\chapter{Density functional theory}


\section{General formalism}

\subsection{Hohenberg-Kohn theorems}

The foundation of density function theory is given by the following two Hohenberg-Kohn theorems \cite{hk_theorem}.
\\
\begin{addmargin}[2em]{2em}
{\bf{Theorem I}}:
\begin{addmargin}[1.5em]{0em}
{\it{The non-degenerate ground state density $\rho({\bf{r}})$ of interacting particles is uniquely determined by the external potential $V({\bf{r}})$}}.\footnote{
\begin{proof}
Suppose there exists another external potential $V'({\bf{r}})$ that gives $\rho({\bf{r}})$.
Denoting the hamiltonians $\hat{H}$ and $\hat{H}'$ repectively such that $\hat{H} |\Psi\rangle = E |\Psi\rangle$ and $\hat{H}' |\Psi'\rangle = E' |\Psi'\rangle$, the density can now be written as $\rho ({\bf{r}})= |\Psi({\bf{r}})|^2 = |\Psi'({\bf{r}})|^2$ by assumption.
Now, 
\begin{eqnarray}
\langle \Psi' | \hat{H} | \Psi' \rangle - \langle  \Psi' | \hat{H}' | \Psi' \rangle &=&  \smallint d{\bf{r}} \left( V({\bf{r}}) - V'({\bf{r}}) \right) \rho({\bf{r}})   \nonumber \\
\langle \Psi | \hat{H}' | \Psi \rangle -  \langle \Psi | \hat{H} | \Psi \rangle &=& \smallint d{\bf{r}} \left( V'({\bf{r}}) - V({\bf{r}}) \right) \rho({\bf{r}})   \nonumber
\end{eqnarray}
therefore $E + E' = \langle \Psi' | \hat{H} | \Psi' \rangle + \langle \Psi | \hat{H}' | \Psi \rangle > \langle \Psi | \hat{H} | \Psi \rangle + \langle \Psi' | \hat{H}' | \Psi' \rangle = E + E'$ due to the variational principle, resulting in a contradiction!
\end{proof}
}
\end{addmargin}
%
{\bf{Corollary I}}:
\begin{addmargin}[1.5em]{0em}
{\it{
The non-degenerate ground state density $\rho({\bf{r}})$ is mapped one-to-one to the many-particle wavefunction $\Psi({\bf{r}})$ of the interacting system.
}}
\end{addmargin}
%
{\bf{Theorem II}}:
\begin{addmargin}[1.5em]{0em}
{\it{
There exists a functional $E[\rho]$ that depends only on the density $\rho({\bf{r}})$, such that the minimum $E[\rho_0]$ is exactly the ground state energy.
}}
\end{addmargin}
%
\end{addmargin}

\noindent \\
\noindent Instead of solving the Schr\"odinger equation for $N$ interacting particles, the ground state density and energy can be obtained exactly from a 3-dimensional minimization of the corresponding functional $E[\rho]$ if the latter is known.
In practice, one has to resort to reasonable approximation schemes for $E[\rho]$.
This drastic reduction in computational costs without much penalty to the accuracy is the main reason why density functional theory is used as a workhorse in the field of conventional material science today

\subsection{Hohenberg-Kohn formalism}

Following Hohenberg and Kohn (HK) \cite{hk_theorem}, the exact ground state energy $E$ and ground state density $\rho({\bf{r}})$ can be determined by minimizing an energy functional of the density:
%
\begin{equation}
\label{functional}
 E\left[\rho\right] =  \int d\mathbf{r}\,  V( \mathbf{r})
                       \rho(\mathbf{r})  +
                       F[\rho];
\end{equation}
%
where the first term is the potential energy due to the external potential $V(\mathbf{r})$.
The second term is an unknown but {\em universal} functional which includes the interaction and kinetic energies, but does not explicitly depend on $V( \mathbf{r})$. 
The generalization to potentially spin-polarized systems one introduces separate 
densities of the two spin components
$\rho_\uparrow$ and $\rho_\downarrow$:
%
\begin{equation}
\label{spin_dependent_functional}
 E\left[\rho_\uparrow,\rho_\downarrow\right] =  \int d\mathbf{r}\,  V( \mathbf{r})
                       \left[ \rho_\uparrow\left(\mathbf{r}\right)  +  \rho_\downarrow\left(\mathbf{r}\right) \right]+
                       F\left[\rho_\uparrow,\rho_\downarrow \right],
\end{equation}
%
An explicit (but rather inaccurate) expression for $F\left[\rho_\uparrow,\rho_\downarrow\right]$ is given by
the Hohenberg-Kohn local spin density approximation (HK-LSDA)
%
\begin{equation}
\label{HK-LSDA}
F\left[\rho_\uparrow,\rho_\downarrow\right] = \int d\mathbf{r}\,  
\epsilon( \rho_\uparrow(\mathbf{r}) , \rho_\downarrow(\mathbf{r})),
\end{equation}
%
where $\epsilon \left( \rho_\uparrow , \rho_\downarrow \right)$ is the ground-state energy density of a homogeneous Fermi gas with the given spin densities\footnote{
The Thomas-Fermi (TF) approximation is recovered  as a mean field approximation  for $\epsilon \left( \rho_\uparrow , \rho_\downarrow \right)$. 
Here, the TF energy density is given by
%
\begin{equation}
\epsilon_{TF} (\rho_\uparrow,\rho_\downarrow) = \frac{3}{5} \rho_\uparrow E_{F\uparrow} + \frac{3}{5} \rho_\downarrow E_{F\downarrow}   \,\, ,
\end{equation}
%
whereby only for the noninteracting case would the spin-up (down) Fermi energy be expressed as
%
\begin{equation}
E_{F\uparrow(\downarrow)} = \frac{\hbar^2 k^2_{F\uparrow(\downarrow)}}{2m}  = \frac{\hbar^2}{2m}  (6\pi^2 \rho_{\uparrow(\downarrow)}  )^{2/3}
\end{equation}
%
}. 
In practice, $\epsilon \left( \rho_\uparrow , \rho_\downarrow \right)$ is obtained from accurate Quantum Monte Carlo calculations, and takes the form of equation \ref{global}. (See appendix \ref{appendix_functional}.)


\subsection{Kohn-Sham formalism}

The kinetic part is usually highly non-local and cannot be treated well under the local approximation. Therefore, Kohn and Sham \cite{ks} proposed a more accurate functional by explicitly including the  exact kinetic energy $T_0$ of non-interacting fermions. What is left is the interaction energy $E_\mathrm{HXC}$, combining the usual Hartree (mean field term) $E_H$ and the exchange-correlation correction $E_{XC}$:
%
\begin{equation}
\label{kohn_sham_functional}
F\left[\rho_\uparrow,\rho_\downarrow\right] = T_0\left[\rho_\uparrow,\rho_\downarrow\right] +  E_\mathrm{HXC}\left[\rho_\uparrow,\rho_\downarrow\right].
\end{equation}
%
A simple yet often reliable treatment of $E_\mathrm{HXC}$ is the \emph{local spin-density approximation} (LSDA)
%
\begin{equation}
\label{LSDAXC}
E_\mathrm{HXC}\left[\rho_\uparrow,\rho_\downarrow\right] = \int d\mathbf{r}\,  
\epsilon_\mathrm{HXC} \left( \rho_\uparrow\left(\mathbf{r}\right) , \rho_\downarrow\left(\mathbf{r}\right) \right),
\end{equation}
%
where the functional is replaced by an integral over the interaction energy density\footnote{
%
\begin{equation}
\epsilon_\mathrm{HXC} \left( \rho_\uparrow , \rho_\downarrow \right) 
 = \epsilon \left( \rho_\uparrow , \rho_\downarrow \right) 
     - \frac{3}{5}\rho_\uparrow  \frac{\hbar^2}{2m}  (6\pi^2 \rho_{\uparrow}  )^{2/3}
     - \frac{3}{5} \rho_\downarrow \frac{\hbar^2}{2m}  (6\pi^2 \rho_{\downarrow}  )^{2/3}
\label{TF}
\end{equation}
}
of a uniform system with the same local density.
The objective is now to minimize the total energy
%
\begin{equation}
\label{total_energy}
E = \int d\mathbf{r} \,\, \Big( \, T_0 \left[ \rho_\uparrow , \rho_\downarrow ; \mathbf{r} \right] \, + \, V (\mathbf{r}) \big( \rho_\uparrow (\mathbf{r}) + \rho_\downarrow (\mathbf{r}) \big)   \, + \,  \epsilon_\mathrm{HXC} \left[ \rho_\uparrow , \rho_\downarrow ; \mathbf{r} \right]  \,\, \Big)
\end{equation}
%
with respect to the spin densities ($\sigma = \uparrow,\downarrow$)
%
\begin{equation}
\small
\rho_\sigma \left( \mathbf{r} \right) = \int \mathcal{D} \mathbf{r}_j \, \delta \left( \mathbf{r} - \mathbf{r}_1 \right)  \left| \prod_{\sigma'=\uparrow,\downarrow} det \left[ \phi_n^{\sigma'} \left( \mathbf{r}_j \right) \right] \right|^2 = \sum_{n\sigma}^\mathrm{occ} \left| \phi_n^\sigma \left( \mathbf{r} \right) \right|^2 \,\, ,
\end{equation}
%
where  $\phi_n^\sigma \left( \mathbf{r} \right)$ are normalized single-quasiparticle spin-orbitals filled to the Fermi level.
Minimizing equation \ref{total_energy} with respect to the complex conjugate quasiparticle spin orbital $\phi_n^{\sigma *} \left( \mathbf{r} \right)$ subjected to the normalization constraint $\int d{\bf{r}}  \left| \phi_n^\sigma \left( \mathbf{r} \right) \right|^2 = 1$, we arrive at the coupled Kohn-Sham (KS) eigenvalue equations \cite{dft_fermigas}
%
\begin{equation}
\label{KS_equations}
\left( -\frac{\hbar^2}{2m} \nabla^2 + V ( {\mathbf r} ) +  V_\sigma^\mathrm{HXC} \left(\rho_\uparrow, \rho_\downarrow; \mathbf{r}\right) \right)  \phi_n^\sigma = \epsilon_n^\sigma \, \phi_n^\sigma
\end{equation}
%
where the Langrange multiplier $ \epsilon_n^\sigma$ is in fact the KS quasiparticle energy.
Here, the hartree-exchange-correlation potential is
%
\begin{equation}
V_\sigma^\mathrm{HXC} \left(\rho_\uparrow, \rho_\downarrow; \mathbf{r}\right) = 
 \frac{\delta }{\delta \rho_\sigma (\mathbf{r}) }
 \left[ \epsilon_\mathrm{HXC} (  \rho_\uparrow,\rho_\downarrow ; \mathbf{r}  ) \right] 
\end{equation}
%
It is convenient to define the KS single-quasiparticle hamiltonian 
%
\begin{equation}
\hat{H}_\mathrm{KS}^\sigma = -\frac{\hbar^2}{2m} \nabla^2 + V_\sigma^\mathrm{eff} \left(\rho_\uparrow, \rho_\downarrow; \mathbf{r}\right)
\end{equation}
%
where the effective potential is $V^{\mathrm{eff}}_\sigma (\rho_\uparrow, \rho_\downarrow; \mathbf{r}) = V (\mathbf{r})  + V^{\mathrm{HXC}}_\sigma (\rho_\uparrow, \rho_\downarrow; \mathbf{r})$.

\section{KS-DFT for fermionic optical lattice}

\subsection{Lattice translational symmetry}


In an optical lattice with periodicity $\mathbf{d}$, {\it i.e.} $V (\mathbf{r} + \mathbf{d} ) = V (\mathbf{r})$, we shall make use of Bloch's theorem \cite{solid_state} to rewrite the KS quasiparticle orbitals as
%
\begin{equation}
\label{Blochstates}
\phi_{n\mathbf{k}}^\sigma \left( \mathbf{r}\right) = e^{2\pi i \mathbf{k} \cdot \mathbf{r}} u_{n\mathbf{k}}^\sigma \left( \mathbf{r}\right)  \,\, ,
\end{equation}
%
with periodic Bloch orbitals $u_{n\mathbf{k}}^\sigma \left( \mathbf{r}\right)$. The wavevectors $\mathbf{k}$ run over the first Brillouin zone of the reciprocal lattice. 
%
Working in the units of lattice spacing $d = \frac{\lambda}{2}$ and recoil energy $E_R = \frac{\hbar^2}{2m} \left( \frac{2\pi}{\lambda} \right)^2$ ($\hbar$ is the reduced Planck constant and $m$ the atomic mass), the coupled Kohn-Sham eigenvalue equations \ref{KS_equations} become 
%
\begin{equation}
\label{kohn_sham}
\left[ \frac{1}{\pi^2} \left( -i\nabla + 2\pi \mathbf{k} \right)^2 + V^\mathrm{eff}_\sigma \left(\rho_\uparrow, \rho_\downarrow; \mathbf{r}\right) \right] 
u_{n\mathbf{k}}^\sigma \left(\mathbf{r}\right) 
=  \epsilon_{n\mathbf{k}}^\sigma  
u_{n\mathbf{k}}^\sigma \left(\mathbf{r}\right)  \,\,\, .
\end{equation}
%
They  must be solved self-consistently with the ground state densities
%
\begin{equation}
 \rho_\sigma \left(\mathbf{r}\right) =
 \sum_{n\mathbf{k}} \left|  u_{n\mathbf{k}}^\sigma \left(\mathbf{r}\right) \right|^2 
 \Theta\left( \mu - \epsilon_{n\mathbf{k}}^\sigma \right)
 \end{equation}
% 
where $\Theta(\cdots)$ is the Heaviside function.
%
The total ground state energy is calculated from the set of all quasiparticle energies $\epsilon_{n\mathbf{k}}^\sigma$ as
\begin{equation}
 E = \sum_{n\mathbf{k}\sigma} \epsilon_{n\mathbf{k}}^\sigma 
 \Theta\left( \mu - \epsilon_{n\mathbf{k}}^\sigma \right) 
\end{equation}
%
up to the Fermi level.
The Bloch orbitals are next expanded in the plane wave basis 
%
\begin{equation}
\label{planewavebasis}
 u_{n\mathbf{k}}^\sigma \left(\mathbf{r}\right) =
 \sum_{\mathbf{G}}  c^\sigma_{n\mathbf{k}} \left( \mathbf{G} \right)
 \exp\left(2\pi i  \mathbf{G}\cdot\mathbf{r} \right) \,\, ,
\end{equation}
%
the KS equation (\ref{kohn_sham}) becomes a coupled set of matrix eigenvalue equations 
%
\begin{equation}
4 \left( \mathbf{G} + \mathbf{k} \right)^2 c^\sigma_{n\mathbf{k}} \left( \mathbf{G} \right)
 + \sum_{ \mathbf{G'} }  V^\mathrm{eff}_\mathbf{G - G'} c^\sigma_{n\mathbf{k}} \left( \mathbf{G'} \right)
 = \epsilon_{n\mathbf{k}}^\sigma c^\sigma_{n\mathbf{k}} \left( \mathbf{G} \right)
\end{equation}
%
for reciprocal vectors $\mathbf{G}$.
%
As a remark, the effective potential possesses both translational and inversion symmetry, and therefore its Fourier components
%
\begin{equation}
V^\mathrm{eff}_\mathbf{G} = \frac{1}{M^3} \int_{\mathrm{unitcell}} V^\mathrm{eff} \left( \mathbf{r} \right) \exp\left(-2\pi i  \mathbf{G}\cdot\mathbf{r} \right) d\mathbf{r}
\end{equation}
%
must be real and related by $V^\mathrm{eff}_\mathbf{G} = V^\mathrm{eff}_\mathbf{-G}$.


\subsection{Simple cubic lattice ($\mathrm{O_h}$ symmetry)}

An isotropic three dimensional optical lattice with simple cubic (sc) geometry possesses $\mathrm{O_h}$ point group symmetry \cite{group_theory}, therefore likewise for the corresponding reciprocal lattice and its band structure \cite{solid_state}.
With $\mathrm{M} \times \mathrm{M} \times \mathrm{M}$ being the dimensions of the lattice, the reciprocal lattice basis vectors read
%
\begin{equation}
\mathbf{b}_1 =  \hat{\mathbf{x}}   \,\, , \,\, \mathbf{b}_2 =  \hat{\mathbf{y}} \,\, , \,\, \mathbf{b}_3 =  \hat{\mathbf{z}}
\end{equation}
%
and the reciprocal lattice vectors are
%
\begin{equation}
\mathbf{G} = \frac{m_1}{M} \mathbf{b}_1 + \frac{m_2}{M} \mathbf{b}_2 + \frac{m_3}{M} \mathbf{b}_3
\end{equation}
%
where $m_1,m_2,m_3 = 0, \pm1, \pm2, \cdots$.
Knowing the point group symmetry of the band structure reduces computational efforts greatly.
The $\mathrm{O_h}$ point group possesses, for instance, inversion (J) and reflection ($\sigma_h$, $\sigma_d$) symmetries \cite{group_theory}, thus giving rise to the following 48-fold degeneracy for the quasiparticle energies $\epsilon_{n\mathbf{k}}^\sigma$:
%
\begin{enumerate}
\item J-symmetry: \\
\begin{equation}
%\epsilon_n(k_x,k_y,k_z) = \epsilon_n(-k_x,-k_y,-k_z)
\epsilon^\sigma_{n(k_x,k_y,k_z)} = \epsilon^\sigma_{n(-k_x,-k_y,-k_z)}
\end{equation} 
\item $\sigma_h$-symmetry: \\
\begin{equation}
\epsilon^\sigma_{n(k_x,k_y,k_z)} = \epsilon^\sigma_{n(\pm k_x,\pm k_y,\pm k_z)}
\end{equation}
\item $\sigma_d$-symmetry: \\
\begin{eqnarray}
\epsilon^\sigma_{n(k_x,k_y,k_z)} &=& \epsilon^\sigma_{n(k_y,k_x,k_z)} \\
\epsilon^\sigma_{n(k_x,k_y,k_z)} &=& \epsilon^\sigma_{n(k_x,k_z,k_y)} \\
\epsilon^\sigma_{n(k_x,k_y,k_z)} &=& \epsilon^\sigma_{n(k_z,k_y,k_x)} \,\, .
\end{eqnarray}
\end{enumerate}
%
Note that the same degeneracy exists likewise for the norm of quasiparticle orbitals $\left| \phi_n^\sigma (\mathbf{r}) \right|^2$, therefore reducing computational efforts further.
\\ \\
\noindent The high-symmetry k-points in the band structure of the sc lattice are 
%
\clearpage
\begin{table}[!h]
\centering
\begin{tabular}{| c | c |}
\hline
Notation & $\mathbf{k}$-point \\
\hline\hline 
$\mathrm{\Gamma}$ & (0, 0, 0) \\
\hline
$\mathrm{X}$ & ($\sfrac{1}{2}$, 0, 0) \\
\hline
$\mathrm{M}$ & ($\sfrac{1}{2}$, $\sfrac{1}{2}$, 0) \\
\hline
$\mathrm{R}$ & ($\sfrac{1}{2}$, $\sfrac{1}{2}$, $\sfrac{1}{2}$) \\
\hline
\end{tabular}
\end{table}
%
\noindent In the event of anistropy along the z-direction, some of the degeneracies within the $\mathrm{O_h}$ point group are lifted, thus resulting in a $\mathrm{D_{4h}}$ point group.
Its symmetry can be easily worked out from the respective character table \cite{group_theory}.


\subsection{Face centered cubic lattice}

A primitive cell, consisting of two adjacent lattice sites within simple cubic (sc) geometry, translates to form a face centered cubic (fcc) lattice, therefore resulting in a body centered cubic (bcc) reciprocal lattice \cite{solid_state}.
With $\mathrm{M} \times \mathrm{M} \times \mathrm{M}$ being the dimensions of the {\bf{optical lattice}}, the reciprocal lattice basis vectors read
%
\begin{equation}
\mathbf{b}_1 = \frac{1}{2} ( \hat{\mathbf{x}} +  \hat{\mathbf{y}} -  \hat{\mathbf{z}} ) \,\, , \,\,  \mathbf{b}_2 = \frac{1}{2} ( \hat{\mathbf{x}} -  \hat{\mathbf{y}} +  \hat{\mathbf{z}} ) \,\, , \,\, \mathbf{b}_3 = \frac{1}{2} ( - \hat{\mathbf{x}} +  \hat{\mathbf{y}} +  \hat{\mathbf{z}} )
\end{equation}
%
and the reciprocal lattice vectors are
%
\begin{equation}
\mathbf{G} = \frac{m_1}{M} \mathbf{b}_1 + \frac{m_2}{M} \mathbf{b}_2 + \frac{m_3}{M} \mathbf{b}_3
\end{equation}
%
where $m_1,m_2,m_3 = 0, \pm1, \pm2, \cdots$.
The band structure still possesses $\mathrm{O_h}$ point group symmetry which facilitates the reduction of computational efforts greatly.
The high symmetry $\mathbf{k}$-points in the band structure of the fcc lattice are
%
\begin{table}[!h]
\centering
\begin{tabular}{| c | c |}
\hline
Notation & $\mathbf{k}$-point \\
\hline\hline 
$\mathrm{\Gamma}$ & (0, 0, 0) \\
\hline
$\mathrm{X}$ & ($\sfrac{1}{2}$, 0, 0) \\
\hline
$\mathrm{W}$ & ($\sfrac{1}{4}$, $\sfrac{1}{2}$, 0) \\
\hline
$\mathrm{L}$ & ($\sfrac{1}{4}$, $\sfrac{1}{4}$, $\sfrac{1}{4}$) \\
\hline
\end{tabular}
\end{table}
%
\\
For two identical sites within the primitive cell of a fcc lattice, its band structure is in fact equivilant to the band structure of a sc lattice.
More specifically in this scenario, each energy band of a sc lattice are folded into two bands of a fcc lattice about some highly symmetric $\mathbf{k}$-planes.


\newpage

\section{Remarks}

\subsection{Validity of Kohn-Sham density functional theory}

The validity of Kohn-Sham density functional theory (DFT) for fermionic optical lattices can be tested through quantitative comparisons with direct Quantum Monte Carlo (QMC) simulations\footnote{
%
We perform fixed-node diffusion Monte Carlo simulations for fermionic optical lattice, extended from the previous study of the homogeneous system \cite{pilati2010}.
To simulate Fermi gases in an optical lattice of sinple cubic geometry, we employ the trial wave function~(\ref{psiT}) using the Bloch states~(\ref{Blochstates}) (obtained by solving the equation~(\ref{KS_equations}) for different lattice depths $V({\bf r})$   without scattering, i.e. $a = 0$) as single-particle orbitals. 
The Bloch states are expanded in a plane-wave basis, as in equation~(\ref{planewavebasis}), using up to $13^3$ states. Unlike other Monte Carlo techniques for the single-band Fermi-Hubbard model which are in principle reliable only for deep lattices, this current version of continuous-space fixed-node diffusion Monte Carlo method allows one to simulate also moderate and shallow lattices. 
It is analogous to a recent bosonic Monte Carlo method based on the ground-state Path-Integral Monte Carlo algorithm, which has been used to perform a continuous-space simulation of the superfluid-to-insulator transition of hard-sphere bosons in optical lattices, going beyond the single-band approximation~\cite{pilati2012}.
%
}.
The excellent agreement, as illustrated in figure \ref{comparison}, demonstrates that DFT calculations with an LSDA functional are extremely reliable in weak and moderate optical lattices, therefore justifying its validity.
%
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{./figures/fig-1.eps}
\caption{{\bf Comparison of DFT results to QMC}. Main panel: energy per particle $E/N$ vs. optical lattice intensity $V_0$, at quarter-filling $n=\rho d^3=0.5$ with scattering length $a=0.04d$. The green curve is the results of Kohn-Sham DFT within the local spin-density approximation, the red points to Fixed-Node Diffusion Monte Carlo simulations. Inset: cross-sectional density profile on one lattice site, particularly at the lattice intensity of $V_0=2.0E_R$.
}
\label{comparison}
\end{figure}
%
\newpage

\subsection{Kohn-Sham band structure}

Finally, we give an example of diagonalizing the Kohn-Sham hamiltonian \ref{kohn_sham} self-consistently for a fermionic optical lattice with lattice depth $V_0 = 3 E_R$, scattering length $a = 0.04d$, and density at half-filling $n=1$, of simple cubic geometry.
The energy band structure is given in figure \ref{band_structure_1}.
%
\begin{figure}[h]
\centering
\includegraphics[width=5.8cm, angle=270]{./figures/band-structure-1.eps}
\caption{Kohn-Sham band structure of fermionic optical lattice with lattice depth $V_0 = 3 E_R$, scattering length $a = 0.04d$, and density at half-filling $n=1$, of simple cubic geometry.
The dotted curves correspond to the band structure of free fermions. 
(The fermi energy is set to be zero by convention.)
The ground state band is well separated from the excited bands, as evident in a finite band-gap $\Delta > 0$.
}
\label{band_structure_1}
\end{figure}
%
\\
\noindent In fact, the band structure $\epsilon_n^\sigma(\mathbf{k})$ completely determines the physics of atomic Fermi gases in weak to moderate optical lattice, in analog to weakly-correlated electrons in material science. 
An important quantity that derives from the band structure is the density of states 
%
\begin{equation}
D^\sigma(\epsilon) = \sum_{n\mathbf{k}} \delta(\epsilon - \epsilon_n^\sigma(\mathbf{k}))
\end{equation}
%
which, for instance, could be used as an input into dynamical mean-field theory simulations \cite{dmft} to improve the accuracy of Kohn-Sham calculations. 

