\chapter{Magnetism in optical lattice}

\section{Introduction}

\begin{quote}
`` ... loadstone attracts iron because it has a soul.''
\quotation{--- Thales of Miletus, {\it{$\sim$ 585 B.C.}}}
\end{quote}
%
The peculiar phenomenon of magnetism has been intriguing mankind ever since the times of Thales from Miletus \cite{fowler}.
It took till the year of 1887 for James Clerk Maxwell to summarize all of classical electromagnetism macroscopically in his four equations \cite{classical_electromagnetism}, which motivated Albert Einstein in his theory of special relativity in 1905 \cite{relativity}.
Its enormous success firmly lay the fundamental basis for the entire electrical and electronic engineering today. 
Microscopically, nature is governed by the laws of quantum mechanics \cite{quantum_mechanics1}, which successfully explain the atomic origin of magnetism.
On the weak side, diamagnetism (paramagnetism) is the consequence of orbital (spin) angular momentum coupling in electrons (unpaired electrons) with external magnetic fields \cite{statistical_mechanics}. 
On the strong side, ferromagnetism is a consequence of exchange interactions between electrons \cite{yosida}.
In a tight binding solid without charge degree of freedom, ferromagnetism has been theoretically explained by the Heisenberg model with qualitative success.
Besides ferromagnetism, other types of magnetism, such as antiferromagetism and ferrimagnetism, have been discovered over the recent decades to exist in some solids at room temperature \cite{fm1_wiki}.
\\ \\
\noindent Not only in solids, exotic ferromagnetism has been recently discovered in 2009 to exist in ultracold atomic Fermi gases at extremely low temperatures below 1 nK \cite{mit}, giving direct experimental evidence to the theoretical prediction for itinerent ferromagnetism described by the Stoner model \cite{yosida}, i.e.
%
\begin{equation}
\hat{H} = \sum_{\mathbf{k},\sigma} \epsilon_\mathbf{k} \hat{c}^\dag_{\mathbf{k}} \hat{c}_{\mathbf{k}} 
 + \frac{1}{2} \frac{U}{N} \sum_{\substack{\mathbf{k}_1\mathbf{k}_2 \\ \mathbf{q} \ne 0}}  \hat{c}^\dag_{\mathbf{k}_1+\mathbf{q}\,\uparrow}  \hat{c}^\dag_{\mathbf{k}_2-\mathbf{q}\,\downarrow}  \hat{c}_{\mathbf{k}_2\,\downarrow} \hat{c}_{\mathbf{k}_1\,\uparrow} \,\, . 
\end{equation}
%
It is obvious that solids favour ferromagnetism over gases.
Indeed, as will be shown later in this chapter, itinerant ferromagnetism of atomic Fermi gases is stabilized by an optical lattice with increasing laser intensity $\mathrm{V_0}$.
Furthermore, calculations from density functional theory recover the antiferromagnetic state towards the Hubbard limit at half-filling, therefore justifying the validation of our theory.
\\ \\
\noindent This chapter is based on one of my publications \cite{dft_fermigas}.


\section{Ferromagnetism in optical lattice}

\subsection{Probing ferromagnetism by Kohn-Sham DFT}

Calculating the ground-state polarization 
%
\begin{equation}
P = \frac{\rho_\uparrow - \rho_\downarrow}{\rho_\uparrow + \rho_\downarrow}
\end{equation}
%
for a range of lattice depths $V_0$, band fillings $n$, and interaction strengths $a/d$ based on Kohn-Sham density functional theory \ref{kohn_sham}, we obtain the phase diagrams shown in figure~\ref{phasediagram}. 
\\
%
\begin{figure}[h]
\includegraphics[width=4.7cm]{./figures/fig-3a.eps}
\hfill
\includegraphics[width=4.7cm]{./figures/fig-3b.eps}
\hfill
\includegraphics[width=4.7cm]{./figures/fig-3c.eps}
\hfill
\caption{{\bf Phase diagrams at fixed optical lattice intensity $V_0$.} The red-color intensity indicates the polarization $P$ for optical lattice depths (a) $V_0=0.5E_R$, (b) $V_0=2E_R$, (c)  $V_0=4E_R$. 
The green and blue curves indicate, respectively, the transitions to partially and fully polarized phases in homogeneous systems ($V_0=0$). The gray and yellow curves indicate the corresponding transitions in the optical lattice. Ferromagnetism dominates in the region of large optical lattice intensity $V_0$ and scattering length $a$, where the non-trivial phase boundary arises due to the Kohn-Sham band theory.
}
\label{phasediagram}
\end{figure}
%
\noindent \\ \\
\noindent In a shallow lattice with $V_0 = 0.5E_R$ (figure \ref{phasediagram}(a)) we see three phases: a paramagnetic phase at weak interactions (white), partially polarized (shown as pink gradations), and fully polarized (ferromagnetic, shown in solid red). The phase boundaries  in this  shallow lattice are similar to those of the homogeneous system $V_0 = 0$~\cite{pilati2010}, indicated by the green and blue lines. In  deeper optical lattices ($V_0 = 2E_R$ in figure \ref{phasediagram}(b) and  $V_0 = 4E_R$ in figure \ref{phasediagram}(c)) polarization sets in at much weaker interactions, indicating that the optical lattice strongly favours itinerant ferromagnetism. 
%
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{./figures/fig-4.eps}
%\includegraphics[width=8cm]{fig-4.eps}
%\includegraphics[width=4.1cm]{fig-4a.eps}
%\hfill
%\includegraphics[width=4.1cm]{fig-4b.eps}
%\hfill
%\includegraphics[width=4.1cm]{fig-4c.eps}
%\hfill
%\includegraphics[width=4.1cm]{fig-4d.eps}
%\hfill
%\includegraphics[width=4.1cm]{fig-4e.eps}
%\hfill
%\includegraphics[width=4.1cm]{fig-4f.eps}
\caption{{\bf Band structure.} Shown are band structures for two lattice depths, $V_0= 2E_R$ in the left column and $V_0=4E_R$ in the right column, and three values of scattering length ($a=0.04, 0.08, 0.16\,d$ from top to bottom) at half-filling $n=1$. The blue and red curve corresponds to the majority and minority spin-component respectively. The black curve is the result for an unpolarized noninteracting gas. Energies are given relative to the chemical potential, shown as a dashed green line at 0. The wave-vector values (given on the $x$-axis in units of scan a curve which goes through the high symmetry points $\mathrm{\Gamma}=(0,0,0)$, $\mathrm{X}=(0, \pi/d ,0)$, $\mathrm{R}=(\pi/d,\pi/d,\pi/d)$ and $\mathrm{M}=(\pi/d,\pi/d,0)$ of the first Brillouin zone.
}
\label{bandstructure}
\end{figure}
%
\noindent \\ \\
\noindent We can see two prominent features due to the presence of an optical lattice. The first is the much bigger extent of the polarized phases, which is due to the higher local density at the potential minima in the optical lattice, which increases the local density beyond the critical value for polarization. 
A second striking effect is the non-monotonic behavior of the phase boundary: there is a large fully polarized region at densities up to half filling ($n \le 1$), which rapidly shrinks at higher filling. 
This phenomenon is due to band structure effects and a gap between up-spin and down-spin subbands. 
\\ \\
\noindent Thus we next calculate the detailed band structure of the interacting system, shown in the left panels of figure~\ref{bandstructure}, for a weak optical lattice ($V_0 = 2E_R$) without a band gap and on the right for  a moderate optical lattice with a band gap ($V_0 = 4 E_R$). 
Weak interactions ($a = 0.04d$) change the band structure only slightly. Increasing the interaction to $a = 0.08d$ (second row) we find a partially polarized state in the deeper lattice: the two spin subbands split and the band structure is substantially changed. At even stronger interaction $a = 0.16d$ (third row) the gas is partially polarized also in the shallower lattice, and becomes fully polarized in the deeper lattice. 
Note that here the fermions are fully polarized up to half band filling $n=1$, since only the up-spin subband gets occupied. Notice also that in the fully polarized state the first band is fully occupied and the system is insulating due to the gap between the first and second subbands. 
Filling the bands further puts fermions in the next band with opposite spin, resulting in a partially polarized state. This explains the sharp feature around $n=1$ in the phase diagram in figure \ref{phasediagram}(c). 
To recover full polarization for $n>1$ one needs to increase either  interaction strength or lattice depth to push the energy of the lowest down-spin subband above the second up-spin subband. 


\subsection{Inadequetcy of Hohenberg-Kohn DFT}

%
\begin{figure}
\centering
\includegraphics[width=4.7cm]{./figures/fig-3c.eps}
\includegraphics[width=4.7cm]{./figures/fig-3d.eps}
\caption{{\bf KS-LSDA vs. HK-LSDA phase diagrams at fixed optical lattice intensity $V_0$.} 
The red-color intensity indicates the polarization $P$ for optical lattice depths $V_0=4E_R$ calculated by (a) KS-LSDA, and by (b) HK=LSDA density functional theory. 
The green and blue curves indicate, respectively, the transitions to partially and fully polarized phases in homogeneous systems ($V_0=0$). 
The gray and yellow curves indicate the corresponding transitions in the optical lattice. 
Ferromagnetism dominates in the region of large scattering length $a$, where the non-trivial phase boundary arises due to the Kohn-Sham band theory, which cannot be captured using HK-LSDA.
}
\label{phasediagram_comparison}
\end{figure}
%
\begin{figure}
\centering
\includegraphics[width=7cm]{./figures/fig-2.eps}
\caption{{\bf Density of states}. Results of Kohn-Sham DFT calculations at half-filling $n=\rho d^3=1$ with optical lattice intensity $V_0=3E_R$ and scattering length $a=0.12d$ are shown indicated by the blue (red) symbols for the majority (minority) spin component. The density of states of the interacting gas is compared to that of non-interacting species (black crosses), and that obtained in the HK-LSDA method (green line), which shows no band gap. The Fermi level is at $E=0$. 
}
\label{densityofstates}
\end{figure}
%
Figure \ref{phasediagram_comparison}(a) and (b) illustrate the inadequetcy of Hohenberg-Kohn density functional theory, where it fails to capture the non-trivial phase boundary for moderate optical lattices that arises due to the Kohn-Sham band structure effects.
\\ \\
\noindent The complete absence of band structure effects in HK-LSDA also largely handicaps itself to estimate correctly the density of states of the system as illustrated in figure \ref{densityofstates}.
%

\newpage

\section{Antiferromagnetism in optical lattice}

\begin{figure}[h]
\centering
\includegraphics[width=14cm]{./figures/fig-5.eps}
%\includegraphics[height=3.14cm]{fig-5a.eps}
%\hfill
%\includegraphics[height=3.1cm]{fig-5b.eps}
\caption
{
{\bf Phase diagram and antiferromagnetic (AF) band structure at half filling $n=\rho d^3 = 1$.} 
Left: Ferromagnetic (antiferromagnetic) phases are indicated by the red-colored polarization (blue-colored staggered polarization). 
As the scattering length $a$ increases, the fermionic optical lattice undergoes phase transitions from an unpolarized to an antiferromagnetic and finally to ferromagnetic phase.
Right: To observe antiferromagnetism, the unit cell has to be doubled, resulting in a face centered cubic (fcc) lattice. A spin-density-wave gap $\Delta_{SDW}$ shows up in the antiferromagnetic state of an optical lattice with laser intensity $V_0 = 4 E_R$ and scattering length $a = 0.08d$.
Here, the high symmetry points are $\Gamma = (0,0,0)$, $X = (0,\pi/d,0)$, $L = (\pi/2d,\pi/2d,\pi/2d)$ and $W = (\pi/2d,\pi/d,0)$.  
}
\label{phasediagram2}
\end{figure}

\noindent To see antiferromagnetism competing with ferromagnetism at half band filling $n=1$ we need to consider a unit cell consisting of two lattice sites, and compare the energies of antiferromagnetic and uniform configurations. We find, as shown in figure \ref{phasediagram2}(a), that antiferromagnetic ordering is preferred at intermediate interaction strengths and half band filling, matching with the single band Hubbard model physics that becomes valid in the upper left hand corner of the shown phase diagram.
\\ \\
\noindent Last but not least, antiferromagnetic symmetry breaking opens up an additional spin-density-wave (SDW) gap $\Delta_\mathrm{SDW}$ in the folded ground-state band as shown in figure \ref{phasediagram2}(b).
This provides experimentalists an indirect method to probe for antiferromagnetism in fermionic optical lattices.

