%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This is a Plain TeX file with 3 figures appended as postscript % % files. Simply TeXing the file will produce a copy without % % figures. In order to print the figures they have to be saved in% % separate files, beginning with a line "%!PS-Adobe-2.0" and % % ending with a line "%%EOF". These files can then be sent to a % % postscript printer. In the present file the postscript files % % are preceded by a line containing - cut here - and the name of % % the file, - fig1.ps - etc., which has to be erased. % % % Title: The Stability of the Peierls Instability % % for Ring-Shaped Molecules % % Authors: Elliott Lieb and Bruno Nachtergaele % % Address: Department of Physics % % P. O. Box 708 % % Princeton, NJ 08544-0708 % % USA % % E-mail: bxn@math.princeton.edu (Bruno Nachtergaele) % % lieb@math.princeton.edu (Elliott Lieb) % % Date created: 26 June 1994 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \magnification=\magstep1 \relax \overfullrule=0pt \hsize=17truecm \vsize=22 truecm \hoffset=0.0truecm \voffset=0.0truecm \parskip=0pt \baselineskip=12pt \def\draftonly#1{\ifx\draft1{\rm \lbrack{\ttraggedright {#1}} \rbrack\quad} \else\fi} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \font\BF=cmbx10 scaled \magstep 3 \font\BBF=cmbx10 scaled \magstep 5 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\eightit=cmti8 \let\1\sp \def\bigstar{{\BF *}} %\headline={\eightrm Lieb and Nachtergaele, Stability of the Peierls %Instability\hfill version of \today\hfill\number\pageno} % THEOREMS : allow items in proclaim \def\lessblank{\parskip=5pt \abovedisplayskip=2pt \belowdisplayskip=2pt } %\outer\def\iproclaim #1. { %plus5pt % \par \vskip10pt\noindent % {\bf #1.\ }\begingroup\interlinepenalty=250\lessblank\it} \def\eproclaim{\par\endgroup\vskip10pt plus5pt \noindent} \def\proof#1{\par\noindent {\bf Proof #1}\ % Use as "\proof:" \begingroup\lessblank\parindent=0pt} %\def\QED {\hfill\endgroup\break % \line{\hfill{\vrule height 1.8ex width 1.8ex }\quad} % \vskip 0pt plus10pt} %\def\QEDnogroup{\hfill\break %\line{\hfill{\vrule height 1.8ex width 1.8ex }\quad} %\vskip 0pt plus 10pt} \def\QED {\hfill {\bf Q.E.D.}\endgroup\break\smallskip} \def\QEDnogroup{\hfill {\bf Q.E.D.} \break\smallskip} \def\ETC {{\hbox{$ \clubsuit {\rm To\ be\ completed}\clubsuit$} }} \def\CHK {{\hbox{$ \spadesuit {\rm To\ be\ checked}\spadesuit$} }} \def\nl{\hfill\break} \def\nll{\hfill\break\vskip 5pt} \def\titem#1\par{\item{$\triangleright$} #1\par \vskip-8pt} %\input xrf.tex %xrd.tex for draft % ************************************* % * This is the file: xrf.tex % ** Tue 04-19-1994 ***************** \catcode`@=11 %%%%%%%%%%%% CROSS REFERENCE PACKAGE %%%%%%%%%%%%%%%%%%%%%%%% \def\ifundefined#1{\expandafter\ifx\csname \expandafter\eat\string#1\endcsname\relax} \def\atdef#1{\expandafter\def\csname #1\endcsname} \def\atedef#1{\expandafter\edef\csname #1\endcsname} \def\atname#1{\csname #1\endcsname} \def\ifempty#1{\ifx\@mp#1\@mp} \def\ifatundef#1#2#3{\expandafter\ifx\csname#1\endcsname\relax #2\else#3\fi} \def\eat#1{} %%%%%%%%%%%%%%%%%%%%%%%% CITATIONS %%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\refno \refno=1 \def\labref #1 #2 #3\par{\atdef{R@#2}{#1}} \def\lstref #1 #2 #3\par{\atedef{R@#2}{\number\refno} \advance\refno by1} \def\txtref #1 #2 #3\par{\atdef{R@#2}{\number\refno \global\atedef{R@#2}{\number\refno}\global\advance\refno by1}} \def\doref #1 #2 #3\par{{\refno=0 \vbox {\everyref \item {\reflistitem{\atname{R@#2}}} {\d@more#3\more\@ut\par}\par}}\vskip\refskip } \def\d@more #1\more#2\par {{#1\more}\ifx#2\@ut\else\d@more#2\par\fi} \let\more\relax \let\everyref\relax % executed before every item \newdimen\refskip \refskip=\parskip \let\REF\labref \def\@cite #1,#2\@ver {\eachcite{#1}\ifx#2\@ut\else,\@cite#2\@ver\fi} \def\citeform#1{\lbrack{\bf#1}\rbrack} \def\cite#1{\citeform{\@cite#1,\@ut\@ver}} \def\eachcite#1{\ifatundef{R@#1}{{\tt#1??}}{\atname{R@#1}}} \def\defonereftag#1=#2,{\atdef{R@#1}{#2}} \def\defreftags#1, {\ifx\relax#1\relax \let\next\relax \else \expandafter\defonereftag#1,\let\next\defreftags\fi\next } %%%%%%%%%%%%%%%citation formats: %%%%%%%%%%%%%%%%%%%%%%%%%% \newdimen\refskip \refskip=\parskip \def\@utfirst #1,#2\@ver {\author#1,\ifx#2\@ut\afteraut\else\@utsecond#2\@ver\fi} \def\@utsecond #1,#2\@ver {\ifx#2\@ut\andone\author#1,\afterauts\else ,\author#1,\@utmore#2\@ver\fi} \def\@utmore #1,#2\@ver {\ifx#2\@ut\and\author#1,\afterauts\else ,\author#1,\@utmore#2\@ver\fi} \def\authors#1{\@utfirst#1,\@ut\@ver} \def\citeform#1{{\bf\lbrack#1\rbrack}} \let\everyref\relax % executed before every item \let\more\relax % executed after every subitem \let\reflistitem\citeform \catcode`@=12 \def\Bref#1 "#2"#3\more{\authors{#1}:\ {\it #2}, #3\more} \def\Gref#1 "#2"#3\more{\authors{#1}\ifempty{#2}\else:``#2''\fi, #3\more} \def\Jref#1 "#2"#3\more{\authors{#1}:``#2'', \Jn#3\more} \def\inPr#1 "#2"#3\more{in: \authors{\eds#1}:``{\it #2}'', #3\more} \def\Jn #1 @#2(#3)#4\more{{\it#1}\ {\bf#2}(#3)#4\more} \def\author#1. #2,{#1.~#2} %% use as "\sameauthor. {}" \def\sameauthor#1{\leavevmode$\underline{\hbox to 25pt{}}$} \def\and{, and} \def\andone{ and} %% use as \noinitial. 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\def\Order{{\rm O}} \def\Re{\mathchar"023C\mkern-2mu e} % redefinition! \def\dim {\mathop{\rm dim}\nolimits} \def\id{\mathop{\rm id}\nolimits} \def\th{\hbox{${}^{{\rm th}}$}\ } % also in text \def\tover#1#2{{\textstyle{#1\over #2}}} \def\tr{\mathop{\rm tr}\nolimits} \def\Tr{\mathop{\rm Tr}\nolimits} \def\undbar#1{$\underline{\hbox{#1}}$} % LETTERS \def\phi{\varphi} % redefinition! \def\epsilon{\varepsilon} % redefinition! \def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\D{{\cal D}} \def\G{{\cal G}} \def\H{{\cal H}} \def\K{{\cal K}} \def\L{{\cal L}} \def\M{{\cal M}} \def\N{{\cal N}} \def\R{{\cal R}} \def\S{{\cal S}} \def\T{{\cal T}} \def\U{{\cal U}} \def\Z{{\cal Z}} \def\E{{\Ibb E}} \def\Eh{{\hat{\Ibb E}}} \def\F{{\Ibb F}} \def\DD{{\Ibb D}} \def\P{{\Ibb P}} \def\up{\uparrow} \def\down{\downarrow} \def\tphi{\tilde{\phi}} \def\tpsi{\tilde{\psi}} \def\Nup{N_\uparrow} \def\Ndown{N_\downarrow} \def\tright{t^{(r)}} \let\tr\tright \def\tl{t^{(l)}} \def\tm{t^{(m)}} \def\btr{t^{(r)*}} \def\btl{t^{(l)*}} \def\btm{t^{(m)*}} %\def\btr{\overline{t^{(r)}}} %\def\btl{\overline{t^{(l)}}} %\def\btm{\overline{t^{(m)}}} \def\cd{c^{\dagger}} \def\c{c^{\mathstrut}} \def\es{{\bf S}} \def\sign{\hbox{sign}} \def\ee{{\cal E}} \def\Ind#1{\hbox{Ind}(\hbox{\eightrm #1})} \def\hc{\hbox{ h.c. }} \def\half{{1/2}} \def\imply{\Rightarrow} \def\textbox#1#2#3{{\hsize= #1cm\vtop to #2cm{ \noindent #3}}} \let\REF\lstref %\lstref for numbers ; \labref for letter labels %\input peierlsr \REF Hue Hue \Gref\par \REF Sal Sal \Bref\par \REF Lon Lon \Jref\par \REF Jon Jon \Jref\par \REF Hub Hub \Jref\par \REF PP PP \Jref\par \REF Pop Pop \Jref\par \REF Lie4 Lie4 \Gref\par \REF SSH SSH \Jref\par \REF KH KH \Jref\par \REF BCM BCM \Gref\par \REF BJ BJ \Gref\par \REF Pei Pei \Bref\par \REF Fro Fro \Jref\par \REF LHS LHS \Jref\par \REF Lab Lab \Jref\par \REF Oos1 Oos1 \Jref\par \REF KL KL \Jref\par \REF Imr Imr \Gref\par \REF Wie Wie \Jref\par \REF Lie3 Lie3 \Jref\par \REF LS LS \Jref\par \REF Lie2 Lie2 \Jref\par \REF Oos2 Oos2 \Jref\par \REF Gar Gar \Bref\par \REF And And \Jref\par \REF Chs Chs \Jref\par \REF Pin Pin \Jref\par \REF BP BP \Jref\par \REF CF CF \Jref\par \REF SKM SKM \Jref\par \REF AN AN \Gref\par \REF BC BC \Bref\par \REF DM DM \Jref\par \REF KSSH KSSH \Jref\par \REF MGEW MGEW \Jref\par \REF EWGMO EWGMO \Jref\par \REF FK FK \Jref\par \REF ML ML \Jref\par \REF PVZ PVZ \Jref\par \REF SG SG \Jref\par \REF HM HM \Jref\par \REF MR MR \Jref\par \REF UMT UMT \Gref\par \REF Maz Maz \Jref\par \REF TH TH \Jref\par \REF KL2 KL2 \Jref\par \REF BS BS \Jref\par \REF LM LM \Gref\par \REF Lie1 Lie1 \Jref\par \REF DLS DLS \Jref\par \REF FILS FILS \Jref\par \REF KLS KLS \Jref\par % \font\BF=cmbx10 scaled \magstep 3 \vbox to \vsize{ {\baselineskip=12pt %\baselineskip=20pt \line{\hfill 26 June 1994} \vskip 40pt plus40pt \centerline{{\BF The Stability of the Peierls Instability}} \vskip10pt \centerline{{\BF for Ring-Shaped Molecules}} \vskip 30pt plus30pt \centerline{Elliott H. Lieb and Bruno Nachtergaele} \centerline{Department of Physics} \centerline{Princeton University} \centerline{P.O.B. 708, Princeton, NJ 08544} \vskip 20pt plus40pt \vfil \noindent {\bf Abstract}\hfill\break { We investigate the conventional tight-binding model of $L$ $\pi$-electrons on a ring-shaped mol\-e\-cule of $L$ atoms with nearest neighbor hopping. The hopping amplitudes, $t(w)$, depend on the atomic spacings, $w$, with an associated distortion energy $V(w)$. A Hubbard type on-site interaction as well as nearest-neighbor repulsive potentials can also be included. We prove that when $L=4k+2$ the minimum energy $E$ occurs either for equal spacing or for alternating spacings (dimerization); nothing more chaotic can occur. In particular this statement is true for the Peierls-Hubbard Hamiltonian which is the case of linear $t(w)$ and quadratic $V(w)$, i.e., $t(w)=t_0-\alpha w$ and $V(w)=k(w-a)^2$, but our results hold for any choice of couplings or functions $t(w)$ and $V(w)$. When $L=4k$ we prove that more chaotic minima {\it can\/} occur, as we show in an explicit example, but the alternating state is always asymptotically exact in the limit $L\to\infty$. Our analysis suggests three interesting conjectures about how dimerization stabilizes for large systems. We also treat the spin-Peierls problem and prove that nothing more chaotic than dimerization occurs for $L=4k+2$ {\it and\/} $L=4k$. } \vfil \noindent {\bf Physics and Astronomy Classification Scheme (1993):} \hfill\break \class 31.10.+z General theory of electronic structure, electronic transitions, and chemical bind\-ing* \class 31.15.+q General mathematical and computational developments* \class 36.20.Hb Macromolecules and polymer molecules; Configurations (bonds, dimensions)* \class 63.20.Kr Phonons and vibrations in crystal lattices; Phonon-electron interactions* \class 75.10.Lp General theory and models of magnetic ordering; Band and itinerant models* \medskip \hrule width2truein \smallskip \noindent Copyright \copyright\ 1994 by the authors. Faithful reproduction of this article by any means is permitted for non-commercial purposes. } \eject } \bgsection 1. INTRODUCTION To derive the shape (and other properties) of molecules from first principles has been an actively pursued goal since the early days of quantum mechanics. Many of the insights into the structure of molecules like benzene and its relatives have, however, been obtained using drastically simplified models. The Schr\"odinger equation for all the nuclei and electrons in such a molecule involves many dozens or even hundreds of degrees of freedom and, therefore, simpler models with a reduced number of degrees of freedom are a necessity. In the case of benzene the introduction of the H\"uckel model \cite{Hue} played an important role. This model is standard textbook material for organic chemistry students (see e.g. \cite{Sal}). London \cite{Lon} used the H\"uckel model to explain the large diamagnetic anisotropy of aromatic compounds and certain other materials quite successfully. A similar approach was used earlier by Jones in his work on bismuth and bismuth alloys \cite{Jon}. In this paper we are interested in ring-shaped molecules of the type $(CH)_{L}$, for {\it even\/} $L$, the so-called {\it annulenes} (sometimes called cyclic polyenes). The H\"uckel model for [$L$]-annulene describes the $L$ $\pi-$electrons (one for each carbon atom) as hopping from one carbon atom to the next (tight-binding approximation). The carbon atoms are located at the $L$ sites of a ring-shaped geometry, so $L+1\equiv 1$. The Coulomb interaction between the electrons is ignored in the H\"uckel model but we will include the Hubbard \cite{Hub} on-site interaction as in the work of Pariser-Parr \cite{PP} and Pople \cite{Pop}, in our study (a nearest neighbour repulsion can also be included). $$ H=-\sum_{j=1}^L\sum_{\sigma=\up,\down} t_j\cd_{j+1,\sigma}\c_{j\sigma} + \hc+ U\sum_{j=1}^L (n_{j\up} -\tover12)(n_{j\down}-\tover12)\quad . \deqno(hubham)$$ Here $\cd_{j+1,\sigma}$ and $\c_{j\sigma}$ are the usual fermion creation and annihilation operators for a particle of spin $\sigma$ at site $j$. The number operators are $n_{j\sigma}=\cd_{j\sigma}\c_{j\sigma}$. The hopping matrix elements are real, but they can be multiplied by complex phases if a magnetic field is present. (In \eq(hubham) we write $(n_{j\sigma} -\tover12)$ instead of the customary $n_{j\sigma}$; the difference amounts only to the addition of trivial terms in the Hamiltonian, but these are convenient because they ensure --- via hole-particle symmetry --- that the expected particle number in the grand canonical ensemble is $\langle N\rangle=L$, i.e., the ``half-filled band''.) The real parameter $U$ represents the strength of the on-site interaction of two particles and the H\"uckel approximation consists in setting $U=0$. The set of $t_j$'s will be denoted by $\{t_j\}$. The dependence of the Hamiltonian on the parameters will be made explicit, where needed, with a notation of the type $H(\{t_j\}), H(\{t_j\},U)$ etc.. For a review of rigorous results on the Hubbard model see \cite{Lie4}. The hopping matrix element $t_j$ is a resonance integral between orbitals at the sites $j$ and $j+1$ and therefore depends on the real space distance $w_j$ between these sites. A typical choice is $t_j=t_0-\alpha w_j$ \cite{SSH}. A more realistic choice would be $t_j=t_0\exp (-\alpha w_j)$ or some other rapidly decaying function of $w_j$. If one adds to the Hubbard Hamiltonian (even in the H\"uckel approximation) a term describing the interaction between the carbon ions (the energy of lattice distortions) one obtains a model that, in spite of its extreme simplicity, serves very well to explain a number of phenomena in annulenes and linear $(CH)_x$ (polyacetylene) \cite{KH}. For a recent review see the references \cite{BCM,BJ}. Although in more refined models the lattice distortions should be treated as quantum mechanical phonons \cite{SSH}, a more common choice, which we will follow here, is to describe them in the Born-Oppenheimer approximation by adding to the Hamiltonian \eq(hubham) a classical potential of the form $$ \sum_{j=1}^L V(w_j) $$ and to minimize, with respect to the $w_j$, the energy functional $$ \lambda_0(H(\{t(w_j)\})) + \sum_{j=1}^L V(w_j)\quad , $$ where $\lambda_0(H)$ denotes the lowest eigenvalue of $H$. Often $V$ is taken to be quadratic, but in this paper we will {\it not\/} assume that $V$ is quadratic or that the dependence of the $t_j$ on the $w_j$ is linear. The Peierls Instability \cite{Pei} (discovered also by Fr\"ohlich \cite{Fro} and seemingly independently by Longuet-Higgins and Salem \cite{LHS}) states that for $L$ sufficiently large and $U$ not too large, the minimum will not be attained in a translation invariant configuration of $w_j$'s. By a straightforward computation (exact in the case $U=0$ or perturbative for small non-zero $U$) one can show that, for any fixed choice of the function $V$, there is an $L_0$ such that for all $L\geq L_0$ the ground state energy of $H$ with $$ w_j= w_0 +(-1)^j\delta\quad , \deqno(dimerw)$$ for small $\delta$, is lower than with the best choice of $w_j=$ constant. In the context of $(CH)_x$ molecules this was discovered independently by Labhart \cite{Lab} and Ooshika \cite{Oos1}. The phenomenon described by \eq(dimerw) is called {\it dimerization\/} in the physics literature. Unfortunately, this word has quite different connotations in the chemistry literature, but there does not seem to be a universally accepted terminology for \eq(dimerw) among chemists. The phrase {\it bond-alternation\/} \cite{LHS} would be a more accurate description as far as chemists are concerned. We are obliged to make a choice here, and we shall use ``dimerization'' --- in the hope that chemists will substitute ``bond-alternation'' for it in their minds. In fact we shall go further and {\it declare a configuration to be dimerized, even if \eq(dimerw) holds with\/} $\delta=0$, i.e., the $w_j$'s are translation invariant. In other words, a dimerized configuration is one with period-two translation invariance, and this includes period one as a special case. This convention, while a bit unusual, conveniently eliminates awkward locutions. For the H\"uckel model ($U=0$) and for $L=2\bmod 4$ (i.e., $L=6,10,14,\ldots$) the dimerization instability was shown very explicitly by Longuet-Higgins and Salem who also estimated the degree of dimerization $\delta$ for realistic values of the parameters \cite{LHS}. The physical mechanism for the occurence of lattice distortions is rather simple and quite universal for electron-lattice systems in one dimension. A lattice distortion of period $1/2k_F$ opens up a gap at the Fermi level, thus lowering the energy of the occupied levels. This was a basic ingredient in Fr\"ohlich's theory of superconductivity. The calculations mentioned above show that for small distortions this lowering of the energy exceeds the positive contribution to the total energy of the lattice distortion itself. In the infinite volume limit the elastic energy per bond is quadratic in the parameter $\delta$ of \eq(dimerw), while the electronic energy per bond decreases by an amount proportional to $\delta^2\log \delta$ for small $\delta$. As far as we know the first author who mentions this logarithmic behavior is Fr\"ohlich in \cite{Fro}. A common feature of the works mentioned above is that they always show that certain instabilities exist. The question of the ``stability of the instabilities'' is usually not raised, much less resolved. The main purpose of this work is to study the stability and instability of lattice distortions and to obtain rigorous statements about the true energy minimizing state. Longuet-Higgins and Salem \cite{LHS} raised the question of the possible occurrence of higher periodicities but expected them to be unimportant. We prove that in the case $L=2\bmod 4$ (an assumption they made for technical reasons) they were right: {\it indeed nothing else than periodicity two occurs for $L=2\bmod 4$, but when $L=0\bmod 4$ other instabilities may, in fact, occur.} For the H\"uckel model with $t=t_0-\alpha w$ and $V(w)=k(w-a)^2$, Kennedy and Lieb \cite{KL} showed that the true minimizing configuration is {\it always\/} of periodicity two, i.e., $w_j$ is of the form \eq(dimerw). Moreover, the minimizing configuration is unique (up to translations). It is remarkable that this result holds for {\it all even\/} $L$, in contrast with what will be proved here for general functions $V$. We note parenthetically that the models we study are also used to describe electrons in mesoscopic metallic rings \cite{Imr}. These rings are typically two orders of magnitude larger than the largest annulenes produced in the laboratory so far, and therefore the magnetic fields needed to obtain a flux through the ring of the order of a flux quantum are much smaller and experimentally accessible. The main issue is to calculate the persistent currents in such a ring threaded by a magnetic flux. Depending on the electron number, among other things, these currents can be paramagnetic or diamagnetic. Although most authors assume uniform hoppings $t$ for this problem, there is an intimate connection between these persistent currents and the stability of dimerized configurations of $t$'s. In order to proceed to a precise formulation of our results it is first necessary to confront some possibly confusing questions about the phases of the $t_j$'s. Let us write $$ t_j=\vert t_j\vert \exp[i\theta_j] $$ with $-\pi< \theta_j\leq \pi$. We note, first, that by a simple unitary gauge transformation of the type $\c_{j\sigma}\to \exp[i\phi_j]\c_{j\sigma}$ and $\cd_{j\sigma}\to \exp[-i\phi_j]\cd_{j\sigma}$, the operators $n_{j\sigma}$ and the energy levels of $H$ are unchanged, but the $t_j$'s change to $t_j\to t_j\exp[i(\phi_j-\phi_{j+1})]$. Consequently, the energy levels depend {\it only} on the ``total flux'', $\Phi$, of the $t_j$'s defined by $$ \Phi= \hbox{argument}(\prod_{j=1}^L t_j)\quad , $$ with $-\pi <\Phi \leq \pi$. The next consideration comes from the physics of our model. The $t_j$'s are real, unless there is a superimposed magnetic field, in the absence of which each $\theta_j=0$ or $\pi$, depending on whether $t_j$ is positive or negative. Both signs can occur. Furthermore, it is understood that the $t_j$'s remain close to some nonzero value in the physically realistic situation. Without loss of generality we can choose this value to be positive. With this in mind, we shall assume that the elastic energy, as a function of $t$, defined by $$ f(\vert t\vert)\equiv\inf\{V(w)\mid t(w)=t\}\quad , $$ depends only on $\vert t\vert$ and not on the sign of $t$. (We will impose below in (1.6) some other very mild physical conditions on $f$.) This is not to say that negative $t$'s will not be permitted. They will --- as we explain next. Our procedure will be to study the total energy $$ \ee_L(\{t_j\})\equiv\lambda_0(H(\{t_j\})) +\sum_{j=1}^L f(\vert t_j\vert)\quad , \deqno(ee) $$ and to evaluate its minimum $$ E_L\equiv \min_{\{t_j\}}\ee_L(\{t_j\})\quad , \deqno(minee)$$ where the minimum is taken over all choices of $\{\vert t_j\vert\}$ {\it as well as\/} the phases $\{\theta_j\}$. Thus, we allow negative $t_j$'s and give them the same elastic energy as $\vert t_j\vert$. Since the $t_j$'s are not expected to pass through zero in the physical case, this is no real restriction. In terms of the $t_j$, dimerization means that the configurations minimzing $\ee_L(\{t_j\})$ are of the form $$ t_j=t_0+(-1)^j\delta \quad . \deqno(dimer)$$ It will turn out that for all $L$ our theorems show that the energy minimzing $t_j$'s are real (modulo a gauge transformation). But for $L=0\bmod 4$ an odd number of them {\it must\/} be negative for an energy minimizer --- thereby precluding a dimerized state. If, on the other hand, we wish to attribute the occurrence of an odd number of negative $t_j$'s to the presence of an external magnetic field then we can say that a dimerized state minimizes the energy in the presence of a field of flux $\pi$. Given that no such field is really present, we will have to conclude that the energy mimimum will not be a dimerized state when $L=0\bmod 4$. In this case we can redefine our problem by restricting the minimum in \eq(minee) to positive $t_j$'s. Do we get a dimerized state then? We do not know the general answer but we have some conjectures about this question. If $f(t)=k(t-a)^2$, as in \cite{KL}, we know that a dimerized state is, indeed, the minimum. The physical conditions we impose on $f$ are $$\eqalign{ &\hbox{i) $f$ is continuous. \qquad\qquad\qquad\hfill}\cr %&\phantom{a} \cr &\hbox{ii) $f(t)\geq Ct$, for large $t$ and for some constant $C>4$. \qquad\qquad\qquad\hfill }\cr }\deqno(conditions)$$ The continuity of $f$ is not really necessary for the validity of our results. In fact in the Appendix (\Lem13/convexf/) we show that $f$ can always be replaced with a convex (and hence continuous) function without changing the minimizing configurations. Property ii) is necessary to have a stable minimum at all, e.g., to prevent the molecule from collapsing to a point. We minimize the ground state energy $\ee_L$ not only with respect to the parameters $t_j$ but {\it also\/} with respect to the number of electrons. In Section 2 we prove that if $U\neq 0$ and all $t_j$ are real, the ground states of \eq(hubham) will in fact necessarily have $N=L$. In this case all results below are equally valid for minimization of the energy with the particle number fixed at half-filling. The condition $U\neq 0$ is necessary in the case of rings of size $L=0\bmod 4$, for with $U=0$ the model with translation invariant $t$'s has ground states with particle number ranging from $L=-2$ to $L+2$, due to zero eigenvalues of the one-particle Hamiltonian. \medskip \noindent {\bf Relation with the Flux-Phase Problem.}\nl The minimization with respect to the flux $\Phi$ of $\ee_L(\{t_j\})$, for fixed $\vert t_j\vert$, is a generalization of the so-called flux-phase problem \cite{Wie,Lie3} for rings. In the case $U=0$ (as in the original formulation of this problem), but arbitrary, fixed, positive $\{t_j\}$, this problem was solved by Lieb and Loss \cite{LS}. The arguments of this paper provide a way of solving the problem for rings and all values of $U$ (\Cor8/fluxphase/). For rings of length $L=2\bmod 4$ the optimal ($\equiv$ energy minimizing) flux is 0. When $L=0\bmod 4$, the optimal flux is $\phi=\pi$. This can be extended to higher dimensions \cite{Lie2} when some geometric periodicity is present. In particular, \cite{Lie2} proves the conjecture that the optimal flux for the Hubbard model on the two-dimenisonal square lattice is $\pi$ through each plaquette. \medskip \noindent {\bf Main Results.\/}\nl The following theorem, which we prove in Section 3, says that, when $L=2\bmod 4$ and the flux is zero and when $L=0\bmod 4$ and the flux is $\pi$, the Peierls instability is itself stable in the sense that no other lattice distortions than period two can occur. The theorem does {\it not\/} state that distortion always happens. Depending on the parameters in the Hamiltonian, the function $f$ in particular, the energy minimzing configuration may or may not be translation invariant. In real molecules both situations occur. Benzene, e.g., has all $C-C$ bond lengths equal (within the precision of today's measurements and in agreement with the analysis of Labhart \cite{Lab} and Ooshika \cite{Oos2}), while for large rings or long chains the Peierls instability will necessarily lead to two different bond lengths. \iproclaim/global/ Theorem (Minimizing configurations). \item{i)} If $L=2 \bmod 4$, the minimum of $\ee_L$ is attained in a dimerized configuration of the form \eq(dimer) in which the $t_j$'s are all positive (or all negative). \item{ii)} If $L=0\bmod 4$, the minimum of $\ee_L$ is attained configuration in which the $\vert t_j\vert$'s satisfy \eq(dimer) but the flux $\Phi$ equals $\pi$. \eproclaim That the cases $L=2\bmod 4$ and $L=0\bmod 4$ behave differently with respect to the flux was noted a long time ago (see e.g. the ``H\"uckel rule'' in \cite{Sal}.) The $U=0$ one-electron problem has zero-energy states when $L=0\bmod 4$ but not when $L=2\bmod 4$. In the first case this leads to diamagnetic, in the second case to paramagnetic response to magnetic fields, as was noted in \cite{Lon} and as is observed experimentally in the annulenes \cite{Gar}. \iproclaim/uniquedim/ Theorem (Uniqueness). If there is only one (up to translation and gauge transformations) dimerized configuration satisfying the conclusion of \Thm/global/, then there are no other energy minimizing configurations. \eproclaim We define three additional minimization problems for the $L=0\bmod 4$ case. \item{$\bullet$} {\it Zero-flux problem:\/} We restrict the minimization in \eq(minee) to those $\{t_j\}$ with zero flux, i.e., $\Phi=0$. \item{$\bullet$} {\it Period-two problem:\/} In addition to $\Phi=0$ we make the further restriction that the $t_j$'s must be dimerized. \item{$\bullet$} {\it The infinite-chain problem:\/} Let $d(x,y)$ denote the dimerized configuration with $t_{2j+1}=x$ and $t_{2j}=y$, and minimize the energy per site defined by $$ e(x,y)=\lim_{L\to\infty} {1\over L} \ee_L(d(x,y))\quad . \deqno(density)$$ The minimum energy in the first case is denoted by $E^0_L$, and the $t_j$'s by $t^0_j$. The second energy is denoted by $E^0_{L,\hbox{\eightrm dimer}}$ and the $t_j$'s by $t^0_{j,\hbox{\eightrm dimer}}$. Clearly $$ E^0_{L,\hbox{\eightrm dimer}} \geq E^0_L $$ and one of our conjectures below concerns the case of equality. The following, however, shows that the three problems become asymptotically the same (to within two powers of $L^{-1}$, instead of merely one) as $L\to\infty$. \iproclaim/asymptotics/Theorem (Asymptotic dimerization). \item{i)} Assume that there is a unique energy minimizing dimerized configuration $d(x,y)$, $x\geq y$, for the infinite-chain problem. Then any sequence of minimizing configurations $\{t^0_j\}$ of the zero-flux problem converges to a dimerized configuration as $L\to\infty$. Namely, $$ \lim_{L\to\infty} \max_j \vert t^0_j - t^0_{j,\hbox{\eightrm dimer}} \vert = \lim_{L\to\infty} \max_j \vert t^0_j - d(x,y)_j \vert = 0 \quad , $$ where we have adopted the convention that $t_1$ is the largest of the $t_j$'s for all configurations. \item{ii)} For any even $L$, the energies satisfy $$ 0\leq E_{L,\hbox{\eightrm dimer}} - E^0_L \leq \hbox{constant}\times L^{-1}\quad . $$ \eproclaim In section 5 we discuss the situation of finite rings of length $L= 0\bmod 4$. Instabilities other than period-two can exist for arbitrarily large rings if the function $f$ is chosen appropriately. An optimally dimerized ring of size $L=0\bmod 4$ can have other instabilities and it seems that the strongest of those actually has wavelength $=L$, and is not, as one could have expected, of period 4 or some other higher periodicity. We illustrate this with an explicit example of a ring of 8 sites for which the minimum is not period 2. A perturbative analysis indicates that the conditions on $f$ for a further instability to occur become more stringent as the system becomes larger. In particular the dimerization itself has to be smaller as $L$ becomes bigger. This leads us to the following two conjectures: \smallskip \noindent {\bf Conjecture 1.} For every $L=0\bmod 4$ there exists a function $f$ (depending on $L$), satisfying conditions i) and ii) in \eq(conditions), such that the energy minimizing configurations for the zero-flux problem are {\it not\/} period two. \smallskip \noindent {\bf Conjecture 2.} For any fixed $f$ satisfying \eq(conditions), and such that there is a unique minimum for the infinite-chain problem, there is a critical size $L_c$ such that for all even $L\geq L_c$ all zero-flux minima are period two. \medskip \noindent {\bf The Spin-Peierls Problem.}\nl Another closely related question is the spin-Peierls problem. For $U$ large, and half-filling, the Hubbard model reduces to the Heisenberg antiferromagnetic spin chain in second order perturbation theory \cite{And}. The spin chain also has an instability when coupled to lattice distortions, as was pointed out by Chesnut \cite{Chs} and by Beni and Pincus \cite{Pin,BP}. More explicitly one looks for the configurations of coupling constants $J_{ij} $ that minimize the smallest eigenvalue of the Hamiltonian: $$ H_\Lambda=\sum_{} J_{ij} \es_{i}\cdot\es_{j} + \sum_{} f(J_{ij})\quad , \deqno(heisham)$$ where the sum is over nearest neighbour pairs in a finite subset $\Lambda$ of the hypercubic lattice, and $f(J_{ij})$ is the elastic energy as before. In one dimension Cross and Fisher \cite{CF} computed the exponent governing the ground state energy density $e(\delta)$ of the system with alternating couplings: $$ J_j =1+(-1)^j \delta\quad , $$ with $J_j\equiv J_{j,j+1}$. They found that, up to logarithmic corrections, $$ \vert e(\delta)-e(0)\vert\sim -\vert\delta\vert^{4/3}\quad . $$ Numerical confirmation of this exponent and evidence for a logarithmic correction was reported in \cite{SKM}. The important point here is that $4/3 <2$, which implies the instability of the translation invariant configurations under period 2 perturbations, provided $f$ has a finite second derivative. It was recently proved that this exponent is identical to a critical exponent of the two dimensional 4-state Potts model \cite{AN}. The question addressed in this paper is again to determine the nature of the minimizing configurations. It turns out that the true minimum is dimerized for any (even) system size and in any dimension. {\it The $2 \bmod 4$ versus $0 \bmod 4$ dichotomy does not arise for the Heisenberg model!} \iproclaim/spinpeierls/ Theorem (Dimerization for spin-Peierls). Let $\Lambda\subset \Ir^d$ be a rectangular box of even size in all coordinate directions and with periodic boundary conditions. Then there is an energy minimizing configuration of $J_{ij}$'s which is of periodicity 2 in all coordinate directions. \eproclaim This theorem prompts one more conjecture. \smallskip \noindent {\bf Conjecture 3.} Suppose that a minimizer for the zero-flux problem for some $f$ and $L=0\bmod 4$ and some $U_0\geq 0$ is a dimerized state. Then for every $U > U_0$ there is a dimerized minimizer for the zero-flux problem with the same $f$ and $L$. In particular, the truth of this conjecture, when combined with the $U=0$ result in \cite{KL}, would imply that, with a quadratic $f$, dimerization always occurs, even in the $L=0\bmod 4$ case. \medskip \noindent {\bf Extensions of the results presented in this paper.} 1) It has been argued that the Hubbard on-site repulsion does not describe the Coulomb interaction between the electrons accurately enough (for a discussion see e.g. the reviews on this topic in \cite{BC} and references therein). While the Hubbard term alone, when not too large, seems to have the effect of enhancing the dimerization \cite{DM}, taking into account some nearest neighbour interaction terms Kivelson, Su, Schriefer, and Heeger \cite{KSSH} found that the Coulomb repulsion in fact suppresses the dimerization in polyacetylene. Without entering into this discussion here we would like to point out that without substantial modification one can prove the results of \Thm/global/ and \Thm/uniquedim/ for a Hamiltonian that includes a repulsive nearest-neighbour interaction of the form $$ \sum_j W(t_j) (n_j-1)(n_{j+1}-1) $$ The $-1$'s are inserted to keep the chemical potential tuned at half-filling. We require $W(t)\geq 0$, but no assumption on the dependence of $W$ on $t$ is needed. Theorem 1 holds unchanged, but we can extend the proof that the grand canonical ground states are necessarily half-filled only under the additional condition that $W(t_{j-1}) + W(t_j) < U_j$, for all $t$'s in the relevant range. Nearest neighbour exchange terms and certain longer range interactions can also be treated. 2) Instead of a homogeneous on-site repulsion it is sometimes more realistic to have different $U$ for different (classes of) sites, e.g. in a periodic manner. Theorem 1 extends to such situations in the following sense: the minimum energy will be attained in a configuration of $t_j$ which is invariant under all reflections through planes that intersect two oppossite bonds on the ring (as indicated by a dashed line in Figure 1) and that leave the configuration of the $U_i$ invariant. As an example consider [18]-annulene. Of the 18 hydrogens in this molecule 6 reside inside the ring and 12 outside, according to the pattern: 1 inside, 2 outside, 1 inside, etc. A possible way of taking the effect of this periodic configuration of H-atoms into account, is to add a periodic one-particle potential to the Hamiltonian. In the present example this potential would take two different values and depend on the site according to the pattern: $(v_i)=(v_1,v_1, v_2, v_1, v_1, \ldots)$. Our general result then implies an energy minimizing configuration of $C-C$ bonds of the form $ABCACBABCACB\ \cdots$. The observed bond lengths in [18]-annulene indeed satisfy this pattern (they are $A=1.419$\AA, $B=1.382$\AA,$ C=B$ \cite{Gar}.) 3) Hubbard models with spin-dependent hoppings, i.e., $t_j= \vert t_j\vert m_j, m_j\in$ SU(2), have been considered in the literature in order to study the effects of spin-orbit coupling \cite{MGEW,EWGMO,FK}. By virtue of the SU(2) gauge symmetry of the Hamiltonian one can diagonalize the spin dependence of the hoppings up to an SU(2) flux, which is defined up to an SU(2) transformation by the matrix $M=m_1 m_2 m_3 \cdots m_L$. The methods of this paper can be used to prove the existence of a minimizer having dimerization and $M=\idty$ when $L=2\bmod 4$ and when $L=0\bmod 4$ a minimizer has $M=-\idty$ and dimerization. This is completely analogous to the U(1) case treated in \Thm/global/. 4) Mattis and Langer \cite{ML} introduced a simple model to study the interaction between the electrons and a phonon instability of the Peierls type as a function of the temperature. Our methods can also be used to prove that in this model the electron state has periodicity two and that its correlations indicate dimerization. For a rigorous study of the Kohn anomaly, which accompanies the phase transition in the Mattis-Langer model, see \cite{PVZ}. Generalizations that are not considered in this work include the interaction between the electrons and lattice in excited states, the effects of doping, and higher dimensional models. Low-lying electronic excited states can induce further (e.g. soliton-like) distortions of the lattice \cite{SSH,SG,HM}. In doped polyacetylene an interesting semiconductor-metal transition occurs \cite{MR}. Away from half-filling, instabilities other than period-two will naturally develop (see e.g. \cite{UMT} and references therein). In two dimensions, e.g., on a square lattice, $2k_F$ instabilities may develop either in the coordinate directions or in the diagonal directions \cite{Maz,TH}. This may be related to the non-period-two instabilities that can occur in rings whose size is a multiple of 4, e.g., in the elementary plaquettes of the square lattice. Another new feature of higher dimensions is that the breaking of translation invariance can occur at non-zero temperature. This has been rigorously shown to occur in the Falicov-Kimball model \cite{KL2,BS} and in the Holstein model \cite{LM}. \bgsection 2. THE GRAND CANONICAL HUBBARD MODEL AT HALF-FILLING In the definition of the energy functional \eq(ee) we used the lowest eigenvalue of the grand canonical Hubbard Hamiltonian at half-filling. Our goal in this section is to prove that the corresponding ground state necessarily has $N=L$ if we assume the $t_j$'s are real (as will later be shown to be the case). It is then obvious that minimization of the energy with respect to the parameters $t_j$ {\it and\/} the number of electrons, produces exactly the same minimizing configurations as the restricted mimimization with the particle number fixed at $N=L$. The arguments in this section are not restricted to one dimension. We believe they are interesting in their own right and can be of interest in a more general context. Therefore, we temporarily consider a more general setup on a general finite lattice $\Lambda$, the number of whose lattice sites is denoted by $\vert\Lambda\vert$. Consider the grand canonical Hubbard Hamiltonian \eq(hubham) for spin 1/2 fermions on $\Lambda$, and with arbitrary chemical potential $\mu$: $$ H=-\sum_{i,j\in\Lambda}\sum_{\sigma=\uparrow,\downarrow} t_{ij}\cd_{j\sigma}\c_{i\sigma} +\sum_{j\in \Lambda} U_j(n_{j\uparrow}-\mu)(n_{j\downarrow}-\mu) $$ The $t_{ij}$ are asssumed (in this section only) to be real and symmetric, i.e $t_{ij}=t_{ji}$. $\Lambda$ is assumed to be connected, i.e., given $i$ and $j$ in $\Lambda$ we can find a sequence $i=i_1,i_2,\ldots,i_m=j$ such that $t_{i_1,i_2}t_{i_2,i_3}\cdots t_{i_{m-1},i_m}\neq 0$. The $U_j$ and $\mu$ are also real parameters. As before the dependence of $H$ on its parameters will be made explicit with a notation of the type $H_\mu$, $H(\{t_{ij}\})$, $H_\mu(\{t_{ij}\}, \{U_j\})$ etc. When $H$ appears without subscript the chemical potential is $\mu=1/2$, corresponding to half-filling. The spin-up and spin-down particle number operators are defined by $$ N_\sigma=\sum_{j\in\Lambda}n_{j\sigma} $$ where $\sigma=\uparrow,\downarrow$ and $n_{j\sigma}= \cd_{j\sigma}\c_{j\sigma}$. Note that $N_\uparrow$ and $N_\downarrow$ each commute with $H$, and hence their eigenvalues are good quantum numbers. $H$ also commutes with the total-spin operators (which generate the global spin rotations) given by $$ S^+=\sum_j \cd_{j\up}\c_{j\down},\quad S^-=\sum_j \cd_{j\down} \c_{j\up},\quad S^3={1\over 2}(N_\up-N_\down) $$ Therefore, by applying $S^+$ or $S^-$ an appropriate number of times, one can transform any ground state of $H$ into a ground state with $$ N_\up-N_\down=\cases{0& if $N$ is even\cr \pm 1 & if $N$ is odd\cr} \deqno(diffN)$$ We say that $\Lambda$ and $t$ are bipartite if the sites of $\Lambda$ can be written as the disjoint union of two subsets; i.e., $\Lambda=\Lambda_A \cup\Lambda_B$, in such a way that the matrix $t$ of hopping matrix elements respects this structure, namely $t_{ij}=0$ if $i\in \Lambda_A$ and $j\in \Lambda_A$ or if $i\in \Lambda_B$ and $j\in \Lambda_B$. In this case the Hamiltonians $H_\half(\{t_{ij}\},\{U_j\})$ and $H_\half(\{t_{ij}\},\{-U_j\})$ are unitarily equivalent, i.e at half-filling ($\mu=\half$) a global sign change of the potential does not affect the spectrum. A unitary transformation implementing this equivalence is the particle-hole transformation mapping $$\eqalign{ \cd_{j\uparrow}\longrightarrow \epsilon(j) \c_{j\uparrow},& \quad \c_{j\uparrow}\longrightarrow \epsilon(j) \cd_{j\uparrow}\cr \cd_{j\downarrow}\longrightarrow \cd_{j\downarrow},& \quad \c_{j\downarrow}\longrightarrow \c_{j\downarrow}\cr }$$ where $\epsilon(j)=1$ for $j\in\Lambda_A$ and $\epsilon(j)=-1$ for $j\in\Lambda_B$. By the same transformation the spin operators are mapped into the so-called {\it pseudospin\/} operators defined by $$ \tilde S^+=\sum_j \epsilon(j)\c_{j\up}\c_{j\down},\quad \tilde S^-=\sum_j \epsilon(j)\cd_{j\down}\cd_{j\up},\quad \tilde S^3={1\over 2}(\vert\Lambda\vert-N_\up-N_\down) $$ These operators therefore also commute with the Hamiltonian, and they also commute with the spin operators and generate an additional SU(2) symmetry. Under the particle-hole transformation $N_\up$ and $N_\down$ are transformed into $\vert\Lambda\vert- N_\up$ and $N_\down$ respectively. It is then obvious that by applying $\tilde S^\pm$ to any ground state we can obtain a ground state with $$ N_\up+N_\down=\cases{\vert\Lambda\vert & if $N-\vert\Lambda\vert$ is even\cr \vert\Lambda\vert \pm 1 & if $N-\vert\Lambda\vert$ is odd\cr} \deqno(sumN)$$ As the spin and pseudospin operators commute \eq(diffN) and \eq(sumN) can be realized simultaneously. For an even bipartite lattice and $\mu=1/2$ this implies that there is a ground state of $H$ with $N_\up=N_\down=\vert\Lambda\vert/2$ or $N_\up=N_\down \pm 1=\vert\Lambda \vert/2$. >From the next lemma it will follow that if in addition all $U_j$ are nonvanishing and of the same sign, {\it all\/} ground states have even particle number and satisfy $N_\up=N_\down=\vert\Lambda\vert/2$. Note that the Lemma itself does not require $\Lambda$ to be bipartite. \iproclaim/spin0/ Lemma. If all $t_{ij}$ are real and $U_j\leq 0$ for all $j$, then the ground state space of $H(\{t_{ij}\},\{U_j\})$ contains a state with $N_\uparrow=N_\downarrow$. If $U_j< 0$ for all $j$, all groundstates satisfy $N_\uparrow=N_\downarrow$. In particular the total number of particles in the ground state is {\it even} and the total spin is {\it zero}. \eproclaim \proof: We can introduce new operators $\hat\c_{i,\sigma}$ defined by $$ \hat\c_{j,\up}=\c_{j,\up},\quad \hat\c_{j,\down}=(-1)^{N_\up} c_{j,\down} $$ $H$ has then the same form in terms of the $\tilde\c$'s as in terms of the $\c$'s, but now the $\hat c^\#_{j,\down}$ commute with the $\hat c^\#_{j,\up}$. Our Hamiltonian is now an operator on $\H_\up\otimes\H_\down$, and can be written in the form $$ H=T\otimes\idty +\idty\otimes T + \sum_{i\in\Lambda} U_j (n_j-\mu)\otimes (n_j-\mu) $$ where $T=-\sum_{i,j\in\Lambda} t_{ij} \cd_j\c_i$. (Because spin $\up$ and spin $\down$ operators act identically on different tensor factors in the Hilbert space, we can omit the spin index.) $T$ and the $n_j$ are real in the canonical basis of localized particles. Therefore we can apply \Lem14/DLS/ of the appendix and the discussion thereafter to conclude that $H$ has a ground state with $N_\up=N_\down$. If $U_j<0$ for all $i$, there cannot be a ground state with $N_\up\neq N_\down$. Indeed the last statement of \Lem14/DLS/ implies that for any ground state $\Omega$ $$ U_j\langle \Omega \mid (n_{j,\up} -\mu)\Omega\rangle = U_j\langle \Omega \mid (n_{j,\down} -\mu)\Omega\rangle $$ If $U_j<0$ this implies $N_\up=N_\down$. \QED \iproclaim/uniqueness/ Lemma. If all $t_{ij}$ are real and bipartite, $\vert\Lambda_A\vert =\vert\Lambda_B\vert$, and either all $U_j>0$ or all $U_j<0$. Then the ground state of $H$ with $\mu=1/2$ is unique and has $\Nup=\Ndown=\vert\Lambda\vert/2$. \eproclaim \proof: It is sufficient to consider the case of all $U_j>0$. The case $U_j<0$ then follows because the particle-hole transformation changes the sign of the $U_j$, and the properties $\Nup=\vert\Lambda\vert/2$ and $\Ndown=\vert\Lambda\vert/2$ are unchanged under the transformation. Let us therefore consider the case $U_j>0$, for all $j$. We first apply \Lem/spin0/ to the particle-hole transformed Hamiltonian which has all $U_j<0$. This tells us that $N_\down=\vert\Lambda\vert-N_\up$ in all ground states. In particular the total number of particles is $\vert\Lambda\vert$ which, by assumption, is even. Then, by Theorem 2 of \cite{Lie1} (the grand-canonical version at half-filling does not need the homogeneity of the potential) and the assumptions on the lattice, $\Nup=\Ndown=\vert\Lambda\vert/2$. [Note: Theorem 2 of \cite{Lie1} requires $U_j=$ constant. That is the case when the interaction is $n_{j\up}n_{j\down}$ as in \cite{Lie1}. If we use the formulation of \eq(hubham), however, with $(n_{j\up}-1/2)(n_{j\down}-1/2)$, the same proof works without the need of constant $U_j$.] \QED For translation invariant $t_{ij}$ on regular lattices the case $U_j\equiv 0$ is of course exactly solvable. It is rather straightforward in such cases to determine the ground state degeneracy ``by hand''. The ground state is unique if and only if the eigenvalues of the matrix $(t_{ij})$ are all non-zero. It is straightforward to extend \Lem/spin0/ to Hamiltonians that include interactions of the form $$ F(\{n_{j,\up}\}) + F(\{n_{j,\down}\}) -\sum_\alpha G_\alpha(\{n_{j,\up}\}) G_\alpha(\{n_{j,\down}\}) $$ for arbitrary real functions $F$ and $G$ of the local particle numbers. One can use this to prove a slightly weaker form of \Lem/uniqueness/ for a class of models with an additional term of the form $$ -\sum_{ij} W_{ij}(n_i-1)(n_j-1) $$ added to the Hamiltonian $H$. For these models our methods also prove that i) if $W_{ij} \geq 0$ and $(U_j-\sum_{i} W_{ij})\geq 0$, for all $j$, then there is a ground state with $N_\up=N_\down=\vert\Lambda\vert/2$. ii) if $W_{ij} \geq 0$ and $(U_j-\sum_{i} W_{ij})> 0$, all ground states have $N_\up=N_\down=\vert\Lambda\vert/2$. \bgsection 3. GENERAL RESULTS BASED ON REFLECTION POSITIVITY;\hfill\break PROOFS OF THEOREMS 1 AND 4 We now return to the half-filled Hubbard model ($N=\vert\Lambda\vert$) on a ring of even length $\vert\Lambda\vert=L$. It will be convenient to express the dependence of the energy functional on $\{t_j\}$ in a more explicit way. We write $$ \ee_L(\tl,\tm,\tr)= \lambda_0(H(\{t_j\})) +\sum_{j=1}^L f(\vert t_j \vert)\quad , $$ where $\tl,\tr$, and $\tm$ denote a partition of the {\it complex\/} $t_j$'s (with $t_j^*$ denoting the complex conjugate of $t_j$) into three groups as follows (also see Figure 1): $$ \tr=(t_1,\ldots,t_{L/2-1}),\quad \tm=(t_L, t_{L/2}),\quad \tl=(t_{L-1},\ldots,t_{L/2+1})\quad . $$ The reflection of a configuration $(t^{(l)},t^{(m)},t^{(r)})$ through the plane intersecting the bonds $\{L,1\}$ and $\{L/2,L/2 +1\}$ is the configuration $(t^{(r)},t^{(m)},t^{(l)})$, i.e., with $t^{(l)}\leftrightarrow t^{(r)}$. Throughout this section we assume a constant potential $U_j=U$. The following lemma can be formulated to include non-constant potentials but this is not needed for our purposes. \iproclaim/RPineq/ Lemma. Assume a constant potential $U_j=U$. If $L=2\bmod 4$, assume that $t_{L/2}$ and $t_L$ are real and nonnegative. If $L=0\bmod 4$, assume that $t_{L/2}\geq 0$ and $t_L\leq 0$. The other $t_j$ are allowed to be arbitrary complex numbers. Then the energy functional satisfies $$ \ee_L(\tl,\tm,\tr)\geq {1\over 2} \left(\ee_L(\tl,\tm,\btl)+ \ee_L(\btr,\tm,\tr)\right)\quad . \deqno(RPineq)$$ \eproclaim \proof: As in the proof of \Lem/spin0/ we consider the Hamiltonian \eq(hubham) as acting on $\H_\uparrow\otimes\H_\downarrow$, which amounts to considering operators referring to opposite spins as commuting. For $\sigma=\uparrow,\downarrow$ define a set of ``spin'' operators using a Jordan-Wigner transformation as follows: $$\eqalign{ S^3_{j,\sigma}&=\cd_{j,\sigma}\c_{j,\sigma} -\tover12\quad , \quad \es_{j,\sigma}=2S^3_{j,\sigma}\quad ,\cr S^+_{j,\sigma}&= \es_{1,\sigma}\cdots\es_{j-1,\sigma} \cd_{j,\sigma}\quad , S^-_{j,\sigma}= \es_{1,\sigma}\cdots\es_{j-1,\sigma} \c_{j,\sigma}\quad .\cr }$$ With these definition and the anticommutation relations of the $\c_{i,\sigma}$ and $\cd_{i,\sigma}$ it is straigthforward to check that $S^3_{i,\sigma},S^+_{i,\sigma}$, and $S^-_{i,\sigma}$ satisfy the standard SU(2) commutation relations and that they commute for different indices $i,\sigma$. In terms of these operators, and using the commutation relations, the Hamitonian can be written as $$\eqalign{ H&=\sum_{j=1}^{L-1}\sum_{\sigma=\uparrow,\downarrow} t_j S^+_{j+1,\sigma}S^-_{j,\sigma}+ \hc + U\sum_{j=1}^L S^3_{j,\uparrow}S^3_{j,\downarrow}\cr &\quad t_L\sum_{\sigma=\uparrow,\downarrow} S^+_{1,\sigma}(\es_{1,\sigma}\cdots\es_{L,\sigma})S^-_{L,\sigma} + \hc \quad .\cr }$$ By performing a rotation by $\pi$ about the 2-axis at every other site and for both $\sigma=\uparrow$ and $\sigma=\downarrow$ we find that this spin Hamiltonian is unitarily equivalent to: $$\eqalign{ \tilde H&=\sum_{j=1}^{L-1}\sum_{\sigma=\uparrow,\downarrow} -t_j S^+_{j+1,\sigma}S^+_{j,\sigma}+ \hc + \sum_{j=1}^L U_j S^3_{j,\uparrow}S^3_{j,\downarrow}\cr &\quad + (-1)^{L/2} t_L\sum_{\sigma=\uparrow,\downarrow} S^+_{1,\sigma}(\es_{1,\sigma}\cdots\es_{L/2,\sigma}) S^+_{L,\sigma}(\es_{L,\sigma}\cdots\es_{L/2+1,\sigma}) + \hc \quad .\cr }$$ Under the conditions stated in the proposition $t_{L/2}\geq 0$ and $(-1)^{L/2} t_L\leq 0$. We can now apply \Lem14/DLS/ of the appendix to $\tilde H$ with $$\eqalign{ A&=-\sum_{j=1}^{L/2-1}\sum_{\sigma=\uparrow,\downarrow} t_j S^+_{j+1,\sigma}S^+_{j,\sigma}+ \hc + U \sum_{j=1}^{L/2}S^3_{j,\uparrow}S^3_{j,\downarrow}\quad ,\cr B&=-\sum_{j=L/2+1}^{L-1}\sum_{\sigma=\uparrow,\downarrow} t_j S^+_{j+1,\sigma}S^+_{j,\sigma} + \hc + U\sum_{j=L/2 +1}^L S^3_{j\up}S^3_{j\down}\quad ,\cr C_1\otimes C_1&= t_{L/2} S^+_{L/2 +1,\up}S^+_{L/2,\up}\quad ,\cr C_2\otimes C_2&= \vert t_L\vert S^+_{1,\up}(\es_{1,\up} \cdots\es_{L/2,\up}) S^+_{L,\up}(\es_{L,\up}\cdots\es_{L/2+1,\up})\quad ,\cr }$$ and six more terms $C_i\otimes C_i$, $i=3,\ldots,8$, which are equal to $C_1\otimes C_1$ and $C_2\otimes C_2$ with $\up$ replaced by $\down$ and the hermitian conjugates of these operators. This gives us the inequality of the lemma for the lowest eigenvalue of $H$. The elastic energy terms in the functional $\ee_L$ add up to the same contribution on both sides of the inequality and therefore can be included trivially. \QED \noindent {\bf Proof of Theorem 1 :}\nl First we prove that the minimum of $\ee_L$ is attained in a configuration of $t_j$ with total flux $\Phi$ when $L=2\bmod 4$, and with $\Phi=\pi$ when $L=0\bmod 4$. Let $\{t_j\}$ be minimizing. Due to the invariance of the ground state energy of $H$ under gauge transformations, the energy functional $\ee_L$ defined in \eq(ee) depends only on $\{\vert t_j\vert\}$ and $\Phi$. Hence, for any set of $t_j$'s there are $t_j^\prime$ such that $\ee_L(\{t_j\})= \ee_L(\{t_j^\prime\})$ and $t_{L/2}^\prime$ and $t_L^\prime$ are real and have the correct sign for application of \Lem/RPineq/. The two configurations in the right side of \eq(RPineq) have flux $\Phi= \hbox{argument}(t_{L/2}^\prime t_L^\prime)$ which, by assumption, takes the values stated in the theorem. Because $\{t_j\}$ is minimizing, \eq(RPineq) must be an equality. Therefore the two configurations on the right side must be minimizers of $\ee_L$, but then both have the correct flux, $0$ or $\pi$. Next we show that in both cases the minimum of $\ee_L$ is attained in a configuration with $\{\vert t_j\vert\}$ dimerized. For any configuration $(\tl,\tm,\tr)$ of real $t_j$'s and $\tm$ of the right signs, \eq(RPineq) implies that either $(\tl,\tm,\tl)$ or $(\tr,\tm,\tr)$ has at least as low an energy and has at least as many pairs of identical $\vert t_j \vert$'s. In particular this shows that $\ee_L$ is minimized in a configuration with $\vert t_j \vert =\vert t_{j+2}\vert $ for $j=L/2-1$ and $j=L-1$. Because $\ee_L$ is translation invariant the argument above can be repeated to show that there is a minimizing configuration with $\vert t_j \vert =\vert t_{j+2} \vert$ for all $j$, which is the desired result. \QEDnogroup \Lem/RPineq/ also permits us to solve the flux phase problem for even rings with $U\neq 0$. \iproclaim/fluxphase/Corollary. Let $U_j\equiv U$ and let $\{t_j\}$ be a fixed configuration of nonnegative $t_j$'s. Then the minimum of $\ee_L$, varying over the total flux $\Phi$ alone, is attained for $\Phi=0$ if $L=2\bmod 4$ and for $\Phi=\pi$ if $L=0\bmod 4$. \eproclaim \noindent {\bf Proof of Theorem 4 :}\nl The Heisenberg antiferromagnet on an arbitrary bipartite lattice is reflection positive \cite{DLS}. This implies, via \Lem14/DLS/, the following inequality, analogous to \Lem/RPineq/: $$\eqalign{ &\lambda_0(H_{\hbox{\eightrm Heis.}}(\{J_{ij}^{(l)}, J_{ij}^{(m)},J_{ij}^{(r)}\}) \cr &\quad \geq {1\over 2}\left( \lambda_0(H_{\hbox{\eightrm Heis.}}(\{J_{ij}^{(l)}, J_{ij}^{(m)},J_{ij}^{(l)}\}) +\lambda_0(H_{\hbox{\eightrm Heis.}}(\{J_{ij}^{(r)}, J_{ij}^{(m)},J_{ij}^{(r)}\}) \right)\quad,\cr }$$ where $\{J_{ij}^{(l)},J_{ij}^{(m)},J_{ij}^{(r)}\}$ denotes a partition of the coupling constants into three groups: the $J_{ij}^{(l)}$ are the couplings on the bonds to the left of any reflection plane of the lattice, the $J_{ij}^{(m)}$ are the couplings on the bonds intersected by the reflection plane, and the $J_{ij}^{(r)}$ are the coupling to the right of the plane. Using this inequality Theorem 4 is then proved in the same way as Theorem 1, but this time there is no need to distinguish between $L=0\bmod 4$ and $L=2\bmod 4$ --- or even to restrict ourselves to one dimension. \QEDnogroup \bgsection 4. ASYMPTOTIC DIMERIZATION FOR $L=0\bmod 4$;\hfill\break PROOFS OF THEOREMS 2 AND 3 Even though finite rings of length $L=0\bmod 4$ can do more complicated things than dimerize (see Section 5 for a discussion of counterexamples), they do dimerize asymptotically. To be precise, for a fixed $f$, satisfying conditions i) and ii) of \eq(conditions), for which the infinite ring has a unique energy minimizing configuration among the configurations of period 2, any sequence of minimizers for finite rings has to approach that dimerized configuration uniformily when $L\to\infty$, i.e., any nearest pair of $\vert t_j\vert$'s converges to the pair occurring in the infinite volume dimerized configuration. The aim of this section is to prove this statement, which is \Thm/asymptotics/ i). At the same time we will also supply the proofs of \Thm/uniquedim/ and \Thm/asymptotics/ ii). In contrast to the previous section, here we will always consider minimization of the energy over arbitrary configurations of $\vert t_j\vert$, but with constant flux $\Phi$. For this purpose it is convenient to introduce an explicit notation for the energy functionals at fixed $\Phi$: $$ \ee^\Phi_L(\{t_j\})=\ee_L(\{e^{i\Phi/L} t_j\})\quad . $$ Here, and in the rest of this section, it is assumed that all $t_j$ are nonnegative. Note that the $t_j$'s now play the role of the $\vert t_j\vert$'s. For any $x$ and $y$ let $d(x,y)$ denote the configuration with $t_j = x$ for $j$ odd and $t_j=y$ for $j$ even, and define the energy density $e$ of a period 2 configuration $d(x,y)$ as in \eq(density). Note that $e$ is independent of $\Phi$. In order to remove the degeneracy due to the obvious symmetry under cyclic permutations of the $t_j$'s, we will henceforth adopt the convention that the maximum value of all $t$'s is attained by $t_1$. A first ingredient in the proof of \Thm/asymptotics/ is the following energy estimate, which by itself implies \Thm/asymptotics/ ii). Let $E_L^\Phi$ denote the value of $\ee^\Phi_L$ in a minimizing configuration $t^\Phi_{L,j}$. Under the general condition \eq(conditions) ii) on $f$ we can find a constant $K$, depending only on $f$, such that $\vert t^\Phi_{L,j}\vert\leq K$. \iproclaim/energyestimate/ Lemma. Assuming only condition ii) on $f$ in \eq(conditions), we have for all $-\pi <\Phi \leq \pi$ $$\cases{ \vert E^0_L- E^\Phi_L\vert \leq 2\Phi^2 K/L & if $L=2\bmod 4$\cr \phantom{a} & \cr \vert E^\pi_L- E^\Phi_L\vert \leq 2(\pi-\Phi)^2 K/L & if $L=0\bmod 4$\cr }$$ and $$\cases{ \vert E^\pi_L - \ee^\Phi_L(t^0_L)\vert \leq 4\pi^2 K/ L & if $L=0\bmod 4$\cr \phantom{a} & \cr \vert E^\Phi_L - \ee^\Phi_L(t^\Psi_L)\vert \leq 8\pi^2 K/ L & for all $\Psi$ and all even $L$.\cr }$$ \eproclaim Because $\ee^\pi_L$ attains its minimum $E^\pi_L$ in a dimerized configuration ($L=0\bmod 4$), this lemma implies, for $\Phi=0$, that even if the true minimum is not dimerized, its energy differs from the best dimerized configuration by $\Order(1/L)$, which is 2 orders down from $E^0_L=\Order(L)$. \proof: First note that $H(\{t_j\})$ has real matrix elements and therefore has a ground state $\psi^0_L$ which is {\it real}, and hence such that $\langle \psi^0_L\mid \cd_{j+1,\sigma} \c_{j,\sigma}\mid\psi^0_L\rangle$ is also real. Using a gauge transformation that makes $H(\{\exp(i\pi/L)t_j\})$ real one can also see that $H(\{\exp(i\pi/L)t_j\})$ has a ground state $\psi^\pi_L$ such that $$ \langle \psi^\pi_L\mid \cd_{j+1,\sigma} \c_{j,\sigma}\mid\psi^\pi_L\rangle = e^{i\pi/L} R_{j,\sigma}\quad , $$ with {\it real} $R_{j,\sigma}$. Let $\{t^0_{L,j}\}$ be any minimizing configuration of $\ee^0_L$, and let $\psi^0_L$ be a real ground state of $H(\{t^0_{L,j}\})$. Then, by the variational principle, $$\eqalign{ E^\Phi_L &\leq \langle \psi^0_L\mid H(\{\exp(i\Phi/L)t^0_{L,j}\}) \mid\psi^0_L\rangle + \sum_j f(t_j^0)\cr &=\langle \psi^0_L\mid H(\{\exp(i\Phi/L)t^0_{L,j}\}) \mid\psi^0_L\rangle + \sum_j f(t_j^0)\cr &\quad +\sum_{j=1}^L\sum _{\sigma=\up,\down} (e^{i\Phi/L} -1)t^0_{L,j} \langle \psi^0_L\mid \cd_{j+1,\sigma}\c_{j,\sigma} \mid\psi^0_L\rangle +\quad \hc \quad .\cr }$$ The sum of the first two terms in the right side of the equality above is $E^0_L$. Using the reality of $\psi^0_L$ the last term can be estimated by $4\vert 1-\cos ( \Phi/L)\vert\sum_j \vert t^0_{L,j}\vert$. As $\cos x\geq 1-x^2/2$, this proves the first estimate of the Lemma. The other inequalities are derived in the same way. \QED The next lemma is an application of a version of the {\it Abstract Chessboard Estimate} \cite{FILS} which is given in the Appendix. \iproclaim/compareaveragedim/ Lemma. For any configuration of $t_j\geq 0$, with $d(t_j,t_{j+1})$ described in \eq(density), $$\eqalign{ {1\over L}\sum_{j=1}^{L} \ee^0_L(d(t_j,t_{j+1})) \leq \ee^0_L(\{t_j\}) & \quad\hbox{if } L=2\bmod 4\quad ,\cr {1\over L}\sum_{j=1}^{L} \ee^\pi_L(d(t_j,t_{j+1})) \leq \ee^\pi_L(\{t_j\}) &\quad\hbox{if } L=0\bmod 4\quad .\cr }\deqno(compareaveragedim)$$ \eproclaim \proof: Let $\Phi=0$ if $L=2\bmod 4$ and $\Phi=\pi$ if $L=0\bmod 4$. We apply \Thm15/chessboardestimate/ (with $D=\Rl^2$) to the following $F$: $$ F(a_1,\dots,a_{L})=\exp(-\ee^\Phi_L (t_1, \ldots,t_{L}))\quad , $$ where $a_j=(t_j,s_j)\in\Rl^2$. The involution $R$ is defined by $R(t,s)=(s,t)$. The domain $\D$ is taken to be the set of all $(a_1,a_2,\ldots,a_{L})$ of the form $((t_1,t_2),(t_2,t_3),\ldots,(t_{L},t_1))$. The necessary invariance properties of $\D$ are trivially satisfied. The cylicity of $F$ follows from the invariance of $\ee^\Phi_L$ under cyclic permutations of the $t_j$. The inequality of \Lem/RPineq/ translates into Schwarz's inequality for $F$ on $\D$. This becomes obvious with the definitions: $$\eqalign{ a_1=(t_1,t_2) &\quad b_1=(t_{L},t_1)\cr a_2=(t_2,t_3) &\quad b_2=(t_{L-1},t_{L})\cr \vdots &\quad \vdots\cr a_{L/2}=(t_{L/2},t_{L/2+1}) &\quad b_{L/2}=(t_{L/2+1},t_{L/2+2}) \quad .\cr }$$ Indeed, one then has: $$\eqalign{ \exp(-\ee^\Phi_L(t_1,\ldots,t_{L})) &=F(a_1,a_2,\ldots,a_{L/2},b_{L/2},\ldots, b_1)\cr \exp(-\ee^\Phi_L(t_1,\ldots,t_{L/2},t_{L/2+1},t_{L/2},\ldots,t_2)) &=F(a_1,a_2,\ldots,a_{L/2},R a_{L/2},\ldots, R a_1)\cr \exp(-\ee^\Phi_L(t_1,t_{L},\ldots,t_{L/2+2},t_{L/2+1},\ldots,t_{L})) &=F(R b_1, R b_2,\ldots, R b_{L/2},b_{L/2},\ldots, b_1)\cr \exp(-\ee^\Phi_L(t_{L},t_{L-1},\ldots,t_{L/2+1},t_{L/2+2}, \ldots,t_1)) &=F(b_1, b_2,\ldots, b_{L/2},R b_{L/2},\ldots,R b_1)\cr }$$ The implication of \Thm15/chessboardestimate/, in terms of $\ee^\Phi_L$ then reads: $$ \ee^\Phi_L(t) \geq{1\over L}\sum_{j=1}^{L} \ee^\Phi_L(t_j,t_{j+1},t_j,t_{j+1},\ldots)\quad , $$ which is the desired inequality. \QED \iproclaim/comparedim/ Lemma. For the given $f$ let $K$ be a constant such that $\vert t^\Phi_{L,j}\vert\leq K$, for all $L$. For $L=0\bmod 4$, let $\{t^0_{L,j}\}$ be any minimizing configuration of $\ee^0_L$. Then $$ \vert\ee^\pi_L(d(t^0_{L,j},t^0_{L,j+1}))-E^\pi_L\vert \leq 4K \pi^2\quad , $$ for all $j=1,2,\ldots, L$. \eproclaim \proof: As $L=0\bmod 4$ we can apply \Lem/compareaveragedim/ to $\ee^\pi_L$ and with $t_j=t^0_{L,j}$. We then have $$ {1\over L}\sum_{j=1}^{L}\ee^\pi_L(d(t^0_{L,j},t_{L,j+1})) \leq \ee^\pi_L(t^0_L)\quad . $$ By \Lem/energyestimate/ the right side is bounded by $E^\pi_L+4\pi^2 K/L$, and hence $$ {1\over L}\sum_{j=1}^{L}\left\{\ee^\pi_L(d(t^0_{L,j},t^0_{L,j+1})) -E^\pi_L\right\}\leq {4\pi^2 K \over L}\quad. $$ As each of the terms in the sum is non-negative, the statement of the lemma follows. \QED \iproclaim/energydensity/ Lemma. i) The following limit defines a continuous function $e$: $$ \lim_{L\to\infty} \ee^\Phi_L(d(x,y))=e(x,y) $$ The convergence is uniform on compact subsets of $\Rl^2$.\nl ii) If $e$ attains its minimum at $(x_0,y_0)$ and $(y_0,x_0)$ and nowhere else, then there exists a sequences $x_L\to x_0$ and $y_L\to y_0$ such that for all $L=0\bmod 4$, $d(x_L,y_L)$ is a minimizing configuration for $\ee^\pi_L$, and such that $$ \lim_{L\to\infty}{1\over L}\ee^\pi_L(d(x_L,y_L))=e(x_0,y_0)\quad . $$ \eproclaim \proof: The arguments involved in proving this lemma are fairly standard. We therefore only give a sketch of the proof. i) We first show that there exists a concave function $\epsilon(x,y)$, satisfying $\epsilon(x,y)\leq 2\vert x\vert + 2\vert y\vert +\vert U\vert$, and such that $$ \epsilon(x,y)=\lim_{L\to\infty}{1\over L} \lambda_0(H_L(d(x,y))) \quad . $$ For brevity we put $G_L= \lambda_0(H_L(d(x,y)))$. By the variational principle and the fact that $\Vert c_{j\sigma}\Vert =1$ one immediately gets that $G_L$ is ``almost subadditive'': $$ G_{L_1+L_2}\leq G_{L_1} + G_{L_2} + 6(\vert x\vert +\vert y\vert) $$ and therefore $G_L + 6(\vert x\vert +\vert y\vert)$ is subadditive. This sequence is also almost monotone: $$ L_1\geq L_2 \Rightarrow G_{L_1}\leq G_{L_2} + \hbox{Constant} $$ because $G_L\leq 0$. It follows that the limit of $G_L/L$ exists and as a limit of concave functions the limit is also concave. The bounds are trivial but they imply continuity and therefore the convergence is uniform on compacts. From \Lem/energyestimate/ it is obvious that $e$ does not depend on $\Phi$. ii) From the conditions on $f$ \eq(conditions) one has an apriori bound $K$ on the minimizing $t_j$: $\vert t^\pi_{j,L}\vert \leq K$. By \Thm/global/, $\ee^\pi_L$ always has a dimerized minimizer $d(x_L,y_L)$, and by compactness there always exists a subsequence $(x_{L_k}, y_{L_k})$ converging to $(x_*,y_*)$. By the continuity of $e^\pi$ and the uniform convergence on compacts $$ \left\vert {1\over L_k}\ee^\pi_{L_k}(d(x_{L_k},y_{L_k}))-e(x_*,y_*) \right\vert\to 0 \quad . $$ As the $d(x_{L_k},y_{L_k})$ are minimizers it follows that $e(x_*,y_*)=e(x_0,y_0)$ and hence, by assumption, $(x_*,y_*)=(x_0,y_0)$ or $(x_*,y_*)=(y_0,x_0)$. Without loss of generality we can assume $x_0\geq y_0$. Consider then the new sequence $d(\tilde x_L,\tilde y_L)$ of minimizers with $(\tilde x_L, \tilde y_L)=(x_L,y_L)$ or $(y_L,x_L)$, and such that $\tilde x_L\geq \tilde y_L$ for all $L$. Then all convergent subsequences $(\tilde x_{L_k}, \tilde y_{L_k})$ of $(\tilde x_L, \tilde y_L)$ converge to the same limit $(x_0,y_0)$ and hence the sequence is convergent. \QED \noindent {\bf Proof of \Thm/asymptotics/ i):}\nl Let $d(x_L,y_L)$ be a sequence of minimizing configurations for $\ee^\pi_L$ and such that $x_L$ and $y_L$ converge to $x_0$ and $y_0$ respectively as $L\to\infty$. The existence of such a sequence is guaranteed by \Lem/energydensity/. First we estimate $e(t^0_{L,j},t^0_{L,j+1})$ for a minimizing configuration $t^0_L$ of $\ee^0_L$ and any \break $j=1,\ldots, L$. $$\eqalign{ &\vert e(t^0_{L,j},t^0_{L,j+1})-e(x_0,y_0)\vert\cr &\leq \left\vert e(t^0_{L,j},t^0_{L,j+1})-{1\over L} \ee^\pi_L(d(t^0_{L,j},t^0_{L,j+1}))\right\vert\cr &\quad + {1\over L}\left\vert\ee^\pi_L(d(t^0_{L,j},t^0_{L,j+1})) -\ee^\pi_L(d(x_L,y_L))\right\vert +{1\over L}\left\vert\ee^\pi_L(d(x_L,y_L)) -e(x_0,y_0)\right \vert \quad .\cr }$$ All three terms on the right side vanish as $L\to\infty$, uniformly in $j$: the first term by the uniform convergence on compact sets given by \Lem/energydensity/ i), the second by \Lem/comparedim/, and the third term by \Lem/energydensity/ ii). The theorem now follows from the uniqueness of the dimerized minimum and compactness. Indeed, suppose one can find arbitrarily large $L_k$ such that $$ \vert t^0_{L_k,j}-x\vert +\vert t^0_{L_k,j+1} - y\vert \geq \epsilon \quad \hbox{ and } \quad \vert t^0_{L_k,j}-y\vert + \vert t^0_{L_k,j+1} - x\vert\geq \epsilon \quad , $$ for some $\epsilon >0$. By compactness and the continuity of $e$ there would then be a limit point $(x^\prime, y^\prime)$ of $(t^0_{L_k, j}, t^0_{L_k, j+1})$, for which $e(x^\prime,y^\prime)=e(x_0,y_0)$. As $(x^\prime,y^\prime)$ as at least distance $\epsilon$ away from $(x_0,y_0)$ this would imply the existence of another minimizer for $e$ which is ruled out by assumption. \QEDnogroup \noindent {\bf Proof of \Thm/uniquedim/ :}\nl Let $\Phi=0$ if $L=2\bmod 4$ and $\Phi=\pi$ if $L=0\bmod 4$. Denote by $d(x,y)$ the unique dimerized minimizer of $\ee^\Phi_L$. Let $\{t^\Phi_j\}$ be any minimzer of $\ee^\Phi_L$. As $\ee^\Phi_L$ attains its minimum in $\{t^\Phi_L\}$, the inequality of \Lem/compareaveragedim/ is an equality for $t_j=t^\Phi_j$. Therefore, all configurations on the left side of \eq(compareaveragedim) must be minimizers. These configurations are all dimerized and therefore (by our uniqueness assumption) must all be equal to $d(x,y)$. \QEDnogroup \bgsection 5. COUNTEREXAMPLES: RINGS OF LENGTH $L=0\bmod 4$\hfill\break HAVING MINIMIZERS AT ZERO FLUX THAT ARE NOT PERIOD TWO We will only discuss counterexamples with $U=0$. It is obvious, then, that counterexamples also exist with small non-vanishing $U$. However, the presence of a not too large on-site repulsion tends to enhance the dimerization, while nearest neighbour repulsions can reduce it. We have not investigated the effects of interactions on the new instabilities discussed in this section. Our result for the spin-Peierls problem, \Thm/spinpeierls/, which can be interpreted as the $U\to\infty$ limit \cite{And}, seems to indicate that, when $U$ is very large, the translation symmetry can break down from period one to at most period two. This seems natural because the electrons are more mobile when $U$ is small and, under this condition, other deformations of the lattice can become favorable if the flux is not optimal. We interpret distortions that break the periodicity two as an attempt of the system to mimic the energy lowering effect of a nonvanishing flux. This is also a possible interpretation of the lattice distortions found in two dimensions \cite{Maz,TH}. Kennedy and Lieb \cite{KL} gave an example of a function $f$ (involving a quartic term) which gives rise to a minimum for the model on a square ($L=4$) which does {\it not\/} have periodicity two. Longuet-Higgins and Salem pointed out the possibility of other than period two instabilities a long time ago \cite{LHS}, but they expected these to be unimportant and thought the period two instability would always dominate. They only considered the case $L=2\bmod 4$ and therefore did not notice that other instabilities can, in fact, occur when $L=0\bmod 4$. Our aim in this section is twofold. First we present a family of functions $f$ that lead to non-period-two minimizers of the zero-flux problem for the ring of eight sites. Then we discuss what instabilities other than period two occur. From the discussion it will become clear how to construct many more examples of non-period-two minima. The properties of the new instabilities, which we study partly only numerically, support our conjectures stated in the introduction. \medskip \noindent {\bf Counterexample for $L=8$.}\nl For simplicity we will present just one family of functions $f$ chosen to yield simple values for the minimizing dimerized configuration. It is easy to generalize this example in several directions. Define a function $f(t)$, for $t\geq 0$, as follows (see Figure 2 for the graph of $f$.) $$ f(t)=\cases{ {1\over 10} + {1\over 5} (t-1) +{1\over 10} (1-t)^2 &if $0\leq 4-a$\cr \phantom{a} & \cr {1\over 10} + {1\over 5} (t-1)+{1\over 10}(1-t)^2 +{1\over 3a^2}(t-4+a)^3 &if $t\geq 4-a$\quad .\cr }\deqno(f)$$ Here $a$ can be any number strictly between $0$ and $4-5/\sqrt{2}\cong .4645$. Consider first configurations $t_j$ of the form $t_{2j+1}=x$, $t_{2j}=y$. We always assume $x\geq y\geq 0$. The ground state energy of eight electrons on a ring of eight sites described by the Hamiltonian \eq(hubham) with $U=0$ is then $4 e(x,y)$ where $e(x,y)$ is given by $$ e(x,y)= -x-\sqrt{x^2+y^2}\quad . $$ For the 8-ring with $f$ given in \eq(f), the optimum period two configuration is then determined by the values of $x_0\geq y_0$ that minimize $$ h(x,y)\equiv e(x,y) + f(x) + f(y) \quad . $$ It is trivial to see that $x_0>0$, and as $h$ is differentiable away from $x=0$ and $h(x,y)\to +\infty$ as $x$ or $y\to +\infty$, $(x_0,y_0)$ must therefore be a critical point of $h(x,y)$ solving the equations $$\eqalign{ 1+ {x\over \sqrt{x^2+y^2}}&=f^\prime(x)\cr {y\over \sqrt{x^2+y^2}}&=f^\prime(y)\quad .\cr }$$ A straightforward calculation shows that $(x,y)=(4,3)$ is the unique minimum. Next we study the stability of the relative minimum $(x_0,y_0)$ under a class of perturbations which break the period two symmetry. For future use we do this for arbitrary $L=0\bmod 4$, $L\geq 8$. More specifically we consider configurations of the form $t_{2j+1}=x$, $t_{2j}=y+\epsilon V_{j+1}$, $j=1,\ldots, N\bmod N$, with $L=2N$. As we do not want to introduce a flux, the $V_j$ must be real. One can use analytic perturbation theory to compute the energy as a function of $x,y$ and $\epsilon$. Up to corrections of $\Order(\epsilon^3)$ one finds $$\eqalign{ \hbox{Kin. Energy } =&-2\sum_{n=1}^N \lambda_n - \epsilon \hat V (0)\sum_{n=1}^N {y+x\cos(\alpha n)\over \lambda_n}\cr &+\epsilon^2\sum_{n=1}^N\left({\vert\hat V (0)\vert^2 (y+x\cos(\alpha n))^2\over \lambda_n^3} -{\sum_{m\neq 2n\bmod N} \vert\hat V(m)\vert^2 \over \lambda_n} \right) -\epsilon^2{\vert \hat V(0)\vert^2 \over \vert x-y\vert}\cr &-4\epsilon^2 \sum_{n,m;\lambda_n \neq\lambda_m} \vert \hat V(n-m)\vert^2 {P_{n,m} \over \lambda_n(\lambda_n^2-\lambda_m^2)} \quad ,\cr }\deqno(delta)$$ where $$ \lambda_n=\sqrt{x^2+y^2+2xy\cos(\alpha n)},\quad \alpha=2\pi/N \quad . $$ and $$ P_{n,m}= y^2 + x^2 \cos(\alpha(n+m)/2))^2 +2xy \cos(\alpha(n+m)/2) \cos(\alpha(n-m)/2) \quad . $$ $\hat V$ is the Fourier transform of $V_j$: $\hat V(q)= (1/N)\sum_j \exp (2\pi ijq/N) V_j$. Note that the coeffient of $\epsilon^2$ is diagonal in $\hat V$. Up to terms $\Order(\epsilon^3)$, the elastic energy is $$ N\left( f(y) +f(x) + \epsilon f^\prime(y)\sum_j V_j+ \tover12 \epsilon^2 f^{\prime\prime}(y)\sum_j V_j^2\right) \quad . $$ In a critical point $(x,y)$ in the set of period two configurations, the terms proportional to $\epsilon$ in the electron energy will cancel the ones in the elastic energy. A perturbation $V$ will then lower the energy for small $\epsilon$ if the total (electron plus elastic energy) coefficient of $\epsilon^2$ is negative, which means that the optimal dimerized configuration is not the true minimum. Call the coefficient of the $\epsilon^2$ contribution to the electron energy $-\Delta(x,y)$. Note that this coefficient is always negative. There is an instability whenever $$ \Delta(x_0,y_0) > {N\over 2}f^{\prime\prime}(y_0) \quad . \deqno(instab)$$ We now return to the model on the 8-ring and show that indeed the energy can be lowered by perturbing the dimerized minimum $(x_0,y_0)$ found above. For the 8-ring ($N=4$) there are four independent perturbations $V_i$ to consider: $$\eqalign{ V=(1,1,1,1),&\quad V=\sqrt{2}(1,0,-1,0)\quad ,\cr V=(1,-1,1,-1),&\quad V=\sqrt{2}(0,1,0,-1)\quad .\cr }$$ The first, $V=(1,1,1,1)$, respects the period 2 and in the optimum dimerized configuration it cannot lower the energy. The second and the fourth have the same energy because they are translates of each other. We have normalized the $V$'s such that $\sum_i V_i^2=4$ (in general we will use the normalization $\sum_i V_i^2=N$.) Let $-\Delta_0,-\Delta_1,-\Delta_2$ be the coefficients of $\epsilon^2$ for the first, second and third choice of $V$ respectively. It is convenient to factor out $x$ as a scale factor and to use the ratio $r=y/x$ as the relevant variable, $0\leq r\leq 1$. The explicit expressions for the $\Delta$'s are: $$\eqalign{ x\Delta_0&={2\over (1+r^2)^{3/2}}\cr x\Delta_1&= 4 - {2\over \sqrt{1+r^2}}\cr x\Delta_2&=2 \quad .\cr }$$ In our example both $\Delta_1$ and $\Delta_2$ satisfy the criterion for instability \eq(instab): $$ \vert \Delta_1(4,3)\vert = {3\over 5}>2f^{\prime\prime}(3)= {2\over 5},\qquad \vert \Delta_2(4,3)\vert = {1\over 2}>2f^{\prime\prime}(3)= {2\over 5} \quad . $$ Of course, $\Delta_0(4,3)$ does not satisfy \eq(instab), indicating that (4,3) is a stable relative minimum within the set of period two configurations (indeed $\Delta_0(4,3)=32/125 < 2/5$). We conclude that for the choice of $f$ defined in \eq(f) the energy minimizing configuration of the 8-ring is not period two. \medskip \noindent {\bf Other than period-two instabilities for arbitrary $L =0\bmod 4$.\/}\nl The behavior of the coefficients of $\epsilon^2$ in \eq(delta) for large $N$ gives support to our conjectures 1 and 2 stated in the introduction. Let $-\Delta_k$ denote the coefficient of $\epsilon^2$ for $V= V^{(k)}(q)$, where $\hat V^{(k)}(q)=(\delta_{q,k}+\delta_{q,N-k})/\sqrt{2}$, $k\neq 0,N/2$, $\hat V^{(0)}(q)=\delta_{q,0}$, $\hat V^{(N/2)}(q)=\delta_{q,N/2}$. The perturbations are normalized such that their contributions to the elastic energy is identical. $\Delta_0$ and $\Delta_1$ are given by: $$\eqalign{ \Delta_0&= \sum_{n=1}^N {x^2\sin(\alpha n)^2\over \lambda_n^{3}} \quad ,\cr \Delta_1&= \sum_{n=1}^N {1\over\lambda_n}+ \sum_{n=1}^N\left({1\over \lambda_n}-{1\over\lambda_{n+1}}\right) {P_{n,n+1}\over xy (\cos(\alpha n) -\cos(\alpha (n+1)))}\quad .\cr }$$ The stability of a relative minimum $(x_0,y_0)$ within the set of period two configurations requires that $\Delta_0 < (N/2)f^{\prime\prime}(y_0)$. When $\Delta_0 < (N/2)f^{\prime\prime}(y_0)< \Delta_1$ such a relative minimum is unstable against the perturbation $\epsilon V^{(1)}$, which has wavelength $=L$. Numerical investigation of these expressions reveals the following: for $r\equiv y/x <1$ but close enough to 1 (depending on $N$), one has $\Delta_1 >\Delta_0$. This is illustrated in Figure 3. Note that the perturbations with $k >1$ are all stable. We showed in the previous subsection that for $L=8$ this situation can be realized with a function $f$ satisfying the conditions i and ii) stated in the introduction. It is reasonable to expect that this can be done for all $L$ that are a multiple of 4, but the larger $L$ becomes the more stringent are the restrictions on $f$. In particular, if one fixes $f$ we expect that for large enough systems the long wavelength instability will not occur and the system will remain in a period two configuration. \bgsection A. Appendix \noindent {\bf 1) Discontinuous and non-convex $f$\/}\nl We have assumed throughout this paper that $f$ was a continuous function satisfying some simple conditions as stated in the introduction \eq(conditions). Whether or not $f$ is continuous we can associate a function $\overline f$ with $f$ having the following properties: (a) $\overline f$ is convex (i.e., $\overline f(\lambda t +(1-\lambda)t^\prime) \leq \lambda \overline f(t) + (1-\lambda)\overline f(t^\prime)$ for all $0\leq \lambda \leq 1$ and all $t,t^\prime \geq 0$; (b) $\overline f(t) =\sup\{g(t)\, :\, g \hbox{ is convex and } g(t^\prime)\leq f(t^\prime) \hbox{ for all } t^\prime\geq 0\}$. This function $\overline f$ is called the convex hull of $f$ and is automatically continuous for $t>0$. The following lemma shows that all we really need is that $\overline f$ (not $f$) satisfy conditions i) and ii) of \eq(conditions). \iproclaim/convexf/ Lemma. Let $\{t_j^\Phi\}$ be a minimizing configuration of $\ee^\Phi_L$. Then for all $j$, $f(t^\Phi_j)=\overline f (t^\Phi_j)$, where $\overline f$ is the convex hull of $f$. \eproclaim \proof: Let $j$ be arbitrary and fixed. As a function of $t_j=t$ the energy can be written as $$\eqalign{ \ee(t)&=E(t) + f(t)\cr &=(E(t) + at) + (f(t)-at)=K(t) + g(t) \quad .\cr }$$ We choose $a = (f(t_2)-f(t_1))/(t_2-t_1)$ where $$ t_1=\sup\{st_j^\Phi\mid f(s)\neq \overline{f} (s)\} \quad . $$ As $t$ appears linearly in the Hamiltonian $H$, $K(t)=E(t) + at$ is a concave function of $t$ (i.e., $-E(t)-at$ is convex). By construction we also have that $g(t_j^\Phi)\geq g(t_1)=g(t_2)$. Hence, with $0\leq\alpha\leq 1$ such that $t_j^\Phi=\alpha t_1 + (1-\alpha) t_2$ one has $$ \ee(t_j^\Phi)\geq \alpha \ee(t_1) +(1-\alpha)\ee(t_2) $$ and the inequality is strict if $g(t_j^\Phi)=f(t_j^\Phi) -\overline f (t_j^\Phi)\neq 0$. Therefore the minimium can be attained only at $t_j$ where $f(t_j)= \overline f (t_j)$. \QED \medskip \noindent {\bf 2) The DLS Lemma for ground states}\nl The following lemma is a ground state version of Lemma 4.1 in \cite{DLS}. An example of this ground state version is given in \cite{KLS}. Note that the matrices $A$ and $B$ need not have real matrix elements. It is obvious how to generalize to reflections that involve an additional unitary transformation. When needed one can in fact apply the lemma as stated below after performing a unitary transformation on one half of the system. In this paper we applied the lemma in two different situations. In the proof of \Lem/spin0/ the two copies of the Hilbert space $\H$ correspond to spin $\up$ and spin $\down$. In \Lem/RPineq/ the two spaces refer to the right and left halves of the ring. \iproclaim/DLS/ Lemma. Let $A,B,C_1,\ldots,C_n$ be a collection of $d\times d$ complex matrices ($n$ could be infinite) with the following properties: $A$ and $B$ are Hermitian (i.e., $A=A^\dagger, B=B^\dagger$), for all $i$, $C_i$ is real and $\sum_i C_i\otimes C_i$ is symmetric (as a $d^2\times d^2$ matrix). Let $\lambda_0(A,B)$ denote the lowest eigenvalue of the matrix $$ T(A,B)\equiv A\otimes\idty-\sum_i C_i\otimes C_i +\idty\otimes B \quad . \deqno(TAB)$$ Then $$ \lambda_0(A,B)\geq \tover12\left( \lambda_0(A,A^* )+\lambda_0(B^* ,B)\right) \quad , \deqno(TABineq)$$ where $A^*$ denotes the matrix obtained from $A$ by complex conjugation of the matrix elements. In particular $$ \lambda_0(A,B)\geq \min\left( \lambda_0(A,A^* ),\lambda_0(B^* ,B)\right) \quad . $$ The inequality \eq(TABineq) is strict if for some $i$ $$ \langle\Omega\mid C_i\otimes\idty\mid\Omega\rangle \neq \langle\Omega\mid \idty\otimes C_i\mid\Omega\rangle \quad , \deqno(strict)$$ where $\Omega$ is any eigenvector for the eigenvalue $\lambda_0(A,B)$ (this is not the only case where the inequality is strict.) In particular all ground states $\Omega$ of $T(A,A^*)$ satisfy $$ \langle\Omega\mid C_i\otimes\idty\mid\Omega\rangle = \langle\Omega\mid \idty\otimes C_i\mid\Omega\rangle \quad . $$ \eproclaim \proof: Let $\{e_j\}$ be a orthonormal basis of $\Cx^d$ in which the $C_i$ have real matrix elements. Define $V$ to be the antilinear map leaving the $e_j$ invariant: $Ve_j=e_j$. Then $VAV=A^*$ etc. Let $\Omega$ be an eigenvector of the smallest eigenvalue of the operator \eq(TAB) and denote by $M_{ij}$ its coefficients in the basis $e_i\otimes e_j$: $$ \Omega=\sum_{i,j} M_{ij}e_i\otimes e_j \quad . $$ We can consider $(M_{ij})$ as a matrix and use the eigenvectors $\{u_\alpha\}$ of $M^\dagger M$ and $\{v_\alpha\}$ of $MM^\dagger$ to define two new bases: $$\phi_\alpha=\sum_j \langle e_j\mid u_\alpha\rangle e_j \quad ,\qquad \psi_\alpha=\sum_j \langle e_j \mid v_\alpha\rangle^* e_j \quad . $$ It is then straigthforward to check that $$ \Omega=\sum_\alpha \lambda_\alpha \phi_\alpha\otimes\psi_\alpha \quad . $$ We can obviously assume that $\lambda_\alpha >0$ by absorbing a phase in the definition of $\phi_\alpha$ and by omitting any vanishing terms in the sum. Using the fact that $\{\phi_\alpha\}$ and $\{\psi_\beta\}$ are orthonormal bases the energy of $\Omega$ can be expressed as $$\eqalign{ \langle\Omega\mid T(A,B)\mid\Omega\rangle &= \sum_\alpha \lambda_\alpha^2\left(\langle\phi_\alpha\mid A\mid\phi_\alpha\rangle +\langle\psi_\alpha\mid B\mid\psi_\alpha\rangle\right)\cr &\quad - \sum_i\sum_{\alpha\beta}\lambda_\alpha \lambda_\beta \langle\phi_\alpha\mid C_i\mid\phi_\beta\rangle \langle\psi_\alpha\mid C_i\mid\psi_\beta\rangle \quad .\cr }$$ This energy can directly be compared with the energy of $T(A,A^*)$ and $T(B^* , B)$ in the states $\Omega_1$ and $\Omega_2$ defined by $$\eqalign{ \Omega_1=\sum_\alpha \lambda_\alpha \phi_\alpha\otimes V\phi_\alpha\cr \Omega_2=\sum_\alpha \lambda_\alpha V\psi_\alpha\otimes \psi_\alpha\cr }$$ which have the same norm: $\Vert\Omega\Vert^2=\Vert\Omega_1\Vert^2=\Vert\Omega_2\Vert^2= \sum_\alpha \lambda_\alpha^2$. The average of the energies of $\Omega_1$ and $\Omega_2$ is $$\eqalign{ &\tover12(\langle\Omega_1\mid T(A,A^* )\mid\Omega_1\rangle + \langle\Omega_2\mid T(B^* , B)\mid\Omega_2\rangle)\cr &= \tover12\sum_\alpha \lambda_\alpha^2 \left(\langle\phi_\alpha\mid A\mid\phi_\alpha\rangle +\langle V\phi_\alpha\mid A^* V\mid\phi_\alpha\rangle +\langle V\psi_\alpha\mid B^* V\mid\psi_\alpha\rangle +\langle\psi_\alpha\mid B\mid\psi_\alpha\rangle\right)\cr &- \tover12\sum_i\sum_{\alpha\beta}\lambda_\alpha \lambda_\beta \left(\langle\phi_\alpha\mid C_i\mid\phi_\beta\rangle \langle V\phi_\alpha\mid C_i V\mid\phi_\beta\rangle \quad +\langle \psi_\alpha\mid C_i\mid\psi_\beta\rangle \langle V\psi_\alpha\mid C_i V\mid\psi_\beta\rangle\right).\cr } \deqno(**)$$ Using $\langle V\phi\mid \psi\rangle=\langle \phi\mid V^\dagger\mid\psi\rangle^*$, for any two vectors $\phi,\psi$, the hermiticity of $A$ and $B$ and the reality of the $C_i$, it is straightforward to verify: $$\eqalign{ \langle V\phi_\alpha\mid A^* V\mid\phi_\alpha\rangle &=\langle \phi_\alpha\mid A \mid\phi_\alpha\rangle\cr \langle V\phi_\alpha\mid C_i V\mid\phi_\beta\rangle &=\langle \phi_\alpha\mid C_i \mid\phi_\beta\rangle^*\quad ,\cr }$$ etc. As $\lambda_\alpha >0$ we can then apply $2\vert u v\vert\leq \vert u\vert^2+\vert v\vert^2$ to each term in the double sum of \eq(**) to obtain $$ \langle\Omega\mid T(A,B)\mid\Omega\rangle \geq \tover12\left(\langle\Omega\mid T(A,A^* )\mid\Omega\rangle +\langle\Omega\mid T(B^* ,B)\mid\Omega\rangle\right) $$ from which \eq(TABineq) directly follows by the variational principle. The inequality is strict if for some $i$ and some $\alpha, \beta$, $\langle \phi_\alpha\mid C_i \mid\phi_\beta\rangle \neq \langle \psi_\alpha\mid C_i \mid\psi_\beta\rangle$, for which \eq(strict) is a sufficient condition. When $A=B^*$, \eq(TABineq) is an equality and so $\langle \phi_\alpha\mid C_i \mid\phi_\beta\rangle \neq \langle \psi_\alpha\mid C_i \mid\psi_\beta\rangle$ for all $i,\alpha,\beta$ in that case. \QED >From the proof of the lemma it also follows that any Hamiltonian of the form $T(A,A^*)$, has a ground state $\Omega$ satisfying $$ \langle\Omega\mid X\otimes X^* \mid\Omega\rangle \geq 0 \quad . $$ for all observables $X$ on $\H$. The same state therefore also satisfies $$ \langle\Omega\mid X\otimes \idty \mid\Omega\rangle =\langle\Omega\mid \idty\otimes X^* \mid\Omega\rangle \quad . $$ In particular if $[X\otimes\idty,T(A,A^* )]=0$, and $X=X^\dagger$, there will be a ground state $\Omega$ such that $X\otimes \idty \Omega =\idty\otimes X^* \Omega = x\Omega$ for some eigenvalue $x$ of $X$. (This also follows from application of \eq(TABineq) with $A$ replaced by $A+ c(X-x_1)^2$ and $B$ by $A^* + c (X^* -x_2)^2$, where $x_1$ and $x_2$ are the eigenvalues of $X\otimes\idty$ and $\idty\otimes X^*$ in some ground state of $T(A,A^*)$. \bigskip \noindent {\bf 3) The Abstract Chessboard Estimate(\cite{FILS})\/}\nl \iproclaim/chessboardestimate/ Theorem. Let $D$ be a set, let $R:D\to D$ be an involution on $D$ ($R^2=$ identity), and let $F:\D\to\Cx$ be a complex valued function on a domain $\D\subset D^{\times 2N}$ that has the following invariance properties: $$\eqalign{ (a_1,\ldots,a_{2N})\in\D &\imply (a_2,\ldots, a_{2N},a_1)\in\D \quad ,\cr (a_1,\ldots,a_N,b_N,\ldots,b_1)\in\D &\imply (a_1,\ldots, a_N, R a_N, \ldots,R a_1), (b_1,\ldots, b_N, R b_N,\ldots R b_1)\in\D %\textbox{6}{.7}{ $(a_1,\ldots, a_N, R a_N, \ldots,R a_1)$ and %\hfill\break $(b_1,\ldots, b_N, R b_N,\ldots R b_1)\in\D$\quad .} \cr }$$ Assume $F$ obeys $$ F(a_1,\ldots,a_{2N})=F(a_2,\ldots,a_{2N},a_1) $$ and $$\vert Fa_1,\ldots,a_N,b_N,\ldots,b_1)\vert^2\leq F(a_1,\ldots, a_N, R a_N, \ldots, R a_1) F(b_1,\ldots , b_N, R b_N,\ldots, R b_1) \quad . $$ Then $\Vert a\Vert\equiv \vert F(a,R a, A,\ldots,R a)\vert^{1/2N}$ satisfies $$ \vert F(a_1,\ldots,a_{2N})\vert \leq \prod_{j=1}^{2N} \Vert a_i\Vert \quad . $$ \eproclaim The proof of the Theorem is exactly the same as in \cite{FILS} Theorem 4.1. 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Simon "Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions" J. Stat. Phys. @18(1978)335-383 \REF FILS FILS \Jref J. Fr\"ohlich, R. Israel, E.H. Lieb, B. Simon "Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models" Commun. Math. Phys. @62(1978)1-34 \REF KLS KLS \Jref T. Kennedy, E.H. Lieb, B.S. Shastry "Existence of N\'eel order in some spin 1/2 Heisenberg antiferromagnets" J. Stat. Phys. @53(1988)1019-1030 \vfill\eject \centerline{\bf FIGURE CAPTIONS} \bigskip\noindent {\bf Fig. 1.} A ring containing an even number of sites, labeled $1,\ldots,L$. The hopping matrix element $t_j$ is associated with the bond $\{j,j+1\}$. Reflection through the plane indicated by the dashed line interchanges the role of the hoppings labeled $\tl$ and the ones labeled $\tr$ (see Lemma 7). \bigskip\noindent {\bf Fig. 2.} Graph of the function $f(t)$ defined in \eq(f) for $a= .46 $. Recall that large $t$ corresponds to small separation $w$ and vice versa. \bigskip\noindent {\bf Fig. 3.} Graph of the coefficients $x\Delta_k$ as a function of the ratio $r=y/x$ for $L= 20 $ and $k=0,\ldots,5$. An instability can occur when $\Delta_k(r) > \Delta_0(r)$. 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567 y(.)g(.)1095 568 y(.)1097 569 y(.)1098 570 y(.)1100 571 y(.)g(.)1102 572 y(.)1104 574 y(.)1105 575 y(.)1107 576 y(.)g(.)1109 577 y(.)1111 578 y(.)1112 579 y(.)1114 580 y(.)g(.)1116 581 y(.)1118 582 y(.)1120 583 y(.)1121 584 y(.)g(.)1123 585 y(.)1125 586 y(.)1127 587 y(.)1128 588 y(.)g(.)1130 589 y(.)1132 590 y(.)1134 591 y(.)1135 593 y(.)g(.)1137 594 y(.)1139 595 y(.)1141 596 y(.)1143 597 y(.)g(.)1144 598 y(.)1146 599 y(.)1148 600 y(.)i(.)e(.)1151 601 y(.)1153 602 y(.)1155 603 y(.)1157 604 y(.)g(.)1159 605 y(.)1160 606 y(.)1162 607 y(.)1164 608 y(.)g(.)1166 609 y(.)1167 610 y(.)1169 611 y(.)1171 612 y(.)g(.)1173 613 y(.)1174 614 y(.)1176 615 y(.)1178 616 y(.)g(.)1180 617 y(.)1183 618 y(.)1185 619 y(.)g(.)1187 620 y(.)1190 621 y(.)1192 623 y(.)g(.)1195 624 y(.)1197 625 y(.)1199 626 y(.)g(.)1202 627 y(.)1204 628 y(.)1206 629 y(.)1266 368 y Fd(k)15 b Ff(=)f(0)1266 56 y Fd(k)h Ff(=)f(1)1266 581 y Fd(k)h Ff(=)f(2)1266 524 y Fd(k)h Ff(=)f(3)1266 723 y Fd(k)h Ff(=)f(4)1266 652 y Fd(k)h Ff(=)f(5)1207 858 y Fd(r)538 37 y(x)p Ff(\001)608 44 y Fc(k)632 37 y Ff(\()p Fd(r)q Ff(\))12 1194 y Fb(Fig.)24 b(3.)e Ff(Graph)14 b(of)i(the)f(co)q(e\016cien)o (ts)g Fd(x)p Ff(\001)815 1201 y Fc(k)855 1194 y Ff(as)g(a)h(function)f(of)h (the)f(ratio)g Fd(r)h Ff(=)e Fd(y)r(=x)i Ff(for)f Fd(L)f Ff(=)f(20)12 1254 y(and)20 b Fd(k)h Ff(=)f(0)p Fd(;)8 b(:)g(:)g(:)h(;)f Ff(5.)34 b(An)20 b(instabilit)o(y)g(can)g(o)q(ccur)g(when)g(\001)1162 1261 y Fc(k)1186 1254 y Ff(\()p Fd(r)q Ff(\))i Fd(>)e Ff(\001)1370 1261 y Fa(0)1392 1254 y Ff(\()p Fd(r)q Ff(\).)35 b(Note)22 b(that)e(this)12 1313 y(condition)c(is)g(satis\014ed)g(only)h(in)f(a)h(small) f(region)g(of)h Fd(r)i Ff(not)e(m)o(uc)o(h)e(smaller)g(than)i(1,)g(and)f (only)12 1373 y(for)g Fd(k)f Ff(=)f(1,)i(whic)o(h)g(corresp)q(onds)e(to)j(an) f(instabilit)o(y)g(of)h(w)o(a)o(v)o(elength)e Fd(L)h Ff(in)h(real)f(space.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF