% the paper contains one figure. To print it it must be compiled % in postscript by using the dvips driver. Otherwise one gets % a blank space, instead of the figure. The figure file % is however always produced automatically in postscript and it % can be printed separately. The name of the figure file will be % fig1abczwk.ps !! \newcount\driver \newcount\mgnf \newcount\tipi \mgnf=0 \driver=1 \tipi=2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newskip\ttglue %%cm completo \def\TIPITOT{ \font\dodicirm=cmr12 \font\dodicii=cmmi12 \font\dodicisy=cmsy10 scaled\magstep1 \font\dodiciex=cmex10 scaled\magstep1 \font\dodiciit=cmti12 \font\dodicitt=cmtt12 \font\dodicibf=cmbx12 \font\dodicisl=cmsl12 \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\seirm=cmr6 \font\seibf=cmbx6 \font\seii=cmmi6 \font\seisy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\dodicitruecmr=cmr10 scaled\magstep1 \font\dodicitruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\seitruecmr=cmr6 \font\seitruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% definizioni per 10pt %%%%%%%% \textfont\truecmr=\tentruecmr \scriptfont\truecmr=\seventruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\tentruecmsy \scriptfont\truecmsy=\seventruecmsy \scriptscriptfont\truecmr=\fivetruecmr \scriptscriptfont\truecmsy=\fivetruecmsy %%%%% cambio grandezza %%%%%% \def \ottopunti{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\seirm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\seii \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\seisy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\seibf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\seirm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\seitruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\eighttruecmsy \scriptfont\truecmsy=\seitruecmsy }\let\nota=\ottopunti} \newfam\msbfam %per uso in \TIPITOT \newfam\truecmr %per uso in \TIPITOT \newfam\truecmsy %per uso in \TIPITOT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%cm ridotto \def\TIPI{ \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seitruecmr=cmr6 \textfont\truecmr=\tentruecmr \textfont\truecmsy=\tentruecmsy %%%%% cambio grandezza %%%%%% \def \ottopunti{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \textfont1=\eighti \textfont2=\eightsy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\seirm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\seitruecmr %\textfont\truecmsy=\eighttruecmsy }\let\nota=\ottopunti} %%am \def\TIPIO{ \font\setterm=amr7 %\font\settei=ammi7 \font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7 %%%%% cambiamenti di formato %%% \def \settepunti{\def\rm{\fam0\setterm}% passaggio a tipi da 7-punti \textfont0=\setterm %\textfont1=\settei \textfont2=\settesy %\textfont3=\setteit %\textfont\itfam=\setteit \def\it{\fam\itfam\setteit} \textfont\bffam=\settebf \def\bf{\fam\bffam\settebf} \normalbaselineskip=9pt\normalbaselines\rm }\let\nota=\settepunti} %%Scelta dei caratteri \newskip\ttglue \ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi %%%%%%%%% GRAFICA % % Inizializza le macro postscript e il tipo di driver di stampa. % Attualmente le istruzioni postscript vengono utilizzate solo se il driver % e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o ???? ( \driver=2); % qualunque altro valore di \driver produce un output in cui le figure % contengono solo i caratteri inseriti con istruzioni TEX (vedi avanti). % %\ifnum\driver=0 \special{ps: plotfile ini.ps global} \fi %\ifnum\driver=1 \special{header=ini.ps} \fi \newdimen\xshift \newdimen\xwidth \newdimen\yshift % % inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro % occupi la posizione (#1,#2) % \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} % % Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in % #4.ps; in questo disegno si possono introdurre delle stringhe usando \ins % e mettendo le istruzioni relative nell'argomento #3. % Il file #4.ps contiene le istruzioni postscript, che devono essere scritte % presupponendo che l'origine sia nell'angolo inferiore sinistro della % scatola, mentre per il resto l'ambiente grafico e' quello standard. % #5 deve essere della forma \eq("nome simbolico"). % % Le istruzioni postscript possono essere inserite nel file che contiene % l'istruzione \insertplot, racchiudendole fra le istruzioni \initfig{#4} % e \endfig; inoltre ogni riga deve cominciare con "write13<" e deve finire % con ">". In questo modo si crea il file #4.ps relativo alla figura. % \def\insertplot#1#2#3#4#5{\par% \xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2% \yshift=#2 \divide\yshift by 2% \line{\hskip\xshift \vbox to #2{\vfil% \ifnum\driver=0 #3 \special{ps::[local,begin] gsave currentpoint translate}% \special{ps: plotfile #4.ps} \special{ps::[end]grestore}\fi \ifnum\driver=1 #3 \special{psfile=#4.ps}\fi \ifnum\driver=2 #3 \ifnum\mgnf=0\special{#4.ps 1. 1. scale} \fi \ifnum\mgnf=1\special{#4.ps 1.2 1.2 scale}\fi %\fi }\hfill \raise\yshift\hbox{#5}}} \fi }\hfill}\line{\hfil\hbox{#5}\hfil}\vskip.3truecm} \def\initfig#1{% \catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2 \openout13=#1.ps} \def\endfig{% \closeout13 \catcode`\%=14\catcode`\{=1 \catcode`\}=2\catcode`\<=12\catcode`\>=12} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% FORMATO \ifnum\mgnf=0 \magnification=\magstep0\hoffset=0.cm \voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=24.truecm \baselineskip=14pt % plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 \fi \ifnum\mgnf=1 \magnification=\magstep1\hoffset=0.cm \voffset=-1truecm\hoffset=-.5truecm\hsize=16.5truecm \vsize=24.truecm \baselineskip=14pt % plus0.1pt minus0.1pt \parindent=12pt \lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt \def\ds{\displaystyle}\def\st{\scriptstyle}\def\sst{\scriptscriptstyle} \font\seven=cmr7 \fi %%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ps=\psi \let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\theta\let\thth=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\ee\varepsilon \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega \let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon %%%%%%%%%%%%%%%%%%%%% Numerazione pagine \def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or aprile \or maggio \or giugno \or luglio \or agosto \or settembre \or ottobre \or novembre \or dicembre \fi/\number\year;\,\the\time} %\newcount\tempo %\tempo=\number\time\divide\tempo by 60} \setbox200\hbox{$\scriptscriptstyle \data $} \newcount\pgn \pgn=1 \def\foglio{\number\numsec:\number\pgn \global\advance\pgn by 1} \def\foglioa{A\number\numsec:\number\pgn \global\advance\pgn by 1} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglio\hss} %\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm %\foglioa\hss} % %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2}% \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula)% \SIA e,#1,(\veroparagrafo.\veraformula) % \global\advance\numfor by 1% \write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} \def\FU(#1)#2{\SIA fu,#1,#2 } %\def\etichettaa(#1){(A\veroparagrafo.\veraformula)% %\SIA e,#1,(A\veroparagrafo.\veraformula) % \def\etichettaa(#1){(A\veraformula)% \SIA e,#1,(A\veraformula) % \global\advance\numfor by 1% \write15{\string\FU (#1){\equ(#1)}}% \write16{ EQ #1 ==> \equ(#1) }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} % \global\advance\numfig by 1% \write15{\string\FU (#1){\equ(#1)}}% \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){% {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){\gwidth=\hsize \divide\gwidth by 2% {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} \footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm \foglio\hss} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1 \write16{#1 non e' (ancora) definito}% \else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi \openout15=\jobname.aux %\newcount\pinclude \newcount\pcount %\def\include#1{\pinclude=0 %\openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi %\openout15=#1.aux %\input #1 %\pcount=\count0 \advance\pcount by 1 %\write15{\string\ifnum% %\string\pinclude =1 % %\string\pageno = \number\pcount \string\fi} %\closeout15 } % %\def\includeaux#1{\pinclude=1 %\openin13=#1.aux \ifeof13 \immediate\write16{#1.aux does not exist}% %\else \input #1.aux \closein13 \fi } %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% DEFINIZIONI LOCALI \let\ciao=\bye \def\fiat{{}} \def\pagina{{\vfill\eject}} \def\\{\noindent} \def\bra#1{{\langle#1|}} \def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} \def\ie{\hbox{\it i.e.\ }} \def\eg{\hbox{\it e.g.\ }} \let\ii=\int \let\ig=\int \let\io=\infty \let\i=\infty \let\dpr=\partial \def\V#1{\vec#1} \def\Dp{\V\dpr} \def\oo{{\V\o}} \def\OO{{\V\O}} \def\uu{{\V\y}} \def\xxi{{\V \xi}} \def\xx{{\V x}} \def\yy{{\V y}} \def\kk{{\V k}} \def\zz{{\V z}} \def\rr{{\V r}} \def\pp{{\V p}} \def\acapo{\hfill\break} \def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}} \def\otto{{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}} \def\guida{\ ...\ } \def\Z{{\bf Z}}\def\R{{\bf R}}\def\tab{}\def\nonumber{} \def\mbox{\hbox}\def\lis#1{{\overline#1}}\def\nn{{\V n}} \def\Tr{{\rm Tr}\,}\def\EE{{\cal E}} \def\Veff{{V_{\rm eff}}}\def\Pdy{{P(d\psi)}}\def\const{{\rm const}} \def\RR{{\cal R}} \def\NN{{\cal N}} \def\ZZ#1{{1\over Z_{#1}}} \def\OO{{\cal O}} \def\GG{{\cal G}} \def\LL{{\cal L}} \def\DD{{\cal D}} \def\WW{{\cal W}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\CC{{\cal C}} \def\fra#1#2{{#1\over#2}} \def\ap{{\it a priori\ }} \def\rad#1{{\sqrt{#1}\,}} \def\={{\equiv}}\def\ch{{\chi}} \let\figini=\initfig \let\figfin=\endfig \let\0=\noindent \def\*{\vglue0.3truecm} %\let\\=\noindent \def\mod{\,\hbox{mod}\,} \def\pf{p_{\sst F}} \vglue1.cm \centerline{\bf Disorder in the 1D spinless Holstein model.} \vglue1.cm \centerline{G. Benfatto,\footnote{${}^1$}{\nota Dipartimento di Matematica, $II^a$ Universit\`a di Roma, Via della Ricerca Scientifica, 00133, Roma, Italia.} G. Gallavotti,\footnote{${}^2$}{\nota Dipartimento di Fisica, $I^a$ Universit\`a di Roma, P.le Moro 2, 00153 Roma, Italia.}J. L. Lebowitz\footnote{${}^3$}{\nota Dept. of Mathematics and Physics, Rutgers University, New Brunswick, NJ 08903, USA.}} \vglue1.5cm \0{\it Abstract: We investigate a spinless fermion system on a one dimensional lattice interacting locally with the optical modes of a quantized phonon field: the Holstein model. The system is shown to have a disordered ground state, for small enough coupling, at any density. This is in contrast to the non quantized phonon case, the static Holstein model, which at half filling has an ordered ground state for all couplings.} \vglue1.cm \0{\it\S 1 Model and results.} \vskip0.5cm\numsec=1\numfor=1 The hamiltonian is the sum of a free lattice fermion hamiltonian $H_F$, a free phonon field hamiltonian $H_B$ and a local fermion phonon interaction $V$: % $$\eqalign{ H=&\left\{ -\fra{\hbar^2}{2m l}\sum_\xx [a^+_\xx a^-_{\xx+l} +a^+_\xx a^-_{\xx-l}+(\m_0-2)n_\xx] \right\}+\cr &+\left\{ -\fra{\hbar^2}{2\s l}\sum_\xx\fra{\dpr^2}{\dpr\f_\xx^2}\, +\,\fra{l\s}2\sum_\xx [\o^2\f_\xx^2 +\fra{c^2}{l^2}(\f_\xx-\f_{\xx+l})^2] \right\}+\cr &+\left\{ \fra{\hbar^2}{2m l}\sum_\xx (-\n+ \l\f_\xx\,)(n_\xx -\fra12) \right\}\cr}\Eq(1.1)$$ % where $\xx$ is a point on the lattice with spacing $l$, $\xx=j l,\ j\in (-\fra{L}{2l},\fra{L}{2l}]$, $j$ integer. Periodic boundary conditions are imposed at $\pm \fra{L}2$ (by identifying such points). The $a^\pm_\xx$ are creation and annihilation operators for a spinless fermion at $\xx$, $n_\xx=a^+_\xx a^-_\xx$ and $a^+_x a^-_y + a^-_y a^+_x = \d_{xy}$. The field $\f_\xx$ represents a quantized bosonic field, corresponding to a discretized vibrating string with linear density $\s$, optical frequency $\o$ and maximum wave propagation speed $c$. The physical meaning of $\f_\xx$ is that of deformation of the crystal cell sitting at $\xx$.\\ The parameter $m$ is a scale parameter fixing the bare fermion mass. The fermions chemical potential is $\m=\m_0+\n$. The reason for writing $\m$ in this form is that, since the early works on the theory of Fermi systems, [LW], it has been realized that it is more natural to study the properties of such systems when the interaction strength $\l$ is varied at fixed Fermi momentum rather than at fixed $\m$. For the free particle system, corresponding to $\l=0$, the Fermi momentum $\pf$ is defined as the value of the momentum where the momentum distribution, which is the Fourier transform of the one particle reduced density matrix or equal time pair Schwinger function, has a discontinuity. This manifests itself in the one particle density matrix having an oscillatory decay $G_0(\xx)\simeq |\xx|^{-1}\sin \pf |\xx|$ and $\pf$ is obtained from $\m$ by the relation $\m=2(1-\cos l\pf)$. For $\l\ne0$ a Fermi momentum $P(\m,\l)$ can still be defined via the position of the singularity of the Fourier transform of the one particle reduced density matrix. In particular, for $\l$ small we shall prove that the one point reduced density matrix has an oscillating leading asymptotic behaviour $G(\V x)$ proportional to $|\V x|^{-1-2\h}(O(\l)+\sin P\V |x|)$ for some $\h>0$. We now define $\m_0\=2(1-\cos l \pf)$ and choose $\n$ such that $P(\m,\l)=\pf$. This defines $\n$ as a function of $\pf$ and $\l$ with $\n = 0$ when $\l = 0$. It follows from our analysis (see [BGPS] and below) that $P$ is smooth in $\m,\l$ for $\m$ near any prefixed $\m_0\in (0,2)$ and $\l$ small enough (how small depends on the value of $\m_0$). Therefore we can write $P=\pf+c_1 (\m-\m_0)+b_1\l+\ldots$ and $c_1\ne0$ (in fact $c_1=(2l\sin l\pf)^{-1}$). Setting $P=\pf$ then yields $\n=d_1\l+\ldots$ with $d_1=b_1/c_1$. We note that, according to the (formal) Luttinger theorem [L], fixing $P(\m,\l)$ is equivalent to fixing the physical density $\r=\r(\m_0+\n,\l)$, \ie $P$ is independent of $\l$ if $\r$ is fixed: in fact $P=\p\r$. The (formal) extension of this theorem (proved formally in [LW], [L] and formulated there as the "conservation of the Fermi surface volume" at constant density) to cover the present case is discussed, for completeness, in the appendix. In general the value of $\n$ is a complicated function of $\l$, which can only be determined order by order in perturbation theory. There is however an exception; in fact, setting $\e_\xx=(-1)^{\xx/l}$, the unitary transformation $a^\pm_\xx\to \e_\xx a^\mp_\xx$, $\f_\xx\to-\f_\xx$ maps the hamiltonian with chemical potential $\m=2$ and $\n=\n_0$ into that with $\m=2$ and $\n=-\n_0$ and a state with density $\r$ into one with density $1-\r$. Hence we see that, if $\n=0$, there must be a ground state with density $\fra12$. Furthermore the hamiltonian has other symmetries; namely translation invariance and reflection (parity) symmetry. Thus, if we suppose that the ground state is unique (a property that we expect but do not prove) then by applying to it the above three symmetries we see that $G(\V x)=\media{a^-_{\V x} a^+_0}=0$ for $\V x/l$ even. Hence if one can prove, as we claim here, the existence of $P$ such that $G(\V x)$ is proportional to $|\V x|^{-1-2\h}(O(\l)+\sin P\V |x|)$, it follows that $P=\p/2l$ (otherwise, for $\xx/l$ even and large enough, $\sin P\xx$ could be of order $1$), so that the Luttinger theorem is automatically satisfied. The units can be fixed so that: $$\hbar=m=l=m\s\o^2=1\Eq(1.3)$$ \0Setting $b=c\o^{-1}, \s_0^2=\s$, $H$ becomes: $$\eqalign{ 2H=&\left[\sum_\xx (-a^+_\xx a^-_{\xx+1}-a^+_\xx a^-_{\xx-1} +(2-\m)a^+_\xx a^-_\xx)\right]+\cr &+\left\{-\fra1{\s_0^2}\sum_\xx\fra{\dpr^2}{\dpr \f_\xx^2}\,+\,\sum_\xx(\f_\xx^2+ b^2\,(\f_\xx-\f_{\xx+1})^2)\right\}+\cr &+\left(\sum_\xx (-\n+\l\f_\xx)\,(a_\xx^+ a^-_\xx-\fra12)\right)\cr}\Eq(1.4)$$ \0with the half filled band case corresponding to $\m=2,\n=0$. \vglue.5cm \0{\it\S 1.1 The Static Holstein Model.} \vskip0.25cm If $\s_0=+\io,\ b=0$ the model \equ(1.4) is the static Holstein model with ``magnetic field'' equal to $\l/4$, [LM]. The ground state problem is now equivalent to the computation of the fermionic energy $E(\l,\n;\f)$ in the presence of the external field $\f_\xx$ and then finding the field $\f$ minimizing: $$E(\l,\n;\f)+\fra12\sum_\xx \f_\xx^2\Eq(1.5)$$ \0Calling $\EE(\l,\n)$ the minimum value, the corresponding density is obtained by setting $\r-\fra12=-2 \dpr_\n\EE(\l,\n)$ which, if $\r$ is given \ap, is an equation determining $\n$ as a function of $\l,\r$. The case $\m=2,\n=0,\r=\fra12$, the so called ``half filled band'' case, has been solved by [LM], following the methods used by [KL] for the case in which the field $\f_\xx$ can only take values $\pm1$ (the Falikov--Kimball model, see also [MM]). It is shown in [LM] that the minimizing field is $\f_\xx=(-1)^\xx f$ where $f$ is a suitable constant, which can be easily computed by remarking that $E(\l,\n;(-1)^\xx f)$ can be evaluated by the Bloch waves techniques. One finds that $\l\ne0$ implies $f\ne0$. This means that, if $\l\ne0$, there are two non translation invariant ground states, in which the field $\f_{\V x}$ has a periodic structure with period $2$. This is interpreted by saying that at half filling the atoms of the system acquire a crystalline ordering with a period $2$, at any non zero coupling strength; \ie the Peierls instability. They, thus, behave as if they were non interacting particles immersed in a periodic potential with period $2$. It follows from this that the one particle energies are split into two bands, the first corresponding to the momenta $|k|<\fra\p2$, and the second to the larger $k's$. The two bands are separated by a gap $\k(\l)$ and, as a consequence, the fermions are in a filled band state (because the Fermi momentum is $\pf=\fra\p2$ for $\r=1/2$, by the above symmetry). Consequently the two point equal time Schwin\-ger function for the fermions, $G(\xx)=\lim_{t\to 0^-} G(\xx,t)$, decays at large separation as, % $$|G(\xx)|\simeq e^{-\k(\l)|\xx|}\Eq(1.6)$$ % where $\k(\l)$ is the energy gap at momentum $\pf=\p/2$. Note that the energy gap $\k(\l)\tende{\l\to0}0$, and that at $\l=0$ one has instead: $G(\xx) = G_0(\xx) = -(\p \xx)^{-1}\sin \pf\xx$. \vglue.5cm \0{\it\S 1.2 The Dynamic Holstein Model.} \vskip0.25cm The above picture holds {\it for all} values of $\l\ne0$ at half filling ($\r=\fra12,\n=0$) when the phonon field is treated classically, by setting $\s_0^{-2}=0$ in (1.3) from the beginning. We shall show that the results in [BGPS] imply that, if $\s_0<\io$, \ie if quantum effects are not neglected, then for {\it any density} $\r\in (0,1)$, and $\l$ small enough (depending on $\s_0,b$), the decay of $G(\xx,0)$ is given by: % $$G(\xx,0)\simeq \fra{G_0(\xx,0)}{|\xx|^{2\h(\l)}}\Eq(1.7)$$ % with $\h(\l)$ analytic in $\l^2$; $\h(\l)=a \l^4+O(\l^6)$ if $b\ne0$, or $\h(\l)=a'\l^6+O(\l^8)$ if $b=0$, with $a,a'\ne0$. This is clearly incompatible with a periodic minimizing state of the field $\f_x$, showing that the Holstein model ground state for a spinless fermion system is ``disordered'' at small coupling, \ie there is no long range order in $<\f_\xx \f_\yy>$ or $$ as $|\xx-\yy|\to \io$. The situation could change at large coupling, where the ground state could again be ordered: but this is outside the domain of applicability of our techniques. The maximum value of $\l$, $|\l|< \l_m(\s_0)$, for which we can prove \equ(1.7) goes to zero as $\s_0\to\io$. The problem of fermions with spin is, of course, much more interesting. By the same argument given below this case can also be reduced to the problem of fermions with spin, interacting with a short range potential. However the fermionic interaction is attractive: and while this has no consequences in the spinless case it has profound consequences in the case with spin. Unfortunately the spinning case is not understood in the sense needed here to draw any conclusion, see [BM]. \pagina \vskip1.cm \0{\it\S 2 Reduction to a continuum problem.} \vskip0.5cm \numsec=2\numfor=1\pgn=1 When $\l=\n=0$ the phonon and fermion fields are independent. Their respective Schwinger functions (imaginary time Green functions, see [FW] for definitions) can be computed via the Wick rule from the two point functions $S_\L(x)$ and $g_\L(x)$, where $\L\= [0,\b]\times (-\fra{L}2,\fra{L}2]$ and $x=(t,\xx)\in\L$: % $$\eqalign{ S_\L(x)=&\fra1{\b L}\sum_{e^{i k_0\b}=1}\sum_{e^{i\kk L}=1,|\kk|<\p} \fra{e^{-i k_0t-i\kk\xx}}{\s_0^2 k_0^2+1+ b^2 e_B(\kk)}\cr g_\L(x)=&\fra1{\b L}\sum_{e^{i k_0\b}=-1}\sum_{e^{i\kk L}=1,|\kk|<\p} \fra{e^{-i k_0t-i\kk\xx}}{-ik_0+e_F(\kk)}\cr}\Eq(2.1)$$ % where $k=(k_0,\kk)$, and: % $$e_B(\kk)=2(1-\cos\kk),\qquad e_F(\kk)=(\cos \pf-\cos \kk) \Eq(2.2)$$ % The summation rule over $k_0$ is to take the limit of $\sum_{|k_0|0$ which is true in the present case as well, provided $0< \pf <\p$, as we suppose (``positive density smaller than close packing''). Therefore we can perform the decomposition of the propagator in the same way as in [BGPS], see (13) in [G1], by writing: % $${1\over -i k_0+e_F(\kk)} = \fra{1-e^{-k_0^2+e_F(\kk)^2}}{-i k_0+e_F(\kk)} +\fra{e^{-k_0^2-e_F(\kk)^2}}{-i k_0-e_F(\kk)}\Eq(2.7)$$ % \equ(2.7) and \equ(2.1) generate a decomposition of the propagator into a sum of an {\it ultraviolet part}, $g_{u.v.}(x)\=$ $g^{(>0)}(x)$, and of an {\it infrared part}, $g_{i.r.}(x)\=$ $g^{(\le0)}(x)$. The decomposition \equ(2.7) allows us to represent $\psi^\pm_x$ as sums of two independent grassmannian variables, that we denote $\psi^{(>0)}_x$ and $\psi^{(\le0)}_x$ with respective propagators $g^{(>0)}$ and $g^{(\le0)}$. The integration over $\psi^{(>0)}$ can be controlled by perturbation expansions, as in [BGPS], and we shall integrate over it. The remaining integral for the evaluation of the partition function, \ie for the denominator of \equ(2.5), then becomes: % $$\ig P_F(d\psi^{(\le0)}) e^{- V^{(0)}(\psi^{(\le0)})} \Eq(2.8)$$ % with: % $$\eqalign{ V^{(0)}(\psi)=& -\fra{\l^2}2\ig dx\ dy\, S(x-y)\,\psi^+_x\psi^+_y\psi^-_y\psi^-_x+\cr &-\l^2\ig dx dy\, S(x-y) g^{(>0)}(y-x)\psi^+_x\psi^-_y+\cr &-(\bar\n +\l^2 g^{(>0)}(0))\ig dx\,\psi^+_x\psi^-_x+ \sum_{n=1}^\io\ig dx_1\ldots dx_{2n}\cdot\cr &\cdot W_{2n}(x_1\ldots x_{2n})\psi^+_{x_1}\ldots\psi_{x_n}^+\psi_{x_{n+1}}^-\ldots\psi^-_{x_{2n }}\cr}\Eq(2.9)$$ % where the kernels $W_{2n}(x_1\ldots)$ are analytic in $(\l^2,\bar\n)=r$ for $|r|<\e$ (for some $\e>0$) and verify the short range property: % $$\ig dx_1\ldots dx_{2n}\,|W_{2n}(x_1\ldots x_{2n})|\, e^{\k d(x_1,\ldots,x_{2n})}< |\L| (D |r|)^{\max(2,n-1)}\Eq(2.10)$$ % where $\k=\fra12\min(\k_1 \s_0^{-1}, \k_2(b))$, see \equ(2.4), and $d(x_1,\ldots)$ is the length of the shortest path connecting all the points $x_1,\ldots$ (regarded as points on the torus $\L$). The main result achieved by \equ(2.9) is that we have ``disposed'' of the ultaviolet part of the problem (in the evaluation of the partition function) and we can say that the problem is reduced to an essentially identical one with a purely infrared propagator and a new $V^{(0)}$ which, to lowest order in the couplings $r=(\l^2,\bar\n)$ has the same form as the original one, as far as the quartic part is concerned, and a slightly different (non local) quadratic part; plus ``higher order terms'' of every degree in the $\psi$ fields. All the terms in \equ(2.9) are well defined, and have a convergent power series in $r=(\l^2,\bar\n)$ if $r$ is small enough. The proof of the above statements is simpler than the corresponding one in [BGPS], see theorem 1, because in the present case there is a natural ultraviolet cut off, at least in the $\xx$ direction, so that no multi-scale decomposition of $g_{u.v.}(x)$ is needed. The non locality of the quartic part in $V^{(0)}$ is not important even for the infrared problem as it does not affect the notion of relevant or marginal and irrelevant operators, which is the notion on which the infrared integration is based when performed via the renormalization group method in [BGPS]. Hence we reach the same conclusions about the partition function and about the Schwinger functions (whose analysis can be performed once the partition function can be estimated in detail, see [BGPS], \S5 and \S6). The beta function is essentially $0$ (as shown in \S7 of [BGPS]) and this allows us to draw the "same" conclusions as in [BGPS]. In particular one can prove the anomalous asymptotic behaviour \equ(1.7) of the two point Schwinger function with a coefficient $\h(\l)$ which is analytic in $\l^2$ and in general is of order $\l^4$. However, if $b=0$, one can see by an explicit calculation that the leading term in the expansion of $\h(\l)$ vanishes, while the following one seems different from zero. Another interesting observation is that the formal ``Luttinger Theorem'' [LW] is valid in our case: it states that the density $\r$ is a function only of $\pf$, that is $\r=\pf/\p$ for any $\l$: see appendix for a discussion of this formal result (for which it would be nice to have a rigorus proof). In the next section we give some details on the recursion procedure that we follow to obtain the estimates, and on how we define the relevant operators: this will follow closely [BGPS] but should be useful to the readers who wish to see where one is going, before plunging themselves into the analysis of our estimates. But neither the estimates nor the proof of the vanishing of the beta function will be reproduced here: they are identical to the corresponding proofs in [BGPS]. % \vskip1.cm \0{\it\S3 The recursive evaluation of the partition function.} \vskip0.5cm \numsec=3\numfor=1\pgn=1 The following is the "standard" anomalous dimension renormalization procedure. It is illustrated in quite a general context, in which one is given \ap an arbitrary sequence of "wave function renormalization" constants: $Z_0\=1,Z_{-1},Z_{-2},\ldots$. The infrared propagator $g^{(\le 0)}(x)$ is decomposed as: % $$g^{(\le0)}(x)=\ig e^{-ikx}\fra{dk}{(2\p)^2}\fra{T_0(k)}{-ik_0+e_F(\kk)}+ \ig e^{-ikx}\fra{dk}{(2\p)^2}\fra{t_{-1}(k)}{-ik_0+e_F(\kk)}\Eq(3.1)$$ % where $t_h(k)= \exp(-2^{-2h}(k_0^2+ e_F(\kk)^2))$ and $T_0(k)=t_0(k)-t_{-1}(k)$. Eq. \equ(3.1) allows us to represent $\psi^{(\le0)}$ as $\psi^{(0)}+\psi^{(\le -1)}$ where $\psi^{(\le-1)}$ has propagator $Z_0^{-1}t_{-1}(k) (-ik_0+e_F(\kk))^{-1}$ and $\psi ^{(0)}$ has a propagator given by $Z_0^{-1}T_0(k)(-ik_0+e_F(\kk))^{-1}$. Let $\lis P_{Z_0}(d\psi^{(0)})$ and $P_{Z_0}(d\psi^{(\le-1)})$ denote the corresponding integrations, the grassmannian integral in \equ(2.8) thus becomes: % $$\lis P_{Z_0}(d\psi^{(0)}) \, P_{Z_0}(d\psi^{(\le-1)})\Eq(3.2)$$ % Using the sequence $Z_j$ one can then write the identity: % $$(Z_0 t_{-1}^{-1}+ Z_{-1}-Z_0)^{-1}= Z_{-1}^{-1} t_{-2}+ Z_{-1}^{-1}\lis\G^{(-1)}\Eq(3.3)$$ % defining $\lis\G^{(-1)}(k)$. Hence if $\psi^{(-1)}$ is a grassmannian field with propagator: % $$\fra1{Z_{-1}}\fra{\lis \G^{(-1)}(k)}{-ik_0+e_F(\kk)}\Eq(3.4)$$ % and $\psi^{(\le-2)}$ is a grassmannian field with propagator $Z_{-1}t_{-2}(k)(-ik_0+e_F(\kk))^{-1}$, independent of $\psi^{(-1)}$ and of $\psi^{(0)}$, then the grassmannian integral \equ(3.2) becomes (formally), up to normalization constants: % $$\eqalign{ &\lis P_{Z_0}(d\psi^{(0)}\lis P_{Z_{-1}}(d\psi^{(-1)}) P_{Z_{-1}}(d\psi^{(\le-2)})\cdot\cr &\cdot e^{(Z_{-1}-Z_0)\ig \psi^{(\le-1)+}_x (\dpr_t+ e(i\dpr_\xx))\psi^{(\le-1)-}_x}\cr}\Eq(3.5)$$ % as is easily checked with some algebra, if $\psi^{(\le-1)}=\psi^{(-1)}+ \psi^{(\le-2)}$. By iteration: % $$\eqalign{ &P(d\psi^{(\le0)})=\left(\prod_{j=0}^h\lis P_{Z_j}(d\psi^{(j)})\right)\cdot P_{Z_h}(d\psi^{(\le h-1)})\cdot\cr &\cdot\exp\sum_{j=0}^{h+1} (Z_{j-1}-Z_j)\ig \psi^{(\le j-1)+}_x (\dpr_t+ e(i\dpr_\xx))\psi^{(\le j-1)-}_x\,dx\cr}\Eq(3.6)$$ % and $\psi^{(\le0)}=\psi^{(0)}+\psi^{(-1)}+\ldots+\psi^{(h)}+\psi^{(\le h-1)}$, with $\psi^{(j)}$ having propagator $Z_{j}^{-1}\lis \G^{(j)}(k)$ $(-ik_0+e_F(\kk))^{-1}\=$ $Z_{j}^{-1}\lis g^{(j)}$ with: % $$\lis \G^{(j)}(k)=t_j-t_{j-1}+ (1-t_j) t_j\,\fra{z_j}{1+z_j t_j}\Eq(3.7)$$ % if $z_j$ is defined by: $Z_j\=(1+z_j) Z_{j+1}$. The integration \equ(2.8) can therefore be performed recursively by setting: % $$\eqalign{ &e^{-V^{(h)}(\sqrt{Z_h}\psi^{(\le h)})}= \ig \prod_{h'=h+1}^0\lis P_{Z_{h'}}(d\psi^{(h')})\cr &e^{-V^{(0)}(\sqrt{Z_0}\psi^{(\le0)}+\sum_{h'=h}^{-1}(Z_{h'}-Z_{h'+ 1})\ig dx\,\psi_x^{(\le h')+}(\dpr_t+ e(i\dpr_\xx))\psi^{(\le h')-}_x}\cr}\Eq(3.8)$$ % so that: % $$\eqalign{ &\ig e^{-V^{(0)}(\psi^{(\le0)})}P(d\psi^{(\le0)})= \ig e^{-V^{(0)}(\sqrt{Z_0}\psi^{(\le0)})}\,P_{Z_0}(d\psi^{(\le0)})=\cr &=\ig \lis P_{Z_h}(d\psi^{(h)})P_{Z_h}(d\psi^{(0) \bra{0} a^-_\xx e^{-tH} a^+_0\ket{0}- \c(t\le 0) \bra{0} a^+_0 e^{+tH} a^-_\xx\ket{0} \Eqa(a1)$$ % where $\chi(A)$ is the characteristic function of the set of points where the condition $A$ holds. Hence its Fourier tranform $\hat G(\kk,k_0)$ is given by: % $$\eqalign{ \hat G(\kk,k_0) &= \int_0^\io \ dt \Big[ e^{ik_0 t} \bra{0} a^-_\kk e^{-tH} a^+_\kk \ket{0} - e^{-ik_0 t} \bra{0} a^+_\kk e^{-tH} a^-_\kk \ket{0}\Big]\cr &= \sum_{n:E_n>0} \int_0^\io \ dt \Big[ e^{-t(E_n-ik_0)} |\bra{0} a^-_\kk \ket{n}|^2 - e^{-t(E_n+ik_0)} |\bra{0} a^+_\kk\ket{n}|^2 \Big] =\cr &= \sum_{n:E_n>0} \Big[ \fra{|\bra{0} a^-_\kk \ket{n}|^2}{-ik_0+E_n} + \fra{|\bra{0} a^+_\kk\ket{n}|^2}{-ik_0-E_n} \Big] \cr}\Eqa(a2)$$ % where $E_n, n=0,\ldots$, are the eigenvalues of $H$ and $\ket{n}$ the corresponding eigenstates. The ground state gives no contribution to the sum in \equ(a2) (so allowing the integration over $t$) because of the conservation of the total momentum, which implies that $\bra{0}a^\pm_\kk \ket{0}=0$, when $\kk\not=0$, and because of a cancellation between the two different terms in the r.h.s. of \equ(a2), when $\kk=0$. The function $\hat G(k)$, $k=(\kk,k_0)$, is often written also in the following form: % $$\hat G(k)= {1\over -ik_0 + e_F(\kk) +\Si(k)} \Eqa(a3)$$ % which defines the {\it self-energy function} $\Si(k)$. As is well known, [LW], $\Si(k)$ can be expressed in perturbation theory as the sum of all connected graphs with two external lines, which are irreducible, that is which do not become disconnected by erasing any internal line. By using \equ(a1), one sees immediately that the density $\r$ is given by: % $$\r = -\lim_{t\to 0^-} G(0,t) = {1\over L}\sum_\kk \lim_{\e\to 0^+} \int {dk_0\over 2\p} \fra{e^{ik_0\e}}{ik_0 -e_F(\kk)-\Si(k)}\Eqa(a4)$$ Let us write % $${1\over \z -e_F(\kk)-\Si(\kk,-i\z)} = {\dpr\over \dpr\z}\log [\z -e_F(\kk)-\Si(\kk,-i\z)] + \fra{\dpr\Si(\kk,-i\z)/\dpr\z}{\z -e_F(\kk)-\Si(\kk,-i\z)} \Eqa(a5)$$ % by choosing the branch of $\log z$ so that $\log z = \log |z| +i \arg{z}$, $0<\arg{z}<2\p$. It was argued in [LW], if we insert the r.h.s of \equ(a5) in the r.h.s. of \equ(a4), the second term gives no contribution, that is: % $${1\over L}\sum_\kk \lim_{\e\to 0^+} \int {dk_0\over 2\p} e^{ik_0\e} \fra{\dpr\Si(k)/\dpr k_0}{ik_0 -e_F(\kk)-\Si(k)} = -{1\over L}\sum_\kk \int {dk_0\over 2\p} \Si(k) {\dpr \hat G(k)\over \dpr k_0} = 0 \Eqa(a6)$$ We recall briefly the arguments given in [LW] to justify \equ(a6). Indeed, our problem is not explicitly considered in [LW], where the interaction between the fermions is local in time; hence only the case $b=0$ is covered by [LW]. However, their arguments extend in a trivial way to our general case. We start from the observation (not easy to prove rigorously) that $\Si(k)$ can be thought as the sum of all graphs (connected and irreducible) with two external lines, such that the internal lines carry the complete propagator $\hat G(k)$ and the following condition is satisfied: there is no proper subgraph with two external lines. We shall call such graphs, as usual, {\it skeleton graphs} and say that a graph is of order $n$, if it contains $n$ four-fermions interacting terms (hence it has $2n$ vertices). Let $\Si_n(k)$ be the sum of all skeleton graphs of order $n$ and let $Y_n$ be the sum of all vacuum (that is with no external lines) skeleton graphs of order $n$, which can be obtained by closing the two external lines of a graph contributing to $\Si(k)$ with a propagator $\hat G(k)$. Since each graph contributing to $Y_n$ can be obtained in $2n$ different ways from a graph contributing to $\Si(k)$, we have: % $$Y_n = {1\over 2n} {1\over L}\sum_\kk \int {dk_0\over 2\p} \Si(k) \hat G(k)\Eqa(a7)$$ % Moreover, given a graph ${\cal G}$ contributing to $Y_n$, its value $Y_n^{\cal G}$ can be written in the following way: % $$\eqalign{ Y_n^{\cal G} &= \int dk^{(1)}\ldots dk^{(2n)} dq^{(1)}\ldots dq^{(n)} \hat G(k^{(1)}) \cdots \hat G(k^{(2n)}) \cdot \cr &\cdot S(q^{(1)})\ldots S(q^{(n)}) \prod_{i=1}^{2n} \d( k^{(s_i)} -k^{(r_i)} + q^{(b_i)} )\cr} \Eqa(a8)$$ % where $\int dk \= L^{-1}\sum_\kk \int dk_0/(2\p)$ and $k^{(1)},\ldots, k^{(2n)}$ are the (space-time) momenta of the $2n$ fermion lines, $q^{(1)},\ldots, q^{(n)}$ are the momenta of the $n$ phonon lines and, finally, $k^{(s_i)}$ and $k^{(r_i)}$ are the momenta of the fermion entering and leaving the vertex $i$, respectively, while $q^{(b_i)}$ is the momentum of the phonon propagator entering vertex $i$. Moreover, if we consider the sum of all the quantities that we obtain by substituting, in the r.h.s. of \equ(a8), one fermion propagator by its derivative with respect to $k_0$, we get: % $$\eqalign{ &\sum_{j=1}^{2n} \int dk^{(1)}\ldots dk^{(2n)} dq^{(1)}\ldots dq^{(n)} \hat G(k^{(1)}) \cdots \fra{\dpr \hat G}{\dpr k_0^{(j)}}(k^{(j)}) \cdots \hat G(k^{(2n)}) \cdot \cr &\cdot S(q^{(1)})\ldots S(q^{(n)}) \prod_{i=1}^{2n} \d( k^{(s_i)} -k^{(r_i)} + q^{(b_i)} ) =\cr &= \pm \sum_{j=1}^{2n} \int dk^{(1)}\ldots dk^{(2n)} dq^{(1)}\ldots dq^{(n)} \hat G(k^{(1)}) \cdots \hat G(k^{(2n)}) \cdot \cr &\cdot S(q^{(1)})\ldots S(q^{(n)}) ( \fra{\dpr}{\dpr k_0^{(s_j)}} +\fra{\dpr}{\dpr k_0^{(r_j)}} ) \prod_{i=1}^{2n} \d(k^{(s_i)} -k^{(r_i)} + q^{(b_i)} ) = 0\cr} \Eqa(a9)$$ % If we now sum the l.h.s. of \equ(a9) over ${\cal G}$, we get, by an argument similar to that used in order to prove \equ(a7), that: % $$2n \int dk\ \Si(k) \fra{\dpr \hat G}{\dpr k_0}(k) = 0 \Eqa(a10)$$ % so that \equ(a6) is proved. The formal identity \equ(a6) implies that $\r = {1\over L}\sum_\kk \r_\kk$, with % $$\r_\kk = \lim_{\e\to 0^+} \int_{-i\io}^{+i\io} {d\z\over 2\p i} e^{\e\z} {\dpr\over \dpr\z} \log [\z -e_F(\kk)-\Si(\kk,-i\z)]\Eqa(a11)$$ % Moreover, \equ(a2) implies that the integrand in the r.h.s. of \equ(a11) is analytic in all the complex $\z$ plane, except on the real axis, where there are branch points at $\z=\pm E_n$, $n>0$. Therefore, we can deform the integration contour into the contour $\cal C$ of Fig. 1 \initfig{fig1abczwk} \write13 \write13<4 copy exch pop exch sub % x1 dy> \write13<6 1 roll exch pop sub % dy x1 y1 dx> \write13<4 1 roll translate % sposta l'origine; st: dx dy> \write13<2 copy exch atan rotate % rotazione di arctan(dy/dx); st: dx dy> \write13<2 exp exch 2 exp add sqrt % sqrt(dx^2 +dy^2)> \write13<} def> \write13 \write13 \write13 \write13<2 div 0 translate> \write13<15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto> \write13 \write13<> \write13 \write13 \write13 \write13<0 translate> \write13<15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto> \write13 \write13<> \write13 \write13 \write13<> \write13<20 50 280 50 frecciafin 150 10 150 90 frecciafin> \write13<20 30 150 30 2 sfreccia 150 30 150 50 2 sfreccia> \write13<150 50 150 70 2 sfreccia 150 70 20 70 2 sfreccia> \write13 \endfig \vfill\eject \insertplot{300pt}{100pt}{% \ins{155pt}{35pt}{$-\d$}\ins{155pt}{75pt}{$+\d$}% \ins{275pt}{45pt}{${\rm Re\,}\z$}\ins{155pt}{95pt}{${\rm Im}\,\z$}% \ins{25pt}{40pt}{${\cal C}$}}{fig1abczwk}{Fig. 1: Integration Contour for \equ(a11).} \vskip.3truecm \noindent and then we can integrate by parts. We get: % $$\eqalign{ \r_\kk &= \lim_{\h\to 0^+} \fra{\log [-e_F(\kk)-\Si(\kk,-\h)] - \log[-e_F(\kk)-\Si(\kk,\h)]}{2\p i} \ \cr &-\ \lim_{\e\to 0^+} \e\int_{\cal C} {d\z\over 2\p i} e^{\e\z} \log [\z -e_F(\kk)-\Si(\kk,-i\z)] \cr} \Eqa(a12)$$ % It is easy to see that the second term in \equ(a12) vanishes, if the argument of the logarithm has a negative real part for ${\,\rm Re\,}\z \to-\io$, by recalling that we have chosen the branch of $\log z$ with the cut along the positive real axis. We want to show that this is certainly true, if $|\bra{0} a^+_\kk \ket{n}|\to 0$ sufficiently fast, as $E_n\to\io$. We note that ${\,\rm Re\,} [\z -e_F(\kk)-\Si(\kk,-i\z)]$ has the same sign as $-{\,\rm Re\,} \hat G(\kk,-i\z)$; moreover, by \equ(a2), if $\z=u+i\d$: % $$-{\,\rm Re\,} \hat G(\kk,-i\z) = \int_{-\io}^{+\io} \s_\kk(dE) \fra{u-E}{(u-E)^2+ \d^2}\Eqa(a12a)$$ % where $\s_\kk(dE)$ is a probability measure, as it is easy to check. Moreover, if $u\le -u_0$, with $u_0$ large enough, there is an interval $[E_1,E_2]$, such that $|u-E|\ge |u|/2$ for $E\in [E_1,E_2]$ and $\int_{[E1,E_2]} \s_\kk(dE) =1/2$; hence it is easy to show that: % $$-{\,\rm Re\,} \hat G(\kk,-i\z) \le (2\d)^{-1} \int_{-\io}^{u} \s_\kk(dE) - \fra{|u|}{u^2+ 4\d^2}\Eqa(a12b)$$ % which immediately implies that $-{\,\rm Re\,} \hat G(\kk,-i\z)$ is definitely negative for $u\to -\io$, if % $$\lim_{u\to-\io} u \int_{-\io}^u \s_\kk(dE) = 0 \Eqa(a12c)$$ Since, by \equ(a2), the function $\Si(\kk,-i\z)$ is real at $\z=0$, it follows from \equ(a12) that: % $$\r = {1\over L}\sum_\kk \c[e_F(\kk)+\Si(0,\kk) <0] \Eqa(a13)$$ Equation \equ(a13) implies that, in the limit of infinite volume, there is a relation, independent of the strength of the interaction $\l$, between the density and the Fermi momentum $\pf$, defined as the value of $\kk$, such that $e_F(\kk)+\Si(\kk,0)=0$ (that is the value of $\kk$ where the interacting propagator is singular for $k_0=0$). For $\l=0$ we have $\r= \pf/\p$, hence this relation has to be valid for any $\l$. \pagina {\it Acknowledgements:} We thank Nicolas Macris for useful comments. This work has been supported supported in part by NSF Grant DMR 92--13424 4--20946, by CNR-GNF and by MURST 40\%. It is also part of the research program of the European Network on: "Stability and Universality in Classical Mechanics", \# ERBCHRXCT940460. \vskip2.cm \0{\it References} \vskip0.5cm \0[BG1] Benfatto, G., Gallavotti, G.: {\it Perturbation theory of the Fermi surface in a quantum liquid. A general quasi particle formalism and one dimensional systems}, Journal of Statistical Physics, {\bf 59}, 541--664, 1990. \0[BGPS] Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: {\it Beta function and Schwin\-ger functions for a many fermion system in one dimension. 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