Graphene Susceptibility LaTeX Source

\documentclass[aps,12pt,prb]{revtex4}

\begin{document}

\title{Graphene Susceptbility}

\author{R. B. Laughlin}

\affiliation{Stanford}

\date{17 May 08}

\maketitle

\begin{displaymath}
\chi_q(\omega) = \frac{1}{\Omega_{BZ}} \int_{BZ} |{\cal M}_q(k)|^2
\biggl\{ \frac{1}{\hbar \omega - (E_k + E_{k+q}) + i \eta}
+ \frac{1}{-\hbar \omega - (E_k + E_{k+q}) - i \eta} \biggr\}
dk_x dk_y
\end{displaymath}

\begin{displaymath}
|{\cal M}_q(k)|^2 = \biggl[ 1 - {\rm Re}(z_{k+q}^* z_k)/E_k E_{k+q} 
\biggr]/2
\end{displaymath}

\begin{displaymath}
z_k = - t \sum_\nu \exp(i {\bf k} \cdot {\bf r}_\nu )
\; \; \; \; \; \; \; \; \; \; 
E_k = |z_k|
\end{displaymath}

\begin{displaymath}
- \frac{1}{\pi} {\rm Im} \biggl[ \chi_q(\omega) \biggr] 
= \frac{{\rm sgn} (\omega)}
{\Omega_{BZ}} \int_{BZ} |{\cal M}_q(k)|^2
\delta [\hbar \omega - (E_k + E_{k+q}) ]
dk_x dk_y
\end{displaymath}

\begin{displaymath}
{\rm Re} \biggl[ \chi_q(\omega) \biggr] 
= - \frac{1}{\pi} {\cal P} \int_{-\infty}^\infty
\frac{{\rm Im} [\chi_q(\omega') ]}{\omega - \omega'} d\omega'
\end{displaymath}

\end{document}