%Paper: hep-th/9303043
%From: NEPOMECHIE%PHYVAX@ren.IR.Miami.EDU
%Date: 08 Mar 1993 10:08:32 -0500 (EST)
%Date (revised): 31 Mar 1993 13:47:46 -0500 (EST)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                        %
%         To print this document, simply TEX twice                       %
%         (the relevant macro is already included)                       %
%                                                                        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newif\ifproofmode			% true => wide right margin, eqn.
\proofmodefalse				% labels shown; date in headline

\newif\ifforwardreference		% true => allow reference to an
\forwardreferencetrue			% equation that appears later

\newif\ifeqchapternumbers		% true => equations labeled as e.g.
\eqchapternumbersfalse			% (3.7) instead of (7)

\newif\ifsectionnumbers			% true => sections will be numbered.
\sectionnumberstrue			% Must be true if the next are true.

\newif\ifeqsectionnumbers		% true => equations labeled with
\eqsectionnumbersfalse			% section numbers: (4.7) or (3.4.7)
					% The last form occurs if BOTH
					% eqchapter and eqsection numbering
					% is enabled.

\newif\ifchaptersectionnumbers     	% true => section numbers as 3.4
\chaptersectionnumberstrue		%         (chapter.section)

\newif\ifcontinuoussectionnumbers	% true => don't reset numbering in eqs.
\continuoussectionnumbersfalse	% in each section (if
					% \eqsectionnumberstrue)

\newif\ifcontinuousnumbers		% true => don't reset numbering of
\continuousnumbersfalse 		% equations in each chapter

\newif\iffigurechapternumbers		% true => figures labeled as e.g.
\figurechapternumbersfalse		% (3.7) instead of (7)

\newif\ifcontinuousfigurenumbers	% true => don't reset numbering of
\continuousfigurenumbersfalse		% figures in each chapter

\newif\ifcontinuousreferencenumbers     % Normally, all references are
\continuousreferencenumberstrue         % packaged together

\newif\ifparenequations			% Default is parentheses around
\parenequationstrue			% the equations referred to

\newif\ifstillreading			% To check for eof in input

\font\eqsixrm=cmr6			% given these odd names to avoid
\font\eqtwelverm=cmr12			% possible conflict with a user.
\def\marginstyle{\eqsixrm}		% for proofmode labels

\newtoks\chapletter			% for appendices
\newcount\chapno			% chapter number
\newcount\sectno			% section number
\newcount\eqlabelno			% equation #'s
\newcount\figureno			% figure numbers
\newcount\referenceno			% footnote counter
\newcount\minutes			% time of day stamp
\newcount\hours				%   "

\newread\labelfile			% input name for labels
\newwrite\labelfileout			% output "    "    "
\newwrite\allcrossfile			% output "    "  cross references

\chapno=0
\sectno=0
\eqlabelno=0
\figureno=0

% A few definitions made here simply to guarantee upward compatibility.
\def\chapternumbersfalse{\eqchapternumbersfalse}
\def\chapternumberstrue{\eqchapternumberstrue}
\def\continuoussectionnumberingtrue{\continuoussectionnumberstrue}
\def\continuoussectionnumberingfalse{\continuoussectionnumbersfalse}
\def\continuousnumberingtrue{\continuousnumberstrue}
\def\continuousnumberingfalse{\continuousnumbersfalse}
\def\continuousfigurenumberingtrue{\continuousfigurenumberstrue}
\def\continuousfigurenumberingfalse{\continuousfigurenumbersfalse}
\def\continuousreferencenumberingtrue{\continuousreferencenumberstrue}
\def\continuousreferencenumberingfalse{\continuousreferencenumbersfalse}

% The initialization procedure.
%
\def\initialeqmacro{
    \ifproofmode
        \headline{\tenrm \today\ --\ \timeofday\hfill
                         \jobname\ --- draft\hfill\folio}
        \hoffset=-1cm
        \immediate\openout\allcrossfile=zallcrossreferfile
    \fi
    \ifforwardreference
        \openin\labelfile=zlabelfile
        \ifeof\labelfile
        \else
            \stillreadingtrue
            \loop
                \read\labelfile to \nextline
                \ifeof\labelfile
                    \stillreadingfalse
                \else
                    \nextline
                \fi
                \ifstillreading
            \repeat
        \fi
        \immediate\openout\labelfileout=zlabelfile
    \fi}

% BE SURE TO {ENCAPSULATE} THESE CATCODE COMMANDS or the most
% amazing things will happen to your document.

{\catcode`\^^I=9
\catcode`\ =9
\catcode`\^^M=9
\endlinechar=-1
\globaldefs=1

% TAB, SPACE, and CR are to be ignored in the following macros
%    This means that when a space is needed, it is inserted by
%    the \space command.
%    All definitions are made global to get out of the encapsulation.

% This returns either a number or a letter depending on the sign
% of \chapno.  It uses the CURRENT vaule of \chapno.  Typically used
% only by other macros.  The user may however, want to place the chapter
% number (or letter) into the headline, then \chapfolio should be used.
%
\def\chapfolio{			% A <--> -2,  B <--> -3 etc.
    \ifnum \chapno>0 \relax
        \the\chapno
    \else
        \the\chapletter
    \fi}

% This increments \chapno in the correct direction (more positive OR
% more negative).  If as is normal, there is NO continuous numbering
% of equations and figures, those variables are reset.  It is typically
% used only by the other macros, not directly by the user.
%
\def\bumpchapno{
    \ifnum \chapno>-1 \relax
        \global \advance \chapno by 1
    \else
        \global \advance \chapno by -1 \setletter\chapno
    \fi
    \ifcontinuousnumbers
    \else
        \global\eqlabelno=0
    \fi
    \ifcontinuousfigurenumbers
    \else
        \global\figureno=0
    \fi
    \ifcontinuousreferencenumbers
    \else
        \global\referenceno=0
    \fi
    \sectno=0}

\def\bumpsectno{
    \global\advance\sectno by 1 \relax
    \ifeqsectionnumbers
        \ifcontinuoussectionnumbers
        \else
            \global\eqlabelno=0
        \fi
    \fi}

% This is a very awkward way to turn a number into a letter, but it's
% the only way I found that works.  [The difficulty with simpler methods
% occurred in the \write routines.]  Typically used only by other macros.
%
\def\setletter#1{\ifcase-#1 {}  \or\global\chapletter={A}
  \or\global\chapletter={B} \or\global\chapletter={C} \or\global\chapletter={D}
  \or\global\chapletter={E} \or\global\chapletter={F} \or\global\chapletter={G}
  \or\global\chapletter={H} \or\global\chapletter={I} \or\global\chapletter={J}
  \or\global\chapletter={K} \or\global\chapletter={L} \or\global\chapletter={M}
  \or\global\chapletter={N} \or\global\chapletter={O} \or\global\chapletter={P}
  \or\global\chapletter={Q} \or\global\chapletter={R} \or\global\chapletter={S}
  \or\global\chapletter={T} \or\global\chapletter={U} \or\global\chapletter={V}
  \or\global\chapletter={W} \or\global\chapletter={X} \or\global\chapletter={Y}
  \or\global\chapletter={Z}\fi}

% And a non-global version of the above:
%
\def\tempsetletter#1{\ifcase-#1 {}\or{} \or\chapletter={A} \or\chapletter={B}
 \or\chapletter={C} \or\chapletter={D} \or\chapletter={E}
  \or\chapletter={F} \or\chapletter={G} \or\chapletter={H}
   \or\chapletter={I} \or\chapletter={J} \or\chapletter={K}
    \or\chapletter={L} \or\chapletter={M} \or\chapletter={N}
     \or\chapletter={O} \or\chapletter={P} \or\chapletter={Q}
      \or\chapletter={R} \or\chapletter={S} \or\chapletter={T}
       \or\chapletter={U} \or\chapletter={V} \or\chapletter={W}
        \or\chapletter={X} \or\chapletter={Y} \or\chapletter={Z}\fi}

% A utility: it produces a number or a letter, depending on
% the sign of the argument.  Used by other macros for appendices.
% (It is like \chapfolio, but need not refer to the current chapter.)
%
\def\chapshow#1{
    \ifnum #1>0 \relax
        #1
    \else
        {\tempsetletter{\number#1}\the\chapletter}
    \fi}

% In proofmode, it is useful to put today's date and time on each output page.
%
\def\today{\number\day\space \ifcase\month\or Jan\or Feb\or
        Mar\or Apr\or May\or Jun\or Jul\or Aug\or Sep\or
        Oct\or Nov\or Dec\fi, \space\number\year}

\def\timeofday{\minutes=\time    \hours=\time
        \divide \hours by 60
        \multiply \hours by 60
        \advance \minutes by -\hours
        \divide \hours by 60
        \ifnum\the\minutes>9 \relax
     		\the\hours:\the\minutes
 	\else
  		\the\hours:0\the\minutes
	\fi}

\def\initialeqmacros{\initialeqmacro}			% either spelling

% A new chapter comes along so infrequently that it is better to handle
% the advancing of the chapter number by a separate command.  That is
%     \bumpchapno
%
%     \chapfolio		% just the number (or letter)
%     \chaplabel{...}		% set up a label for cross-reference
%
%     \chapref{...}		% in the text, refer to a labeled chapter
%
% For compatability, \chapnum is left in.  I recommend not using it.
%
\def\chapnum{\bumpchapno \chapfolio}

\def\chaplabel#1{
    \ifforwardreference                             % A label to
        \write\labelfileout{                        % reference
        \noexpand\expandafter\noexpand\def          % the chapter:
        \noexpand\csname CHAPLABEL#1\endcsname{\the\chapno}}
    \fi
    \global\expandafter\edef\csname CHAPLABEL#1\endcsname
    {\the\chapno}
    \ifproofmode
        \rlap{\hbox{\marginstyle #1\ }}
    \fi}

\def\chapref#1{                                  % refer to some
    \ifundefined{CHAPLABEL#1}                     % chapter (forward
        ??                                        % or backward):
        \ifproofmode
            \ifforwardreference
            \else
                \write16{ ***Undefined\space Chapter\space Reference #1*** }
            \fi
        \else
            \write16{ ***Undefined\space Chapter\space Reference #1*** }
        \fi
    \else
        \edef\LABxx{\getlabel{CHAPLABEL#1}}
        \chapshow\LABxx
    \fi
    \ifproofmode
        \write\allcrossfile{Chapter\space #1}
    \fi}

% For section numbers use a structure parallel to equation numbers.
% These will add to the section number and print the section number.
%
\def\sectnum{
    \bumpsectno
        \ifchaptersectionnumbers
            \chapfolio.
        \fi
    \the\sectno}

\def\sectlabel#1{
    \bumpsectno
    \ifforwardreference
        \immediate\write\labelfileout{
        \noexpand\expandafter\noexpand\def
        \noexpand\csname SECTLABEL#1\endcsname{\the\chapno.\the\sectno?!}}
    \fi
    \global\expandafter\edef\csname SECTLABEL#1\endcsname
    {\the\chapno.\the\sectno?!}	 			% automatic numbering
    \ifproofmode
        \llap{\hbox{\marginstyle #1\ }}
    \fi
    \ifchaptersectionnumbers
        \chapfolio.
    \fi
    \the\sectno}

\def\sectref#1{                                  % refer to some
    \ifundefined{SECTLABEL#1}                     % section (forward
        ++                                        % or backward):
        \ifproofmode
            \ifforwardreference
            \else
                \write16{ ***Undefined\space Section\space Reference #1*** }
            \fi
        \else
            \write16{ ***Undefined\space Section\space Reference #1*** }
        \fi
    \else
        \edef\LABxx{\getlabel{SECTLABEL#1}}
	\ifchaptersectionnumbers
            \def\LAByy{\expandafter\stripchap\LABxx}
	    \chapshow\LAByy.
	\fi
	\expandafter\stripsect\LABxx
    \fi
    \ifproofmode
        \write\allcrossfile{Section\space #1}
    \fi}

% Various ways to place a number on an equation.
%     \eqnum			% just the equation number: (7) or (3.7)
%     \eqlabel{...}		% also set up a label for reference
%     \eqalignnum  		% used in the context of \eqalignno
%     \eqalignlabel{...}	%  "    "  "     "    "      "
%
%     \eqref{...}		% in text, refer to a labeled equation
%
\def\eqnum{                                    % numbered equation
    \global\advance\eqlabelno by 1              % but no reference.
    \eqno(
    \ifeqchapternumbers
        \chapfolio.
    \fi
    \ifeqsectionnumbers
        \the\sectno.
    \fi
    \the\eqlabelno)}

\def\eqlabel#1{                                % for setting up
    \global\advance\eqlabelno by 1              % a reference to
    \ifforwardreference                     % an equation:
        \immediate\write\labelfileout{\noexpand\expandafter\noexpand\def
        \noexpand\csname EQLABEL#1\endcsname
        {\the\chapno.\the\sectno?\the\eqlabelno!}}
    \fi
    \global\expandafter\edef\csname EQLABEL#1\endcsname
    {\the\chapno.\the\sectno?\the\eqlabelno!}
    \eqno(
    \ifeqchapternumbers
        \chapfolio.
    \fi
    \ifeqsectionnumbers
        \the\sectno.
    \fi
    \the\eqlabelno)
    \ifproofmode
        \rlap{\hbox{\marginstyle #1}}		% marginal note?
    \fi}

\def\eqalignnum{                               % numbered eqalignno
    \global\advance\eqlabelno by 1              % but no reference
    &(\ifeqchapternumbers
        \chapfolio.
    \fi
    \ifeqsectionnumbers
        \the\sectno.
    \fi
    \the\eqlabelno)}

\def\eqalignlabel#1{                   	% for refering to
    \global\advance\eqlabelno by 1 	        % an EQALIGNNO'ED
    \ifforwardreference                     % equation:
        \immediate\write\labelfileout{\noexpand\expandafter\noexpand\def
        \noexpand\csname EQLABEL#1\endcsname
        {\the\chapno.\the\sectno?\the\eqlabelno!}}
    \fi
    \global\expandafter\edef\csname EQLABEL#1\endcsname
    {\the\chapno.\the\sectno?\the\eqlabelno!}
    &(\ifeqchapternumbers
        \chapfolio.
    \fi
    \ifeqsectionnumbers
        \the\sectno.
    \fi
    \the\eqlabelno)
    \ifproofmode
        \rlap{\hbox{\marginstyle #1}}			% marginal note?
    \fi}

\def\dnum{                                     % numbered equation
    \global\advance\eqlabelno by 1              % but no reference.
    \llap{(	 				% (special purpose)
    \ifeqchapternumbers
        \chapfolio.
    \fi
    \ifeqsectionnumbers
        \the\sectno.
    \fi
    \the\eqlabelno)}}
\def\deqnum{\dnum}

\def\dlabel#1{                                 % for setting up
    \global\advance\eqlabelno by 1              % a reference to
    \ifforwardreference                         % an equation:
        \immediate\write\labelfileout{\noexpand\expandafter\noexpand\def
        \noexpand\csname EQLABEL#1\endcsname
        {\the\chapno.\the\sectno?\the\eqlabelno!}}
    \fi
    \global\expandafter\edef\csname EQLABEL#1\endcsname
    {\the\chapno.\the\sectno?\the\eqlabelno!}
    \llap{(
    \ifeqchapternumbers
        \chapfolio.
    \fi
    \ifeqsectionnumbers
        \the\sectno.
    \fi
    \the\eqlabelno)}
    \ifproofmode
        \rlap{\hbox{\marginstyle #1}}		% marginal note?
    \fi}
\def\deqlabel#1{\dlabel#1}

\def\eqref#1{\ifparenequations(\fi
    \ifundefined{EQLABEL#1}***
        \ifproofmode
            \ifforwardreference
            \else
                \write16{ ***Undefined\space Equation\space Reference #1*** }
            \fi
        \else
            \write16{ ***Undefined\space Equation\space Reference #1*** }
        \fi
    \else
        \edef\LABxx{\getlabel{EQLABEL#1}}
	\def\LAByy{\expandafter\stripsect\LABxx}
        \def\LABzz{\expandafter\stripchap\LABxx}
        \ifeqchapternumbers
            \chapshow{\LABzz}.
        \else
            \ifnum \number\LABzz=\chapno \relax
            \else
                \chapshow{\LABzz}.
            \fi
        \fi
	\ifeqsectionnumbers
	    \LAByy.
	\fi
        \expandafter\stripeq\LABxx
    \fi
    \ifparenequations)\fi
    \ifproofmode
        \write\allcrossfile{Equation\space #1}
    \fi}

% These are analogous to \eqnum etc.  They will automatically
% generate figure numbers and in the second case accept your
% label for later reference.
%
\def\fignum{                                   % numbered figure
    \global\advance\figureno by 1\relax         % but no reference.
    \iffigurechapternumbers
        \chapfolio.
    \fi
    \the\figureno}

\def\figlabel#1{				% for setting up
    \global\advance\figureno by 1\relax 	% a reference to
    \ifforwardreference				% a figure:
        \immediate\write\labelfileout{\noexpand\expandafter\noexpand\def
        \noexpand\csname FIGLABEL#1\endcsname
        {\the\chapno.\the\sectno?\the\figureno!}}
    \fi
    \global\expandafter\edef\csname FIGLABEL#1\endcsname
    {\the\chapno.\the\sectno?\the\figureno!}
    \iffigurechapternumbers
        \chapfolio.
    \fi
    \ifproofmode
        \llap{\hbox{\marginstyle #1\ }}\relax
    \fi
    \the\figureno}

\def\figref#1{					% refer to some
    \ifundefined				% figure (forward
        {FIGLABEL#1}!!!!			% or backward):
        \ifproofmode
            \ifforwardreference
            \else
                \write16{ ***Undefined\space Figure\space Reference #1*** }
            \fi
        \else
            \write16{ ***Undefined\space Figure\space Reference #1*** }
        \fi
    \else
        \edef\LABxx{\getlabel{FIGLABEL#1}}
        \def\LABzz{\expandafter\stripchap\LABxx}
        \iffigurechapternumbers
            \chapshow{\LABzz}.\expandafter\stripeq\LABxx
        \else \ifnum\number\LABzz=\chapno \relax
                \expandafter\stripeq\LABxx
            \else
                \chapshow{\LABzz}.\expandafter\stripeq\LABxx
            \fi
        \fi
        \ifproofmode
            \write\allcrossfile{Figure\space #1}
        \fi
    \fi}

% You can refer to a specific page of the text by this macro.
% Simple say \pagelabel{yourlabel}  and you can then refer
% to that page by the name yourlabel.
%
% Note: This label making macro should
%       NOT use an \immediate\write.
%
\def\pagelabel#1{
    \ifforwardreference
        \write\labelfileout{
        \noexpand\expandafter\noexpand\def
        \noexpand\csname PGLABEL#1\noexpand\endcsname{\the\pageno}}
    \fi
    \global\expandafter\edef\csname PGLABEL#1\endcsname{\the\pageno}}

\def\pageref#1{
    \ifundefined
        {PGLABEL#1}***
        \ifproofmode
        \else
            \write16{ ***Undefined\space Page\space Reference #1*** }
        \fi
    \else
        \csname PGLABEL#1\endcsname
    \fi
    \ifproofmode
        \write\allcrossfile{Page\space #1}
    \fi}

% Another similar set of macros for footnotes:
% These are analogous to \eqnum etc.  They will automatically
% generate reference numbers and in the second case accept your
% label for later reference.
%
\def\refnum{                                      % numbered reference
    \global\advance\referenceno by 1\relax         % but no reference.
    \the\referenceno}	                           % Useless, but symmetric

\def\internalreflabel#1{			% for setting up
    \global\advance\referenceno by 1\relax 	% a reference to
    \ifforwardreference				% a reference:
        \immediate\write\labelfileout{\noexpand\expandafter\noexpand\def
        \noexpand\csname REFLABEL#1\endcsname
        {\the\chapno.\the\sectno?\the\referenceno!}}
    \fi
    \global\expandafter\edef\csname REFLABEL#1\endcsname
    {\the\chapno.\the\sectno?\the\figureno!}
    \ifproofmode
        \llap{\hbox{\marginstyle #1\hskip.5cm}}\relax
    \fi
    \the\referenceno}

\def\internalrefref#1{				% refer to some
    \ifundefined				% figure (forward
        {REFLABEL#1}!!!!			% or backward):
        \ifproofmode
            \ifforwardreference
            \else
                \write16{ ***Undefined\space Footnote\space Reference #1*** }
            \fi
        \else
            \write16{ ***Undefined\space Footnote\space Reference #1*** }
        \fi
    \else
        \edef\LABxx{\getlabel{REFLABEL#1}}
        \def\LABzz{\expandafter\stripchap\LABxx}
        \expandafter\stripeq\LABxx
        \ifproofmode
            \write\allcrossfile{Reference\space #1}
        \fi
    \fi}

% When the references are placed at the end, a standard format is used: e.g.
% These are three examples of how to make your own:
%
%\def\reflabel#1{\item{[\internalreflabel{#1}]}}
%\def\reflabel#1{\noindent{\bf \internalreflabel{#1}}.\space}
\def\reflabel#1{\item{\internalreflabel{#1}.}}

% Also, you may want a special format for placing the reference in the body
% of the text. This will make it easier to write your own.
%
%\def\refref#1{[\internalrefref{#1}]}
\def\refref#1{\internalrefref{#1}}

\def\eq{\ifhmode Eq.~\else Equation~\fi}		% most common forms.
\def\eqs{\ifhmode Eqs.~\else Equations~\fi}
\def\Eq{Equation~}
\def\fig{Figure~}
\def\figs{Figures~}

% The following are typical forms for the beginning of sections:
%
% The format is   \sectionhead{The title of the section}
% OR              \sectionheadlabel{The title of the section}{alabel}
%
% The choice of \vskip size and font is typical.
% An alternative format is commented out.  This presents an underlined
% header.
%
%\def\sectionhead#1{
%    \vskip13pt plus 2pt minus 1pt
%    \centerline{
%        \ifsectionnumbers
%            \sectnum\space
%        \fi
%        $\underline{\smash{\hbox{#1}}}$}
%    \nobreak\vskip1pt\nobreak}
%
%\def\sectionheadlabel#1#2{
%    \ifsectionnumbers
%    \else
%        \message{
%    Using SECTIONHEADLABEL without SECTIONNUMBERSTRUE does not make sense.
%    }
%    \fi
%    \vskip13pt plus 2pt minus 1pt
%    \centerline{
%        \ifsectionnumbers
%            \sectlabel{#2}\space
%        \fi
%        $\underline{\smash{\hbox{#1}}}$}
%    \nobreak\vskip1pt\nobreak}

\def\sectionhead#1{
    \vskip13pt plus 2pt minus 1pt 			% SECTION
    \ifsectionnumbers					% HEADERS
        {\bf\noindent\sectnum\ {#1}}
    \else
        {\bf\noindent#1}
    \fi
    \hfil\break\nobreak}

\def\sectionheadlabel#1#2{
    \ifsectionnumbers
    \else
        \message{
    Using SECTIONHEADLABEL without SECTIONNUMBERSTRUE does not make sense.
    }
    \fi
    \vskip13pt plus 2pt minus 1pt
    {\bf\noindent\sectlabel{#2}\ #1}
    \hfil\break\nobreak
    }

% Utilities for use by other macros
%
\def\getlabel#1{\csname#1\endcsname}
\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\stripchap#1.#2?#3!{#1}			% NOTE FORMAT:
\def\stripsect#1.#2?#3!{#2}			%
\def\stripeq#1.#2?#3!{#3}			% ##.##?##!
						% chapter.section?equation!
}  %%%%% This is the termination of the special \catcode's.

%	To place an equation number on an equation in display mode,
%	simply type
%		\eqnum
%	at the end of the equation.

%	To be able to refer to that equation number in the text, use
%		\eqlabel{alabelofmychoice}
%	instead. What goes into the braces is up to you.

%	To refer to a labeled equation in the text, use
%		\eqref{alabelofmychoice}
%	and it will place the correct, parenthesized number,
%	e.g. (11).
%	Another macro to use in addition to these is
%		\eq
%      	This will simply place the standard phrase, Eq.~ so that
%		\eq\eqref{alabelofmychoice}
%	can become Eq.~(11)
%	This "\eq" macro is convenient because you may change your mind
%	about how you want references to appear. You then need only change
%	the definition of "\eq" to change it everywhere.

%	When using an ALIGNED displayed equation, the syntax is
%	different.  When you use the TeX command "\eqalignno" then
%	instead of  "\eqnum" use
%		\eqalignnum
%	Instead of "\eqlabel" use
%		\eqalignlabel
%	The use of "\eqref" is unaffected by this change.

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%	James Nearing
%	Physics Dept.
%	University of Miami
%	Coral Gables, Florida
%	jnearing@umiami

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\line{\hfill LPTHE-PAR 93-09, UMTG-171}

\vskip 0.2 in


\centerline{\bf Thermodynamics of Integrable Chains with Alternating Spins}
\bigskip

\medskip

\centerline{H. J. de Vega\footnote{$\dagger$}{Laboratoire de Physique
Th\'eorique et Hautes Energies, Tour 16 - 1er. \'etage, Universit\'e
Paris VI, 4, place Jussieu, 75252 Paris Cedex 05, FRANCE},
Luca Mezincescu${}^*$, and
Rafael I. Nepomechie\footnote*{Department of Physics, University of Miami,
Coral Gables, FL 33124, USA}}

\vskip 0.2 in

\bigskip

\centerline{\bf Abstract}

\vskip 0.2 in

We consider a two-parameter $( \bar c \,, \tilde c )$ family of quantum
integrable Hamiltonians for a chain of alternating spins of spin $s=1/2$ and
$s=1$. We determine the thermodynamics for low-temperature $T$ and small
external magnetic field $H$, with $T << H$.
In the antiferromagnetic $( \bar c > 0\,, \tilde c > 0)$ case, the model has
two gapless excitations. In particular,
for $\bar c = \tilde c$, the model is conformally invariant and has central
charge $c_{vir} = 2$. When one of these parameters is zero, the Bethe
Ansatz equations admit an infinite number of solutions with lowest energy.

\vfill\eject

The one-dimensional Heisenberg model, like the hydrogen atom, has served
as the guiding example for a very large body of both experimental and
theoretical work. Progress has recently been made on closely related models,
consisting of chains with {\it alternating} spins, such as spin 1/2 and spin 1.
On the experimental side, materials (e.g.${}^{\refref{experimental}}$,
[MnCp${}_2^*$] [TCNE]) have been synthesized which, at
temperatures above a certain transition temperature $T_c$, behave as
one-dimensional ferromagnets of alternating spins.
On the theoretical side, quantum integrable models of chains with alternating
spins have recently been constructed${}^{\refref{devega/woynarovich}}$. In this
Letter we investigate the thermodynamics of a two-parameter family of such
integrable models. Depending on the values of the parameters, we find both
antiferromagnetic and ferromagnetic behavior. When one of these parameters is
zero, the Bethe Ansatz equations admit an infinite number of solutions with
lowest energy.

We consider a system of $N$ spins
${1\over 2}\vec \sigma_2 \,, {1\over 2}\vec \sigma_4 \,, \cdots \,,
{1\over 2}\vec \sigma_{2N}$
of spin 1/2 and $N$ spins $\vec s_1 \,, \vec s_3 \,,$  \break
$\cdots \,, \vec s_{2N-1}$  of spin 1 in an external magnetic field
$H (\ge 0)$ with the Hamiltonian ${\cal H}$ given by
$$ {\cal H} = \bar c  \bar {\cal H} + \tilde c \tilde {\cal H}
- H S^z \,, \eqlabel{hamiltonian} $$
where
$S^z =\sum_{n=1}^N {1\over 2} \sigma^z_{2n} + \sum_{n=1}^N s^z_{2n-1}$,
$$\eqalignno{
  \bar {\cal H} &= -{1 \over 9} \sum_{n=1}^{N} \left( 2 \vec \sigma_{2n} \cdot
\vec s_{2n+1} + 1 \right) \left( 2 \vec \sigma_{2n+2} \cdot
\vec s_{2n+1} + 3 \right) \,, \eqalignnum \cr
\tilde {\cal H} &=  -{1 \over 9} \sum_{n=1}^{N} \left( 2 \vec \sigma_{2n} \cdot
\vec s_{2n-1} + 1 \right) \left[ \left( 1 +  \vec s_{2n-1} \cdot
\vec s_{2n+1}\right) \left( 2 \vec \sigma_{2n} \cdot
\vec s_{2n+1} + 1 \right) + 2 \right] \,, \eqalignnum \cr} $$
and $\bar c$ and $\tilde c$ are real constant parameters. (In this paper,
bars and tildes are interchanged with respect to Ref.
\refref{devega/woynarovich}.) Note that the Hamiltonian contains both nearest
and next-to-nearest neighbor interactions. We assume periodic boundary
conditions: $\sigma_{2n} \equiv \sigma_{2n + 2N}$ and $s_{2n+1}
\equiv s_{2n +1 + 2N}$.

The corresponding energy eigenvalues are given
by${}^{\refref{devega/woynarovich}}$
$$ E = \bar c  \bar E + \tilde c \tilde E  - H \left( {3\over 2}N - M  \right)
\,, \eqlabel{eigenvalues} $$
where
$$
\bar E = {5\over 3}N - i  \sum_{j=1}^M {d\over d\lambda_j}
\ln \left( \lambda_j + {i\over 2} \over \lambda_j - {i\over 2} \right) \,,
\qquad
\tilde E = - {1\over 2}N - i  \sum_{j=1}^M {d\over d\lambda_j}
\ln \left( \lambda_j + i \over \lambda_j - i \right) \,,
\eqlabel{EE}  $$
where the variables $\lambda_j$ satisfy the Bethe Ansatz (BA) equations
$$
\left( {\lambda_j + {i\over 2} \over \lambda_j - {i\over 2}}
       {\lambda_j + i \over \lambda_j - i} \right)^N
= - \prod_{k=1}^M {\lambda_j - \lambda_k + i \over
\lambda_j - \lambda_k - i } \,, \qquad j = 1, \cdots , M \,. \eqlabel{BA} $$
We consider here a strictly alternating arrangement of spins, with
spins 1/2 at even sites and spins 1 at odd sites. For any other ordering
of the spins, one can construct a corresponding Hamiltonian which has the
same energy eigenvalues and BA equations.

This system of equations admits the same string solutions that are
found for the Heisenberg model. In the thermodynamic limit, the model
is characterized by particle densities $\rho_n (\lambda)$ and hole densities
$\tilde \rho_n (\lambda)$. Following the standard procedure (see, e.g.,
Refs. \refref{tsvelick/wiegmann}, \refref{xxx/spin/s}),
we find that these densities obey the constraints
$$\tilde \rho_n +  \sum_{m=1}^\infty A_{nm} * \rho_{m} = a_n
+ \sum_{l=1}^{min(n,2)} a_{n+3 -2l} \,, \eqlabel{constraint}$$
where
$$a_n(\lambda) = {1\over 2\pi} {n\over \lambda^2 + {n^2\over 4}} \,, \eqnum $$
$$A_{nm}(\lambda) = \delta_{nm}\delta(\lambda)
+ \left(1 - \delta_{nm} \right) a_{|n - m|}(\lambda) + a_{n + m}(\lambda)
+ 2\sum_{l=1}^{min(n,m)-1} a_{|n - m| + 2l}(\lambda) \,, \eqlabel{Anm} $$
and $*$ denotes the convolution
$ \left( f * g \right) (\lambda) = \int_{-\infty}^\infty
d\lambda'\ f(\lambda - \lambda') g(\lambda') \,.$

The thermodynamic Bethe Ansatz (TBA) equations describing the equilibrium
system at temperature $T$ are
$$T \ln \left( 1 + e^{\epsilon_n/T} \right)
= \sum_{m=1}^\infty  A_{nm} * T \ln \left( 1 + e^{-\epsilon_m/T} \right)
-2\pi \left( \bar c a_n + \tilde c \sum_{l=1}^{min(n,2)} a_{n+3-2l} \right)
 + n H \,, \eqlabel{TBA} $$
where
$$\epsilon_n (\lambda) = T \ln \left( \tilde\rho_n (\lambda)/
\rho_n (\lambda)\right) \,. \eqlabel{epsilon}  $$
(The particle and hole densities are understood to be those at equilibrium.)
The equilibrium free energy is given by
$$F = 2N e_0
- N T \int_{-\infty}^\infty  d\lambda\ s(\lambda)
\ln \left[ \left( 1 + e^{\epsilon_1(\lambda)/T} \right)
           \left( 1 + e^{\epsilon_2(\lambda)/T} \right) \right]
\,, \eqlabel{free} $$
where $s(\lambda) = 1/ \left( 2 \ch \pi \lambda \right) \,,$
and
$ 2 e_0 = \bar c \left[ {5\over 3}  - \left( 2\ln 2 + 4 -\pi \right) \right]
       + \tilde c \left[ - {1\over 2} - \left( 6 - \pi \right) \right]
 \,.$

In the high-temperature limit with zero field, these equations give the
expected value for the entropy, namely $S = N \ln 6$. We now
consider small values of $T$ and $H$, with $T << H$. We define
$\varepsilon_n(\lambda) = \lim_{T \rightarrow 0}\ \epsilon_n(\lambda) \,.$
Keeping in mind that $\epsilon_n > 0$ for $H > 0$ and $n \ge 3$, we obtain from
the TBA equations the following system of linear integral equations for
$\varepsilon_1$ and $\varepsilon_2$:
$$\eqalignno{
\varepsilon_1 &= -2\pi \bar c s + s * \varepsilon^+_2  \,, \cr
\varepsilon_2 &= {1\over 2}H -2\pi \tilde c s + h * \varepsilon^+_2
+ s * \varepsilon^+_1 \,, \eqalignlabel{varepsilon} \cr} $$
where $h = s * a_1 \,,$
and the superscript $+$ on a function denotes the positive part
of that function; i.e.,
$\varepsilon^+ \equiv {1\over 2} \left( \varepsilon + |\varepsilon| \right)
\,.$

The qualitative behavior of the solutions depends on the sign of the
parameters $\bar c$ and $\tilde c$, and the various cases must be
studied individually. Let us first consider the
antiferromagnetic case $\bar c > 0$, $\tilde c > 0$. In this case, for $H = 0$
the solutions $\varepsilon_1(\lambda)$ and $\varepsilon_2(\lambda)$ are
readily found, and are seen to be negative for all $\lambda$. Hence, for the
ground state, $\rho_1(\lambda) = \rho_2(\lambda) = s(\lambda) \,,$
and all other particle and hole densities are equal to
zero${}^{\refref{note1}}$.
Thus, the ground state is a ``sea'' of strings of length 1 and a
``sea'' of strings of length 2, in agreement with the alternative analysis of
Ref. \refref{devega/woynarovich}. This corresponds to the antiferromagnetic
ground states of integrable chains${}^{\refref{xxx/spin/s}}$ of
spin $s=1/2$ and $s=1$.

For $H$ small but nonzero, we can solve for $\varepsilon_1$ and $\varepsilon_2$
by generating from \eqref{varepsilon} a system of Wiener-Hopf equations.
To this end, we define $\alpha_n$ to be the zeros of
$\varepsilon_n(\lambda)$, i.e.,
$ \varepsilon_n(\alpha_n) =  0 \,,$  for $n = 1\,, 2 \,.$
We assume
$$\alpha_n = -{1\over \pi} \left[
\ln H  + \ln \kappa_n + O \left( {1\over \ln H} \right) \right]
\,, \eqlabel{kappa} $$
where the constants $\kappa_n$ (which are independent of $H$) have still
to be determined. Introducing the functions
$$S_n(\lambda) = \left\{ \matrix{e^{\pi \alpha_n}
\kappa_n \varepsilon_n (\lambda + \alpha_n)  & \lambda > 0 \cr
0 & \lambda < 0 \cr} \right. \,, \eqlabel{funcS} $$
we obtain (for $H \rightarrow 0$) the following system of Wiener-Hopf equations
$$\eqalignno{
S_1(\lambda) &= -2\pi \bar c \kappa_1 e^{-\pi\lambda}
+ \int_0^\infty d\lambda'\
s(\lambda - \lambda' + \alpha_1 - \alpha_2) S_2(\lambda') \,, \cr
S_2(\lambda) &= {1\over 2} -2\pi \tilde c \kappa_2 e^{-\pi\lambda}
+ \int_0^\infty d\lambda'\ \left[
s(\lambda - \lambda' + \alpha_2 - \alpha_1) S_1(\lambda') +
h(\lambda - \lambda') S_2(\lambda') \right] \,, \cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad \lambda \ge 0 \,.
\eqalignlabel{WHS} \cr} $$

In order to find the leading-order temperature dependence of the free energy,
we must compute the leading correction to the solutions
$\epsilon_n = \varepsilon_n$ of the linearized equations \eqref{varepsilon}.
Hence, we set
$\epsilon_n (\lambda) = \varepsilon_n(\lambda) + \eta_n (\lambda)$
in the TBA equations and expand to leading order in $\eta_n$, as is explained
in Refs. \refref{johnson/mccoy} and \refref{kingston}.
In terms of the functions
$$T_n(\lambda) = \left\{ \matrix{ {6 e^{-\pi \alpha_n}
\over \pi^2 T^2 \kappa_n}
\eta_n (\lambda + \alpha_n)  & \lambda > 0 \cr
0 & \lambda < 0 \cr} \right. \,, \eqlabel{funcT} $$
we obtain (for $H \rightarrow 0$) a second system of Wiener-Hopf equations:
$$\eqalignno{
T_1(\lambda) &=
{s(\lambda + \alpha_1 - \alpha_2)\over S_2'(0)}
+ \int_0^\infty d\lambda'\
s(\lambda - \lambda' + \alpha_1 - \alpha_2) T_2(\lambda') \,, \cr
T_2(\lambda) &= {h(\lambda)\over S_2'(0)} +
{s(\lambda + \alpha_2 - \alpha_1)\over S_1'(0)}
+ \int_0^\infty d\lambda'\ \left[
s(\lambda - \lambda' + \alpha_2 - \alpha_1) T_1(\lambda') +
h(\lambda - \lambda') T_2(\lambda') \right] \,, \cr
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad \lambda \ge 0 \,,
\eqalignlabel{WHT} \cr} $$
where $ S_n'(0) = d S_n/ d\lambda \Big\vert_{\lambda = 0^+}$.

Both systems \eqref{WHS} and \eqref{WHT} involve the same
$2 \times 2$ matrix kernel, which in Fourier space is given by
$$\hat K (\omega) = \left(
\matrix{ 0 &  e^{-i \omega
\left( \alpha_1 -\alpha_2 \right)} \hat s(\omega)\cr
e^{i\omega \left( \alpha_1 - \alpha_2 \right)}
\hat s(\omega)  & \hat h(\omega) \cr} \right) \,.
\eqlabel{kernel} $$
(Our convention for Fourier transforms is that
$\hat f(\omega) = \int_{-\infty}^\infty d\lambda\ e^{i \lambda \omega}
f(\lambda)$.)
Results from Ref. \refref{gohberg/krein} imply that the following factorization
exists
$$ \left( 1 - \hat K(\omega) \right)^{-1} = G_+ (\omega)\ G_- (\omega)
\,, \qquad\qquad -\infty < \omega < \infty \,,
\eqlabel{factorization} $$
where $G_+ (\omega)$ and $G_+^{-1} (\omega)$ are analytic in the upper-half
complex $\omega$ plane with $G_+ (+\infty) = 1$, and (for $\omega$ in the
lower-half plane) $G_-(\omega) = G_+(-\omega)^T \,.$

Using standard Wiener-Hopf methods, we  conclude that the solutions of
Eqs. \eqref{WHS} and \eqref{WHT} are given (in matrix notation) by
$$\hat S(\omega) = {i\over 2}
\left( {1\over \omega + i0} - {1\over \omega + i\pi} \right)
G_+(\omega) G_-(0) \left( \matrix{ 0 \cr
                                   1 \cr} \right)
 \,, \eqlabel{Ssolution} $$
and
$$\hat T (\omega) = \left( G_+(\omega) - 1 \right)
\left( \matrix{ 1/S_1'(0) \cr
                1/S_2'(0) \cr} \right) \,, \eqlabel{Tsolution} $$
where
$$ S'(0) = - \lim_{|\omega| \rightarrow \infty} \omega^2 \hat S(\omega)
= {\pi \over 2}
G_-(0) \left( \matrix{ 0 \cr
                       1 \cr} \right)
\,. \eqlabel{derivative} $$
Moreover, the parameters $\kappa_n$ introduced in \eq\eqref{kappa} are given by
$$\kappa = \left( \matrix{ \kappa_1 \cr
                           \kappa_2 \cr} \right)
= {1\over 4\pi} \left( \matrix{ 1/\bar c & 0 \cr
                                  0      & 1/\tilde c \cr} \right)
G_-(-i\pi )^{-1} G_-(0) \left( \matrix{ 0 \cr
                                        1 \cr} \right) \,.
\eqnum $$

In order to calculate the free energy per site $f= F/2N$, we substitute
$\epsilon_n = \varepsilon_n + \eta_n$ into the expression \eqref{free}
for the free energy, as is explained in Refs. \refref{johnson/mccoy}
and \refref{kingston}. We obtain
$$f = e_0 - H^2 A  - {\pi^2 T^2 \over 6} B \,, \eqnum $$
with
$$ A = \kappa^T\ \hat S(i\pi) \,, \qquad
   B = \kappa^T\ \left[ \hat T(i\pi)
+ \left( \matrix{ {1/ S_1'(0)} \cr
                  {1/ S_2'(0)} \cr}  \right) \right]  \,.
\eqnum $$

For $\bar c \ne \tilde c$, the quantities $A$ and $B$ cannot be explicitly
evaluated without having explicit expressions for the factors
$G_+(\omega)$ and $G_-(\omega)$, which we have not yet found.

For the special case $\bar c = \tilde c \equiv c$, the quantities $A$ and $B$
can be readily evaluated, and we conclude (for $T << H$ )
$$f = e_0 - {1\over 4\pi^2 c} H^2 - {1\over 6 c}T^2 \,. \eqlabel{final} $$
It follows that the magnetic susceptibility and specific heat, to lowest
order, are given by
$$
\chi = - {\partial^2 f\over \partial H^2}
\Big\vert_T = {1\over 2\pi^2 c} \,, \qquad
C_H = -T {\partial^2 f\over \partial T^2}
\Big\vert_H = {1\over 3 c}T \,,   \eqnum $$
respectively.

This model has${}^{\refref{devega/woynarovich}, \refref{note}}$ two gapless
excitations, with corresponding speeds of sound $\bar v = 2\pi \bar c$ and
$\tilde v = 2\pi \tilde c$. Evidently, the case $\bar c = \tilde c$ is
the unique case for which the two speeds of sound coincide, and
the model is conformally invariant. For a critical chain, the low-temperature
free energy per site is given by${}^{\refref{cardy}, \refref{affleck1}}$
$$f = e_0 - {\pi c_{vir}\over 6 v_s} T^2 + \cdots \,, \eqnum $$
where $c_{vir}$ is the central charge of the Virasoro algebra and $v_s$ is
the speed of sound. Therefore, from \eqref{final} we see that $c_{vir} = 2$.
Presumably it is no coincidence that precisely for the conformally invariant
case, explicit expressions for $G_+(\omega)$ and $G_-(\omega)$ are not
needed to evaluate the free energy.

We now consider the case $\bar c = 0$, $\tilde c > 0$. For this case, there
is a one-parameter $(\alpha)$ family of lowest-energy states. Indeed,
consider the following one-parameter family of densities:
$$
\rho_1 = \alpha s \,, \qquad \tilde \rho_1 = (1 - \alpha) s \,, \qquad
\rho_2 = s + (1 - \alpha) s*s \,,  \qquad\qquad
0 \le \alpha \le 1 \,, \eqnum $$
and all other particle and hole densities are equal to zero. These densities
obey the constraints \eqref{constraint}, and give (independently of
the value of $\alpha$) the same lowest value for the energy. Moreover, to
leading order in $N$, the spin is $S^z = 0$, and the entropy is
$S = -{N\over 2}\left[ \alpha \ln \alpha + (1 - \alpha )\ln (1 - \alpha)
\right]$. In particular, for $\alpha \ne 0 \,, 1$,  the entropy is nonzero
and is proportional to $N$, implying an infinite degeneracy of states.
This degeneracy is consistent with the fact that, above the $\alpha =1$
vacuum, there are excitations (namely, holes in the sea of real roots) which
have {\bf zero energy} and non-zero momentum${}^{\refref{devega/woynarovich}}$.
We are not aware of any other model with such properties${}^{\refref{note2}}$.

We speculate that the system can be brought to these various states by first
preparing the system at finite $T$ and $H$, and then approaching the origin
$(T=0 \,, H=0)$ of the $(T \,, H)$-plane from appropriate directions.
The ground state of the system is the state which is reached by approaching
the origin along the line $H=0$. Unfortunately, we cannot determine
the particular value of $\alpha$ corresponding to this state,
since this would entail computing
$\lim_{T \rightarrow 0}\ \lim_{H \rightarrow 0} \epsilon_1/T$, while
we know how to calculate only for $T << H$ ${}^{\refref{note3}}$.
By approaching the origin of the $(T \,, H)$-plane along the line $T=0$,
the state with $\alpha = 0$ is reached.

We have calculated the free energy for small values of $T$ and $H$,
with $T << H$. The calculation is similar to the one above,
except that now one must take into account that $\varepsilon_1 (\lambda)$
does not have a zero. We find that, to leading order, the free energy per site
is given by
$$f = e_0 +  {1\over 4\pi^3 \tilde c} H^2 \ln H +
{1\over 12 \pi \tilde c} T^2 \ln H \,. \eqnum $$
Contrary to appearance, this result does {\it not} imply that
$f$ diverges for $H \rightarrow 0$, since in the region where the calculation
is valid ($T << H$), the last term is finite. This result is nevertheless
unusual, since it implies that $C_H/T$ is a function of $H$.

Similar results are obtained for the case $\tilde c = 0$, $\bar c > 0$.
For the cases $\bar c < 0$, $\tilde c > 0$ and $\bar c > 0$, $\tilde c < 0$ ,
the model either has a ferromagnetic ground state and a finite gap, or
it has no gap, depending on the precise values of $\bar c \,, \tilde c \,,$
and $H$.
For the cases $\bar c \,, \tilde c \le 0 \,,$ the model is ferromagnetic.


\bigskip

We thank L. Susskind for a very helpful discussion and F. Zuo for bringing
Ref. \refref{experimental} to our attention. One of us (H J de V) thanks the
Department of Physics of the University of Miami for the warm hospitality
extended to him. This work was supported in part by the National Science
Foundation under Grant No. PHY-92 09978.

\bigskip

Note added: After this work was completed, we received a preprint
(Ref. \refref{martins}) which discusses chains of alternating spin 1/2 and
spin $s$. However, that paper considers neither the effect of an external
magnetic field, nor the behavior of the models away from the conformally
invariant point (i.e., for $\bar c \ne \tilde c$).

\bigskip

\noindent
{\bf References}

\vskip 0.2truein

\reflabel{experimental}
G.T. Yee, J.M. Manriquez, D.A. Dixon, R.S. McLean, D.M. Groski, R.B. Flippen,
K.S. Narayan, A.J. Epstein, and J.S. Miller, Adv. Mat. {\it 3}, 309 (1991).

\reflabel{devega/woynarovich}
H.J. de Vega and F. Woynarovich, J. Phys. {\it A25}, 4499 (1992).

\reflabel{tsvelick/wiegmann}
A.M. Tsvelick and P.B. Wiegmann, Adv. in Phys. {\it 32}, 453 (1983).

\reflabel{xxx/spin/s}
H.M. Babujian, Nucl. Phys. {\it B215}, 317 (1983);
L.A. Takhtajan, Phys. Lett. {\it 87A}, 479 (1982).

\reflabel{note1}
By definition, the set of ground state densities should be determined through
\eq\eqref{epsilon} by computing
$\lim_{T \rightarrow 0}\ \lim_{H \rightarrow 0} \epsilon_n/T$,
and then imposing the constraints \eqref{constraint}. Here, we compute
instead $\lim_{H \rightarrow 0}\ \lim_{T \rightarrow 0} \epsilon_n/T$, and
we assume that the two limits commute. If the limits did not commute,
there would be another set of densities which would also satisfy the
constraints \eqref{constraint} and give the same value for the
energy. We believe that for the case $\bar c > 0$, $\tilde c > 0$
such an additional set of densities does not exist.

\reflabel{johnson/mccoy}
J.D. Johnson and B.M. McCoy, Phys. Rev. {\it A6}, 1613 (1972).

\reflabel{kingston}
L. Mezincescu and R.I. Nepomechie, in {\it Quantum Groups, Integrable Models
and Statistical Systems}, ed. by J. Le Tourneux and L. Vinet (World
Scientific), in press.

\reflabel{gohberg/krein}
I.C. Gohberg and M.G. Krein, in {\it American Mathematical Society
Translations, Series 2, Vol 14} (American Mathematical Society, 1960) 217.

\reflabel{note}
We define the momentum here as one-half the log of the two-site shift operator.
This yields the stated values for the speeds of sound, and a system of length
$2N$ (see Eq. (25)). In Ref. 2, the log of the two-site shift operator is used,
which leads to half these values for the speeds and length. We thank M. Martins
for a discussion on this point.

\reflabel{cardy}
H.W.J. Bl\"ote, J.L. Cardy and M.P. Nightingale,
Phys. Rev. Lett. {\it 56}, 742 (1986).

\reflabel{affleck1}
I. Affleck, Phys. Rev. Lett. {\it 56}, 746 (1986).

\reflabel{note2}
Implicit in this discussion is the conventional assumption that a set of
densities which satisfies the constraints \eqref{constraint} and has lowest
energy corresponds to an eigenstate of the Hamiltonian.

\reflabel{note3}
In contrast to the case $\bar c > 0$, $\tilde c > 0$, here we expect that
$\lim_{T \rightarrow 0}\ \lim_{H \rightarrow 0} \epsilon_1/T \ne
 \lim_{H \rightarrow 0}\ \lim_{T \rightarrow 0} \epsilon_1/T \,. $

\reflabel{martins}
S.R. Aladim and M.J. Martins, ``Critical behavior of integrable mixed
spin chains'' UFSCAR-93-03.

\end

