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% $BNL;RJ*M}3X(B : $B1i=,LdBj(B 11-6 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B : g140828 $BIp@n!!=c(B
% $B:n@.G/7nF|(B : /01/12/25
% $BEE;R%a!<%k(B : g140828@mail.ecc.u-tokyo.ac.jp
%
% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
%
\mondai
{g140828 $BIp@n!!=c(B}
{01/12/25}
{g140828@mail.ecc.u-tokyo.ac.jp}
{11-6}
{$B#2<!85%0%i%U%!%$%H$N(B$\pi ,\pi ^\ast
$$B%P%s%I!J(B11.20$B!K$G!"=E$J$j@QJ,$NCM#s$r#0$H$*$/$H$-!"(B$\Gamma
$$B!"(BK$B!"(BM$BE@$G$N%(%M%k%.!<$NCM$r5a$a$h!#(B
}
{
$B#s!a#0$h$j!"!J(B11.20$B!K<0$O!"(B
\begin{equation}
E = \frac{\varepsilon _{2p} \pm tw(k)}{1}
\end{equation}
$B$H$J$k!#$3$3$G!"(B
\begin{equation}
w(k) = \sqrt {1 + 4\cos \frac{\sqrt 3 k_x a}{2}\cos \frac{k_y a}{2} + 4\cos
^2\frac{k_y a}{2}}
\end{equation}
$B$G$"$k!#$h$C$F!"(B
\begin{equation}
E = \varepsilon _{2p} \pm t\sqrt {1 + 4\cos \frac{\sqrt 3 k_x a}{2}\cos
\frac{k_y a}{2} + 4\cos ^2\frac{k_y a}{2}}
\end{equation}
$B$H$J$k!#(B
$B$3$l$K!"LdBj(B11.5$B$G2r$$$?(B$(k_x ,\ k_y )$$B$NCM!"(B$\Gamma (0,0)$,\
M$\displaystyle(\frac{2\pi
}{\sqrt 3 a},0)$,\\
K$\displaystyle(\frac{2\pi }{\sqrt 3 a},\frac{2\pi }{3a})$$B$rBeF~$9$k$H!"(B
\begin{eqnarray}
 E_\Gamma &=& \varepsilon _{2p} \pm t\sqrt {1 + 4\cos 0\cos 0 + 4\cos ^20}
\nonumber \\
 &=& \varepsilon _{2p} \pm 3t \\
\nonumber
 \end{eqnarray}
\begin{eqnarray}
 E_{\rm M} &=& \varepsilon _{2p} \pm t\sqrt {1 + 4\cos \pi \cos 0 + 4\cos
^20} \nonumber \\
 &=& \varepsilon _{2p} \pm t \\
\nonumber
 \end{eqnarray}
\begin{eqnarray}
 E_{\rm K} &=& \varepsilon _{2p} \pm t\sqrt {1 + 4\cos \pi \cos
\frac{\pi }{3} +
4\cos ^2\frac{\pi }{3}} \nonumber \\
 &=& \varepsilon _{2p} \\
\nonumber
 \end{eqnarray}
$B0J>e$h$jEz$($O!"(B$\varepsilon _{2p} $$B$r%(%M%k%.!<$N86E@$K$*$$$F!"(B
\begin{equation}
E_\Gamma = \pm 3t,\ E_{\rm M} = \pm t,\ E_{\rm K} = 0
\end{equation}
}

