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\@writefile{toc}{\contentsline {chapter}{\numberline {6}Magnetism in optical lattice}{59}{chapter.6}}
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\@writefile{toc}{\contentsline {section}{\numberline {6.1}Introduction}{59}{section.6.1}}
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\@writefile{toc}{\contentsline {section}{\numberline {6.2}Ferromagnetism in optical lattice}{60}{section.6.2}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.1}Probing ferromagnetism by Kohn-Sham DFT}{60}{subsection.6.2.1}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.1}{\ignorespaces {\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. }}{60}{figure.6.1}}
\newlabel{phasediagram}{{6.1}{60}{{\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}{figure.6.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.2}{\ignorespaces {\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 ($a=0.04, 0.08, 0.16\tmspace +\thinmuskip {.1667em}d$ 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. }}{61}{figure.6.2}}
\newlabel{bandstructure}{{6.2}{61}{{\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}{figure.6.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2.2}Inadequetcy of Hohenberg-Kohn DFT}{62}{subsection.6.2.2}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.3}{\ignorespaces {\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. }}{63}{figure.6.3}}
\newlabel{phasediagram_comparison}{{6.3}{63}{{\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}{figure.6.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.4}{\ignorespaces {\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 $E=0$. }}{63}{figure.6.4}}
\newlabel{densityofstates}{{6.4}{63}{{\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 $E=0$}{figure.6.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.3}Antiferromagnetism in optical lattice}{64}{section.6.3}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.5}{\ignorespaces {\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 $W = (\pi /2d,\pi /d,0)$. }}{64}{figure.6.5}}
\newlabel{phasediagram2}{{6.5}{64}{{\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 $W = (\pi /2d,\pi /d,0)$}{figure.6.5}{}}
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