Fermi gas $\epsilon = \epsilon(k_F a, P)$ is a functional in terms of the interaction parameter 
%
\begin{equation}
k_F a=(3\pi^2\rho)^{1/3}a
\end{equation}
%
(where $\rho = \rho_\uparrow + \rho_\downarrow$ is the total density), and the polarization 
%
\begin{equation}
P=\frac{\rho_\uparrow-\rho_\downarrow}{\rho_\uparrow+\rho_\downarrow}  \,\, .
\end{equation}
%
The energy density $\epsilon(k_F a, P)$ interpolates between its small polarization limit $\epsilon_{\mathrm{<}}(k_F a, P)$ and its large polarization limit $\epsilon_{\mathrm{>}}(k_F a, P)$, i.e.
%
\begin{equation}
\label{global}
\epsilon (k_Fa, P)  = \,\, \Big(1 - f_\mathrm{damp}(P) \Big) \epsilon_\mathrm{<} (k_Fa, P) \,\, + \,\,  \Big( f_\mathrm{damp}(P) \Big) \epsilon_\mathrm{>} (k_Fa, P) \,\, \mathrm{.}
\end{equation}
%
through a damping function 
%
\begin{equation}
f_\mathrm{damp}(P) = \frac{1}{2} \Big(1 + \tanh \Big[ 2\pi C_\mathrm{damp1}\left(P-C_\mathrm{damp2}\right) \Big]  \Big)
\end{equation}
%
%
such that the coefficient $C_\mathrm{damp1} = 1.8$ determines the steepness, and $C_\mathrm{damp2}=0.5$ is the center of the transition region. 
Figure~\ref{figapp3} illustrates the energy density~(\ref{global}), the small and large polarization limits.
All raw quantum Monte Carlo data are reported in publication \cite{dft_fermigas}. 
Finally, the hartree-exchange-correlation energy density is obtained by subracting the Thomas-Fermi components \ref{TF}, i.e.
%
\begin{equation}
\epsilon_\mathrm{HXC} (k_Fa, P) = \epsilon (k_Fa, P) - \frac{3}{5} \rho_\uparrow E_{F\uparrow} - \frac{3}{5} \rho_\downarrow E_{F\downarrow}  \,\, .
\end{equation}
%

\section{Analytic behavior at polarization limits}

\noindent At small population imbalance, the energy density is quadratic in the polarization, i.e.
%
\begin{equation}
\label{eossmallP}
\epsilon_{\mathrm{<}} (k_Fa, P)  =  \frac{3}{5}\rho E_F
\left[ \epsilon_0\left(k_Fa\right)  +  \frac{5}{9} P^2\chi^{-1}\left(k_Fa\right) \right] \,\, .
\end{equation}
%
\begin{figure}[tbhp]
\begin{center}
\includegraphics[width=8.0cm]{./figures/fig-S2.eps}
\caption{(color on-line). Energy per volume (circles, left axis) and inverse magnetic susceptibility (squares, right axis) of the unpolarized Fermi gas. Units are the ideal Fermi gas values, $3/5\rho E_F$ and $\chi_0 = 3\rho/(2E_F)$, respectively. Solid lines are the fitting functions (\ref{eos})  in red and (\ref{chi}) in green, while the dashed lines correspond to second order perturbation theory.}
\label{figapp1}
\end{center}
\end{figure}
%
\noindent In units of $\frac{3}{5}\rho E_F = \frac{3}{5}\rho \frac{\hbar^2 k_F^2}{2m}$, the energy of unpolarized Fermi gas is expressed in a Taylor expansion of $(k_Fa)$ as
%
\begin{equation}
\label{eos}
\epsilon_0 (k_Fa) = 1 + C_{E1}(k_Fa)  + C_{E2} (k_Fa)^2 +  C_{E3}(k_Fa)^3 + C_{E4}(k_Fa)^4 \,\, ,
\end{equation}
%
and likewise for the inverse of the magnetic susceptibility
%
\begin{equation}
\label{chi}
{\chi}^{-1}\left(k_Fa\right) = 1 - C_{\chi 1} k_Fa -
C_{\chi 2} (k_Fa)^2
- C_{\chi 3} \left(k_Fa\right)^3.
\end{equation}
%
The coefficients $C_{E1} = 0.3536$, $C_{E2} = 0.1855$, $C_{\chi 1} = 0.6366$ and $C_{\chi 2} = 0.2911$ have been determined using second order perturbation theory~\cite{huang,lee,duine,recati,kanno2}, while we obtain $C_{E3} = 0.307(7)$, $C_{E4} = -0.115(8)$ and $C_{\chi 3} = 0.56(1)$ from a best-fit to the Monte Carlo results with zero or small population imbalance (in the range $k_Fa \leq 1$ and $P \leq 0.5$). 
Figure The functions $\epsilon_0(k_Fa)$ and $\chi^{-1}(k_Fa)$ are shown in figure~\ref{figapp1}.
\\ \\
\noindent At large population imbalance the behavior of a normal Fermi gas is well described by the Landau-Pomeranchuk hamiltonian~\cite{lobo,mora,pilati2010}. 
In this approach, the minority component is regarded as a gas of weakly interacting quasiparticles, called Fermi polarons. 
%
\begin{figure}[tbhp]
\begin{center}
\includegraphics[width=8.0cm]{./figures/fig-S3.eps}
\caption{(color on-line). Chemical potential at zero concentration (circles, left axis) and effective mass (squares, right axis) of the repulsive polaron. Units are $3/5E_{F\uparrow}$ and bare atomic mass, respectively. Solid lines are the fitting functions (\ref{A}) in red and (\ref{effectivemass}) in green. The dashed line is the chemical potential in second order perturbation theory.}
\label{figapp2}
\end{center}
\end{figure}
%
In this limit, the energy density 
%
\begin{equation}
\label{Hpolaron}
e_{\mathrm{>}} (k_{F\uparrow}a, x) = \frac{3}{5} \rho_\uparrow E_{F\uparrow} \Bigg[ 1+ A(k_{F\uparrow}a) x + F(k_{F\uparrow}a) x^2 + \frac{ x^{5/3}}{m^\ast(k_{F\uparrow}a)} \Bigg]
\end{equation}  
%
is characterized by the interaction parameter
%
\begin{equation}
k_{F\uparrow}a = (6\pi^2\rho_\uparrow)^{1/3} a
\end{equation}
%
of the majority component, and the concentration 
%
\begin{equation}
x = \frac{1-P}{1+P}
\end{equation}
%
of the minority compoent.
In units of $\frac{3}{5} \rho_\uparrow E_{F\uparrow}$, $A(k_{F\uparrow}a)$ represents the polaron chemical potential at zero concentration, i.e.
%
\begin{equation}
\label{A}
A (k_{F\uparrow}a) = 
\frac{5}{3} \Big[ C_{A1} (k_{F\uparrow}a)  
+ C_{A2}\left(k_{F\uparrow}a\right)^2 
+ C_{A3} \left(k_{F\uparrow}a\right)^3 \Big] \,\, .
\end{equation}
%
Second order perturbation theory~\cite{mora,cui,recatiprivate} gives the first two coefficients $C_{A1}=0.4244$ and $C_{A2}=0.2026$, while $C_{A3} = 0.105(2)$ results from a best-fit to Monte Carlo data for the energy of a single spin-down impurity immersed in the Fermi sea of spin-up (majority component) particles.
The polaron effective mass $m^\ast$ is extracted from the dispersion relation of an impurity with finite momentum. We parametrize the interaction parameter dependence of this effective mass (in units of the bare atomic mass) as
%
\begin{equation}
\label{effectivemass}
m^\ast(k_{F\uparrow}a) = 
1 + C_{m^\ast 1} (k_{F\uparrow}a)^{C_{m^\ast 2}}
\end{equation}
%
where $C_{m^\ast 1} = 0.0807(50)$ and $C_{m^\ast 2} = 1.59(15)$. 
The functions $A(k_{F\uparrow}a)$ and $m^\ast(k_{F\uparrow} a)$ are shown in fig.~\ref{figapp2}.
%
Parametrizing 
%
\begin{equation}
F (k_{F\uparrow}a) = C_F (k_{F\uparrow}a)^{2}  \,\, ,
\end{equation}
%
the coefficient $C_{F} = 0.419(4)$ is obtained from fitting the Monte Carlo data for a highly imbalanced Fermi gas ($x\lesssim 0.5$) via inserting equation \ref{A} and \ref{effectivemass} into the Landau-Pomeranchuck functional.
\\ \\
\noindent To facilitate the implementation of the functional, we summarize the values of all of the coefficients in the following table.
%
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
$C_\mathrm{damp1} = 1.8$      & $C_{\chi 1} = 0.6366$ & $C_{m^\ast 1} = 0.0807$\\
$C_\mathrm{damp2} = 0.5$   & $C_{\chi 2} = 0.2911$ & $C_{m^\ast 2} = 1.59$\\ \cline{1-1} \cline{3-3}
$C_{E1} = 0.3536$           & $C_{\chi 3} = 0.56$   & $C_F = 0.419$\\ \cline{2-3}
$C_{E2} = 0.1855$           & $C_{A1} = 0.4244$ \\
$C_{E3} = 0.307$            & $C_{A2} = 0.2026$ \\
$C_{E4} = -0.115$           & $C_{A3} = 0.105$ \\
\cline{1-2}
\end{tabular}
\end{center}
%


