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\documentstyle[psfig,html,harvard,12pt,Times,equations,vecmat]{article}
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\title{Critical Exponents for the Metal--Insulator Transition in Disordered
Systems}
\author{Angus MacKinnon\\ 
Blackett Lab., Imperial College, London SW7 2BZ}
\newcommand{\shorttitle}[1]
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    \noindent \rm Short title: {\sl #1}\par
    \medskip}
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   {\noindent \rm Submitted to: {\sl #1}\par
    \medskip}


\def\be{\begin{equation}}
\def\ee#1{\label{#1}\end{equation}}
\def\bes{\begin{subequations}\begin{eqnarray}}
\def\ees#1{\end{eqnarray}\label{#1}\end{subequations}}
\def\ba{\begin{array}}
\def\ea{\end{array}}

\newcommand{\etal}{{\it et al.\ }}
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\makeatother
\newenvironment{tex2html_wrap}{}{}
\begin{document}
\pagestyle{empty}
\newpage

{\samepage \clearpage $W_c$
}


\newpage

{\samepage \clearpage $\Lambda _c$
}


\newpage

{\samepage \clearpage $\half$
}


\newpage

{\samepage \clearpage $2$
}


\newpage

{\samepage \clearpage $0.2\%$
}


\newpage

{\samepage \clearpage $s=\nu=1.54 \pm 0.08$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $s$
}


\newpage

{\samepage \clearpage $\nu$
}


\newpage

{\samepage \clearpage $0.5$
}


\newpage

{\samepage \clearpage $1.5$
}


\newpage

{\samepage \clearpage $1.0$
}


\newpage

{\samepage \clearpage $1\%$
}


\stepcounter{section}
\newpage

{\samepage \clearpage \begin{equation}H = \sum_i \epsilon_i |i><i| + \sum_{i\not=j} V_{ij}|i><j|
\label{eq:1}\end{equation}
}


\newpage

{\samepage \clearpage $V_{ij} = V_0$
}


\newpage

{\samepage \clearpage $V_0 =1$
}


\newpage

{\samepage \clearpage $\epsilon_i$
}


\newpage

{\samepage \clearpage $-{\mathchoice{{\displaystyle{\frac12}}}{{\textstyle{1/2}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}} W < \epsilon_i < +{\mathchoice{{\displaystyle{\frac12}}}{{\textstyle{1/2}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}
W$
}


\newpage

{\samepage \clearpage $\sigma$
}


\newpage

{\samepage \clearpage $W$
}


\newpage

{\samepage \clearpage $W^2 =
12\sigma^2$
}


\newpage

{\samepage \clearpage $a_i$
}


\newpage

{\samepage \clearpage \begin{equation}E a_i = \epsilon_i a_i + \sum_{j\not= i} a_j.
\label{eq:2}\end{equation}
}


\newpage

{\samepage \clearpage $L$
}


\newpage

{\samepage \clearpage $M\times M$
}


\newpage

{\samepage \clearpage $\bi A_i$
}


\newpage

{\samepage \clearpage \begin{equation}E \bi A_n = \bss H_n \bi A_n + \bi A_{n+1} + \bi A_{n-1}
\label{eq:3}\end{equation}
}


\newpage

{\samepage \clearpage $n$
}


\newpage

{\samepage \clearpage $\bss H_n$
}


\newpage

{\samepage \clearpage \begin{subequations}% latex2html id marker 222
\begin{eqnarray}
\left(\begin{array}{l}\bi A_{n+1}\\  \bi A_n\end{array}\right)
&=& \left(\begin{array}{ll}E - \bss H_n&-\bss I\\ 
                \bss I                & 0\end{array}\right)
        \left(\begin{array}{l} \bi A_n\\  \bi A_{n-1}\end{array}\right)\\ 
&=& \prod_{m=1}^{n} \left(\begin{array}{ll}E - \bss H_m&-\bss I\\ 
                 \bss I                & 0\end{array}\right)
        \left(\begin{array}{l} \bi A_1\\  \bi A_0\end{array}\right)\\ 
&=& \bss T_n \left(\begin{array}{l} \bi A_1\\  \bi A_0\end{array}\right).
\end{eqnarray}\label{eq:4}\end{subequations}
}


\newpage

{\samepage \clearpage \begin{equation}\lim_{n\to\infty} \left(\bss T_n^\dagger \bss T_n\right)^{1/n} = \bss M
\label{eq:5}\end{equation}
}


\newpage

{\samepage \clearpage $\bss M$
}


\newpage

{\samepage \clearpage $\bss T_n$
}


\newpage

{\samepage \clearpage $\bss T^\dagger \bss T$
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage $50\%$
}


\newpage

{\samepage \clearpage $2\times M\times
M$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $\lambda_M$
}


\newpage

{\samepage \clearpage $\Lambda = \lambda_M/M$
}


\newpage

{\samepage \clearpage \begin{equation}{\d\ln\Lambda\over\d\ln M} = {\mathop\chi\nolimits}\left(\ln\Lambda\right)
\label{eq:6}\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}\Lambda = {\mathop{\rm f}\nolimits}\left(M/\xi\right)
\label{eq:7}\end{equation}
}


\newpage

{\samepage \clearpage $\xi$
}


\newpage

{\samepage \clearpage $\chi = 0$
}


\newpage

{\samepage \clearpage \begin{equation}% latex2html id marker 248
\ln\Lambda = \ln\Lambda_c + A(\tau - \tau_c)M^\alpha
\label{eq:8}\end{equation}
}


\newpage

{\samepage \clearpage $\tau$
}


\newpage

{\samepage \clearpage $\tau_c$
}


\newpage

{\samepage \clearpage $\Lambda$
}


\newpage

{\samepage \clearpage $A$
}


\newpage

{\samepage \clearpage % latex2html id marker 567
$\alpha$
}


\newpage

{\samepage \clearpage \begin{equation}% latex2html id marker 251
\xi \sim \left|\tau - \tau_c\right|^{1/\alpha}
\label{eq:9}\end{equation}
}


\newpage

{\samepage \clearpage % latex2html id marker 573
$\nu =
1/\alpha$
}


\newpage

{\samepage \clearpage $s = (d-2)\nu$
}


\newpage

{\samepage \clearpage % latex2html id marker 581
$\alpha$
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage $\ln\Lambda$
}


\newpage

{\samepage \clearpage $M$
}


\newpage

{\samepage \clearpage $(\ln\Lambda_c, \tau_c)$
}


\newpage

{\samepage \clearpage \begin{equation}% latex2html id marker 254
\ln\Lambda = A\tau M^\alpha + B(M)
\label{eq:10}\end{equation}
}


\newpage

{\samepage \clearpage $B(M)$
}


\newpage

{\samepage \clearpage % latex2html id marker 607
$\alpha$
}


\newpage

{\samepage \clearpage \begin{figure}\psfig{file=fig1a.ps,angle=270,width=10in}
\end{figure}
}


\newpage

{\samepage \clearpage \begin{figure}% latex2html id marker 119
\psfig{file=fig1b.ps,angle=270,width=10in}
\bigskip
\caption[Scaling Curves]{\label{fig:1}$\Lambda$ \mbox{vs.} $W$, for (a)
rectangular and (b) Gaussian distributions. The data are represented
by dots with differing symbols for different system sizes with $4\le
M\le 12$ increasing in the direction of the arrow.  Each point is
accurate to $0.2\%$.  The lines are fitted using (\ref{eq:10}).}
\end{figure}
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage % latex2html id marker 625
$\alpha$
}


\newpage

{\samepage \clearpage $\chi^2$
}


\newpage

{\samepage \clearpage \begin{equation}% latex2html id marker 257
\chi^2 = \sum_i{\left(A\tau_i M_i^\alpha + B(M_i) -
\ln\Lambda_i\right)^2
\over\sigma^2_i}
\label{eq:11}\end{equation}
}


\newpage

{\samepage \clearpage $i$
}


\newpage

{\samepage \clearpage $\sigma_i$
}


\newpage

{\samepage \clearpage $4\le M\le 12$
}


\newpage

{\samepage \clearpage $s=\nu=1.53\pm 0.04$
}


\newpage

{\samepage \clearpage $s=\nu=1.48\pm 0.05$
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage \begin{figure}% latex2html id marker 133
\psfig{file=fig2.ps,angle=270,width=7in}
\bigskip
\caption[Fitted critical exponents]{\label{fig:2}  Fitted critical exponents for rectangular
(Diamonds) and Gaussian (Squares) distributions.  The absciss{\ae}
represent the smallest system size taken into account (with small
offsets for clarity).  In each group the width of the fitted region is
(from left to right)
$(16.2 \le W \le 16.8)\to(16.3\le W\le 16.7)\to(16.4\le W\le 16.6)$ and
$(21.0\le W\le 21.5)\to(21.05\le W\le 21.45)\to(21.1\le W\le 21.4)$ for
rectangular and Gaussian cases respectively.  The dotted lines represent
the range  $s=\nu=1.54\pm 0.08$.}
\end{figure}
}


\newpage

{\samepage \clearpage $M=4$
}


\newpage

{\samepage \clearpage $s=\nu\approx 1.54\pm0.08$
}


\stepcounter{section}
\newpage

{\samepage \clearpage $E=0$
}


\newpage

{\samepage \clearpage $-6<E<6$
}


\newpage

{\samepage \clearpage \begin{table}[htbp]
\begin{tabular*}{\textwidth}{|l@{\extracolsep{\fill}}c|cc|cc|} \hline
			&&\bf Rectangular	&&\bf Gaussian	&\\  \hline
Exponent		&&$1.515\pm 0.033$	&&$1.484\pm0.048$&\\ 
Disorder Range		&&$16.2\le W\le 16.8$	&&$21.0\le W\le21.5$&\\ 
System Sizes		&&$4\le M\le 12$	&&$4\le M\le 12$&\\ 
$\chi^2$(expected) 	&&$142$			&&$97$		&\\ 
$\chi^2$(fitted)	&&$126$			&&$75$		&\\ 
$W_c$			&&$16.50\pm 0.05$	&&$21.20\pm0.06$&\\ 
$\sigma_c$		&&$4.763\pm 0.015$	&&$6.120\pm0.018$&\\ 
$\Lambda_c$		&&$0.580\pm0.005$	&&$0.580\pm 0.005$&\\  \hline
\end{tabular*}

\end{table}
}


\newpage

{\samepage \clearpage $\mbox{SCC}^*$
}



\end{document}