Content-Type: multipart/mixed; boundary="-------------0003061656364" This is a multi-part message in MIME format. ---------------0003061656364 Content-Type: text/plain; name="00-101.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-101.keywords" many-electron system, renormalization, Feynman diagrams, Anderson model ---------------0003061656364 Content-Type: application/x-tex; name="resfeyn.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="resfeyn.tex" \documentclass[a4paper,12pt]{article} \usepackage{latexsym} \usepackage{amssymb,amsmath,amsthm} \textwidth=39em \textheight 22cm \headheight=0pt \headsep=0pt \oddsidemargin=0pt \evensidemargin=0pt \renewcommand{\theequation}{\thesection.\arabic{equation}} \font\Bf=cmbx12 \font\Rm=cmr12 \font\gross=cmr12 scaled \magstep3 \font\mittel=cmr12 scaled \magstep1 \font\Gross=cmbx12 scaled \magstep2 \font\Mittel=cmbx12 scaled \magstep1 \overfullrule=0pt \newcommand{\beq}{\begin{eqnarray*}} \newcommand{\beqn}{\begin{eqnarray}} \newcommand{\eeq}{\end{eqnarray*}} \newcommand{\eeqn}{\end{eqnarray}} \newcommand{\bitem}{\begin{enumerate}} \newcommand{\eitem}{\end{enumerate}} \newcommand{\bmatr}{\begin{array}} \newcommand{\ematr}{\end{array}} %\newcommand{\Bbb}{{\mathbb}} %\newcommand{\1cm}{\hskip 1cm} %\newcommand{\blacksquare}{\sqcup\!\!\!\!\sqcap} \newcommand{\up}{\uparrow} \newcommand{\down}{\downarrow} \newcommand{\ts}{\textstyle} \newcommand{\ds}{\displaystyle} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\pt}{\partial} \newcommand{\pro}{\mathop\Pi} \newcommand{\e}{{\bf e}} \newcommand{\h}{{\bf h}} \newcommand{\bj}{{\bf j}} \newcommand{\bk}{{\bf k}} \newcommand{\n}{{\bf n}} \newcommand{\m}{{\bf m}} \newcommand{\p}{{\bf p}} \newcommand{\q}{{\bf q}} \newcommand{\x}{{\bf x}} \newcommand{\y}{{\bf y}} \newcommand{\E}{{\bf E}} \newcommand{\A}{{\bf A}} \newcommand{\B}{{\cal B}} \newcommand{\F}{{\cal F}} \newcommand{\M}{{\cal M}} \newcommand{\N}{{\cal N}} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ep}{\epsilon} \newcommand{\vep}{\varepsilon} \newcommand{\vp}{\varphi} \newcommand{\vph}{\varphi} \newcommand{\Gs}{\Gamma^{\sharp}} \newcommand{\grad}{{\rm grad}} \newcommand{\rot}{{\rm rot}} \newcommand{\Ld}{{\ts {1\over L^d}}} \begin{document} \include{epsf} { \thispagestyle{empty} \baselineskip=16pt $$ $$ \vskip 3cm \centerline{\gross Resummation of Feynman Diagrams } \smallskip \centerline{\gross and the Inversion of Matrices } \bigskip \bigskip \centerline{by} \bigskip \bigskip \centerline{{\mittel Detlef Lehmann}\footnote{ e-mail: lehmann@math.tu-berlin.de}} \centerline{\mittel Technische Universit\"at Berlin} \centerline{\mittel Fachbereich Mathematik Ma 7-2} \centerline{\mittel Sta{\ss}e des 17. Juni 136} \centerline{\mittel D-10623 Berlin, GERMANY} \vskip 4.5cm \noindent{\bf Abstract:} In many field theoretical models one has to resum two- and four-legged subdia\-grams in order to determine their behaviour. In this article we present a novel formalism which does this in a nice way. It is based on the central limit theorem of probability and an inversion formula for matrices which is obtained by repeated application of the Feshbach projection method. We discuss applications to the Anderson model, to the many-electron system and to the $\vp^4$-model. In particular, for the many-electron system with attractive delta-interaction, we find that the expectation value of the Hubbard-Stratonovich field for small momentum $q$ has a delta-function singularity instead of the commonly expected $1/q^2$-type singularity. \bigskip \bigskip %\centerline{\it (preliminary version)} \vfill \eject } % % % E I N L E I T U N G % % \pagenumbering{arabic} \baselineskip=16pt \section{Introduction} \bigskip \indent The computation of correlation functions in field theoretical models is a difficult problem. In this article we present a novel approach which applies to models where a two point function can be written as $$ S(x,y)=\int [P+Q]_{x,y}^{-1}\> d\mu(Q)\>. \eqno (1.1) $$ Here $P$ is some operator diagonal in momentum space, typically determined by the unperturbed Hamiltonian, and $Q$ is diagonal in coordinate space. The functional integral is taken with respect to some probability measure $d\mu(Q)$ and goes over the matrix elements of $Q$. $[\;\cdot\;]_{x,y}^{-1}$ denotes the $x,y$-entry of the matrix $[P+Q]^{-1}$. Our starting point is always a model in finite volume and with positive lattice spacing in which case the operator $P+Q$ and the functional integral in (1.1) becomes huge- but finite-dimensional. In the end we take the infinite volume limit and, if wanted, the continuum limit. \par Our treatment is based on the following identity which is obtained by repeated application of the Feshbach formula (Lemma 3.2 below). It is proven in Theorem 3.3. Let $B=(B_{kp})_{k,p\in\M}\in \mathbb C^{N\times N}$, $\M$ some index set, $|\M|=N$ and let $$G(k):=\left[ B^{-1}\right]_{kk} \eqno (1.2)$$ Then one has $$G(k)={1\over\ds B_{kk}+\sum_{r=2}^N (-1)^{r+1} \!\!\!\!\!\!\!\!\! \sum_{p_2\cdots p_{r}\in\M\setminus\{k\}\atop p_i\ne p_j} \!\!\!\!\!\!\! B_{kp_2}G_k(p_2)B_{p_2 p_3}\cdots B_{p_{r-1} p_r} G_{kp_2\cdots p_{r-1}}(p_r)B_{p_r k} }\eqno (1.3)$$ where $ G_{k_1\cdots k_j}(p)=\left[ (B_{st})_{s,t\in \M\setminus \{k_1\cdots k_j\}}\right]^{-1}_{pp}$ is the $p,p$ entry of the inverse of the matrix which is obtained from $B$ by deleting the rows and columns labelled by $k_1,\cdots,k_j$. In Section 2 we apply this formula to a matrix of the form $B=$ self adjoint + $i\vep\,{I\!d}$, which, for $\vep\ne 0$, has the property that all submatrices $ (B_{st})_{s,t\in \M\setminus \{k_1\cdots k_j\}}$ are invertible. \par There is also a formula for the off-diagonal inverse matrix elements. It reads $$\left[B^{-1}\right]_{kp}=-G(k) B_{kp} G_k(p)+ \sum_{r=3}^N(-1)^{r+1} \!\!\!\!\!\!\! \!\!\! \sum_{t_3\cdots t_r\in\M\setminus\{k,p\}\atop t_i\ne t_j} \!\!\!\!\!\! G(k) B_{kt_3} G_k(t_3) B_{t_3t_4} \cdots B_{t_rp}G_{kt_3\cdots t_r}(p) \eqno (1.4)$$ These formulae also hold in the case where the matrix $B$ has a block structure $B_{kp}=(B_{k\sigma,p\tau})$ where, say, $\sigma,\tau\in \{\up,\down\}$ are some spin variables. In that case the $B_{kp}$ are small matrices, the $G_{k_1\cdots k_j}(p)$ are matrices of the same size and the 1/$\cdot$ in (1.3) means inversion of matrices, see Theorem 3.3 below. \par The paper is organized as follows. In the next section we demonstrate the method by applying it to the averaged Green function of the Anderson model. The Schwinger-Dyson equation for that model reads $G^{-1}=G_0^{-1}+\Sigma(G_0)$ where $\Sigma(G_0)$ is the sum of all two-legged one-particle irreducible diagrams. Application of (1.3) leads to an integral equation $G^{-1}=G_0^{-1}+I(G)$ where $I(G)$ is the sum of all two-legged graphs without two-legged subgraphs. The latter equation has two advantages. First, $\Sigma$ is the sum of one-particle irreducible diagrams, but these diagrams may very well have two-legged subdiagrams and usually these are the diagrams which produce anomalously large contributions. And second, the propagator for $I(G)$ is the interacting two point function $G$, which, for the Anderson model, is more regular than the free two point function $G_0$ which is the propagator for the diagrams contributing to $\Sigma(G_0)$. More precisely, the series for $I(G)$ can be expected to be asymptotic, that is, its lowest order contributions are a good approximation if the coupling is small, but, usually, the series for $\Sigma(G_0)$ is not asymptotic. \par For the many-electron system and for the $\vp^4$ model repeated application of (1.3,4) amounts to a resummation of two- and four-legged subgraphs. This is discussed in Section 4. In particular, for the many-electron system with attractive delta-interaction we find, as one expects, the formation of a BCS gap, but we are also able to obtain information on the expectation value of the Hubbard-Stratonovich field $\la |\phi_q|^2\ra$ which is related to a certain four point function (see (4.11,12) and (4.3,4)). This function has been investigated by several authors [FMRT,CFS,B]. One expects a singularity at $q=0$. The usual argument, based on a Taylor expansion of the effective potential around its minimum predicts a $1/q^2$ behaviour. Our analysis, given in Section 4.1, results in $\la |\phi_q|^2\ra ={|\Delta|^2\over\lambda}\> \delta(q)+{regular}$ where $\Delta$ is the BCS gap and the regular term is bounded for small $q$. In particular, we find that the existence of a gap and a macroscopic $\la |\phi_0|^2\ra=\beta L^d\,{|\Delta|^2\over\lambda}$ enforce each other ($L^d$ volume, $1/\beta=T$ temperature). The proof of the inversion formula is given in Section 3. \bigskip \bigskip % % % S E C T I O N I I % % \section{Application to the Anderson Model} \bigskip \indent Let coordinate space be a lattice of finite volume with periodic boundary conditions, lattice spacing $1/M$ and volume $[0,L]^d$: $$\ts \Gamma=\left\{ x={1\over M}(n_1,\cdots,n_d)\;|\; 0\le n_i\le ML-1\right\}= \left( {1\over M}\Bbb Z\right)^d/(L\Bbb Z)^d \eqno (2.1)$$ Momentum space is given by $$\ts \M:=\Gamma^\sharp=\left\{k={2\pi\over L}(m_1,\cdots,m_d)\;|\; 0\le m_i\le ML-1\right\} =\left( {2\pi\over L}\Bbb Z\right)^d/(2\pi M \Bbb Z)^d \eqno (2.2)$$ We consider the averaged Green function of the Anderson model given by $$\la G\ra(x,x'):=\int \left[ -\Delta-z+\lambda V \right]_{x,x'}^{-1} dP(V) \eqno (2.3)$$ where the random potential is Gaussian, $$dP(V)= \pro_{x\in\Gamma} \ts e^{-{V_x^2\over 2}} {dV_x\over \sqrt{2\pi}}. \eqno(2.4)$$ Here $z=E+i\vep$ and $\Delta$ is the discrete Laplacian, $$\left[ -\Delta-z+\lambda V \right]_{x,x'}= -M^2\sum_{i=1}^d \left( \delta_{x',x+ e_i/M} +\delta_{x',x- e_i/M}-2\delta_{x',x}\right) -z \,\delta_{x,x'} +\lambda V_x \, \delta_{x,x'} \eqno (2.5)$$ By taking the Fourier transform, one has \setcounter{equation}{5} \label{sec2} \beqn \la G\ra(x,x')&=&{\ts {1\over M^d L^d}} \sum_{k\in\M} e^{ik(x'-x)} \la G\ra (k) \label{2.6} \\ \la G\ra (k)&=& \int_{\Bbb R^{N^d}} \ts \left[ a_k\delta_{k,p}+{\lambda\over \sqrt{N^d}} v_{k-p} \right]^{-1}_{k,k}\;dP(v) \label{2.7} \eeqn where $N^d=(ML)^d=|\Gamma|=|\M|$, $dP(v)$ is given by (2.10) or (2.11) below, depending on whether $N^d$ is even or odd, and $$a_k=4M^2\sum_{i=1}^d \sin^2\left[\ts {k_i\over 2M}\right] -E-i\vep \eqno (2.8)$$ The rigorous control of $\la G\ra (k)$ for small disorder $\lambda$ and energies inside the spectrum of the unperturbed Laplacian, $E\in [0,4M^2]$, in which case $a_k$ has a root if $\vep\to 0$, is still an open problem [AG,K,MPR,P,W]. It is expected that $\lim_{\vep\searrow 0}\lim_{L\to\infty} \la G\ra (k)=1/(a_k-\sigma_k)$ where Im$\sigma=O(\lambda^2)$. The integration variables $v_{q}$ in (2.7) are given by the discrete Fourier transform of the $V_x$. In particular, observe that, if $F$ denotes the unitary matrix of discrete Fourier transform, the variables $$v_{q} \equiv(FV)_{q}={\ts {1\over \sqrt{N^d}}}\sum_{x\in\Gamma} e^{-iqx} V_x={\ts \left({M\over L}\right)^{d\over2} \; {1\over M^d}} \sum_{x\in\Gamma} e^{-iqx} V_x \equiv {\ts \left({M\over L}\right)^{d\over2}} \, \hat V_q \eqno(2.9)$$ would not have a limit if $V_x$ would be deterministic and cutoffs are removed, since the $\hat V_q$ are the quantities which have a limit in that case. But since the $V_x$ are integration variables, we choose a unitary transform to keep the integration measure invariant. Observe also that $v_q$ is complex, $v_q=u_q+iw_q$. Since $V_x$ is real, $u_{-q}=u_q$ and $w_{-q}=-w_q$. In order to transform $dP(V)$ to momentum space, we have to choose a set $\M^+\subset\M$ such that either $q\in \M^+$ or $-q\in \M^+$. If $N$ is odd, the only momentum with $q=-q$ or $w_q=0$ is $q=0$. In that case $dP(V)$ becomes $$ dP(v)=e^{-{u_{0}^2\over2}} {\ts {du_{0}\over \sqrt{2\pi}}} \pro_{q\in \M^+} e^{-(u_q^2+w_q^2)} {\ts {du_q dw_q\over \pi}} \eqno (2.10) $$ For even $N$ we get $$ dP(v) =e^{-{1\over2}(u_{0}^2+u_{q_0}^2)} {\ts {du_{0}du_{q_0}\over {2\pi}}} \pro_{q\in \M^+} e^{-(u_q^2+w_q^2)} {\ts {du_q dw_q\over \pi}} \eqno (2.11) $$ where $q_0={2\pi m\over L}$ is the unique nonzero momentum for which ${2\pi\over L}m=2\pi M (1,\cdots,1)- {2\pi\over L}m$. \vspace{0.6cm} Now we apply the inversion formula (1.3) to the inverse matrix element in $$\la G\ra (k)= \int_{\Bbb R^{N^d}} \ts \left[ a_k\delta_{k,p}+{\lambda\over \sqrt{N^{d}}} v_{k-p} \right]^{-1}_{k,k}\;dP(v) \eqno (2.7) $$ We start with the `two loop approximation', which we define by retaining only the $r=2$ term in the denominator of the right hand side of (1.3), $$G(k)\approx {1\over B_{kk}-\sum_{p\in\M\setminus\{k\}} B_{kp} G_k(p) B_{pk} } \eqno(2.12)$$ Thus, let $$\ts G(k):=\left[ a_k\delta_{k,p}+{\lambda\over \sqrt{N^{d}}} v_{k-p} \right]^{-1}_{k,k}=G(k;v,z) \eqno (2.13)$$ In the infinite volume limit the spacing $2\pi/L$ of the momenta becomes arbitrary small. Hence, in computing an inverse matrix element, it should not matter whether a single column and row labelled by some momentum $t$ is absent or not. In other words, in the infinite volume limit one should have $$ \phantom{ {\;\;\;{\rm for}\;\;L\to\infty} } G_t(p)=G(p)\;\;\;{\rm for}\;\;L\to\infty \eqno (2.14) $$ and similarly $G_{t_1\cdots t_j}(p)=G(p)$ as long as $j$ is independent of the volume. We remark however that if the matrix has a block structure, say $B=(B_{k\sigma,p\tau})$ with $\sigma,\tau\in\{\up,\down\}$ some spin variables, this structure has to be respected. That is, for a given momentum $k$ all rows and columns labelled by $k\!\up,\;k\!\down$ have to be eliminated, since otherwise (2.14) may not be true. \par Thus the two loop approximation gives $$ G(k)={1\over a_k+{\lambda\over \sqrt{N^d}} v_0 -{\lambda^2\over N^d}\sum_{p\ne k} v_{k-p}\,G(p)\, v_{p-k} } \eqno (2.15)$$ For large $L$, we can disregard the ${\lambda\over \sqrt{N^d}} v_0$ term. Introducing $\sigma_k=\sigma_k(v,z)$ according to $$G(k)=:{1\over a_k-\sigma_k}\;, \eqno (2.16)$$ we get $$\sigma_k={\ts {\lambda^2\over N^d}}\sum_{p\ne k} {|v_{k-p}|^2 \over a_p-\sigma_p } \approx {\ts {\lambda^2\over N^d}}\sum_{p} {|v_{k-p}|^2 \over a_p-\sigma_p } \eqno (2.17)$$ and arrive at $$\la G\ra (k)=\int {1\over \ds a_k-{\ts {\lambda^2\over N^d}} \sum_{p}^{\phantom{I}} {|v_{k-p}|^2\over a_p- {\ts {\lambda^2\over N^d}} \sum_{t} {|v_{p-t}|^2\over a_{t}-{\lambda^2\over N^d}\Sigma\cdots} } } \;dP(v) \eqno (2.18)$$ Now consider the infinite volume limit $L\to\infty$ or $N=ML\to\infty$. By the central limit theorem of probability ${1\over \sqrt{N^d}}\sum_q\left(|v_q|^2-\la |v_q|^2\ra\right)$ is, as a sum of independent random variables, normal distributed. Note that only a prefactor of $1/\sqrt{N^d}$ is required for that property. In particular, if $F$ is some bounded function independent of $N$, sums which come with a prefactor of $1/N^d$ like ${1\over N^d}\sum_q c_q |v_q|^2$ can be substituted by their expectation value, $$\lim_{N\to\infty}\int F\Bigl({\ts{1\over N^d}} \sum_kc_k |v_k|^2\Bigr)dP(v)=F\Bigl(\lim_{N\to\infty}{\ts{1\over N^d}} \sum_kc_k \la |v_k|^2\ra\Bigr) \eqno (2.19)$$ Therefore, in the two loop approximation, one obtains in the infinite volume limit $$\la G\ra (k)= {1\over \ds a_k-{\ts {\lambda^2\over N^d}} \sum_{p}^{\phantom{I}} {\la |v_{k-p}|^2\ra \over a_p- {\ts {\lambda^2\over N^d}} \sum_{t} {\la |v_{p-t}|^2\ra \over a_{t} -{\lambda^2\over N^d}\Sigma \cdots} } } \;=:\; {1\over a_k-\la \sigma_k \ra } \eqno (2.20)$$ where the quantity $\la \sigma_k \ra$ satisfies the integral equation $$\la \sigma_k \ra= {\ts {\lambda^2\over N^d}}\sum_{p}{ {\la |v_{k-p}|^2 \ra \over a_p-\la \sigma_p\ra }\; } \buildrel L\to\infty\over \to \; {\ts {\lambda^2\over M^d}}\int_{[0,2\pi M]^d} { {d^dp\over (2\pi)^d}} \, {\la |v_{k-p}|^2\ra \over a_p-\la\sigma_p\ra } \eqno (2.21)$$ For a Gaussian distribution $\la |v_q|^2\ra=1$ for all $q$ such that $\la \sigma_k\ra =\la\sigma\ra$ becomes independent of $k$. Thus we end up with $$\la G\ra(k)={1\over 4M^2\sum_{i=1}^d\sin^2\left[ {k_i\over 2M} \right] -E-i\vep-\la\sigma\ra} \eqno (2.22)$$ where $\la\sigma\ra$ is a solution of \setcounter{equation}{22} \beqn \la\sigma\ra&=& {\ts {\lambda^2\over M^d}}\int_{[0,2\pi M]^d} { {d^dp\over (2\pi)^d}} \, {1\over 4M^2\sum_{i=1}^d \sin^2\left[{p_i\over 2M}\right]-z-\la\sigma\ra}\nonumber \\ &=&\lambda^2\int_{[0,2\pi]^d} { {d^dp\over (2\pi)^d}} \, {1\over 4M^2\sum_{i=1}^d \sin^2\left[{p_i\over 2}\right]-z-\la\sigma\ra}\>. \eeqn This equation is of course well known and one deduces from it that it generates a small imaginary part Im$\,\sigma=O(\lambda^2)$ if the energy $E$ is within the spectrum of $-\Delta$. \bigskip We now add the higher loop terms (the terms for $r>2$ in the denominator of (1.3)) to our discussion and give an interpretation in terms of Feynman diagrams. To make the volume factors more explicit, asume that the lattice spacing in coordinate space is $1/M=1$ such that $N=L$. \par For the Anderson model, Feynman graphs may be obtained by brutally expanding \beqn \lefteqn{\int \bigl[a_k\delta_{k,p}+{\ts{\lambda\over \sqrt L^{d}} v_{k-p}} \bigr]^{-1}_{k,k}\;dP =\sum_{r=0}^\infty \int \left( C[VC]^r\right)_{kk} \,dP} \\ && =\sum_{r=0}^\infty {\ts {(-\lambda)^r\over \sqrt{L^d}^r}} \sum_{p_2\cdots p_{r}} {\ts {1\over a_ka_{p_2}\cdots a_{p_{r}} a_k}} \int v_{k-p_2} v_{p_2-p_3}\cdots v_{p_{r}-k} \,dP \nonumber \eeqn For a given $r$, we may represent this by ($c_k:=1/a_k$) \begin{figure}[thb] \centerline{\epsfbox{bild1.eps}} \caption{A string of particle lines with unpaired squiggles (dashed lines)} \label{Figure 1} \end{figure} \noindent The integral over the $v$ gives a sum of $(r-1)!!$ terms where each term is a product of $r/2$ Kroenecker-delta's, the terms for odd $r$ vanish. If this is substituted in (2.24), the number of independent momentum sums is cut down to $r/2$ and each of the $(r-1)!!$ terms may be represented by a diagram \begin{figure}[thb] \centerline{\epsfbox{bild2.eps}} \caption{Lowest order diagrams contributing to (2.24) for $r=2$ and $r=4$ } \label{Figure 2} \end{figure} \noindent where, say, the value of the third diagram is given by ${\lambda^4\over L^{2d}}\sum_{p_1,p_2} {1\over a_k a_{k+p_1} a_{k+p_1+p_2} a_{k+p_2}a_k}$. For short: $$\la G\ra (k)={\rm sum\;of\; all\; two\; legged\; diagrams\,.} \eqno (2.25)$$ Since the value of a diagram depends on its subgraph structure, one distinguishes, in the easiest case, two types of diagrams. Diagrams with or without two-legged subdiagrams. Those diagrams with two-legged subgraphs usually produce anomalously large contributions. They are devided further into the one-particle irreducible ones and the reducible ones. Thereby a diagram is called one-particle reducible if it can be made disconnected by cutting one solid or `particle' line (no squiggle or dashed line), see also figure 3. \begin{figure}[thb] \centerline{\epsfbox{bild3.eps}} \caption{Reducible and irreducible diagrams} \label{Figure 3} \end{figure} \noindent The reason for introducing reducible and irreducible diagrams is that the reducible ones can be easily resummed by writing down the Schwinger-Dyson equation which states that if the self energy $\Sigma_k$ is defined through $$\la G\ra (k)={1\over a_k-\Sigma_k(G_0)} \eqno (2.26)$$ then $\Sigma_k(G_0)$ is the sum of all amputated (no $1/a_k$'s at the ends) one particle irreducible diagrams. Here we wrote $\Sigma_k(G_0)$ to indicate that the factors (`propagators') assigned to the solid lines of the diagrams contributing to $\Sigma_k$ are given by the free two point function $G_0(p)={1\over a_p}$. However, the diagrams contributing to $\Sigma_k(G_0)$ still contain anomalously large contributions, namely irreducible graphs which contain two-legged subgraphs like diagram (c) in figure 3. In the following we show, using the inversion formula (1.3) including all higher loop terms, that all graphs with two-legged subgraphs can be eliminated or resummed by writing down the following integral equation for $\la G\ra$: $$ \la G\ra (k)= {1\over a_k-\sigma_k(\la G\ra)} \eqno (2.27) $$ where \begin{quote} $\sigma_k(\la G\ra)$ is the sum of all amputated two-legged diagrams which do not contain any two-legged subdiagrams, but now with propagators $\la G\ra(k)={1\over a_k-\sigma_k}$ instead of $G_0={1\over a_k}$ \end{quote} \noindent which may be formalized as in (2.35) below. The advantage of this formula is that the series for $\sigma_k(\la G\ra)$ can be expected to be asymptotic, that is, its lowest order contributions are a good approximation if the coupling is small, but, usually, the series for $\Sigma_k(G_0)$ is not asymptotic. Thus, in order to rigorously controll $\la G\ra (k)$, one has to define a suitable space of propagators, to estimate the sum of all amputated two-legged graphs without two-legged subgraphs on that space and then finally to show that the equation (2.27) has a solution on this space. We intend to address this problem in another paper. \bigskip We now show (2.27) for the Anderson model. For fixed $v$ one has $$G(k,v)={1\over a_k-\sigma_k(v)} \eqno (2.28)$$ where $$\sigma_k(v)=\sum_{r=2}^{L^d}(-1)^{r}\!\!\!\!\!\! \sum_{p_2\cdots p_{r} \atop p_i\ne p_j,\> p_i\ne k} \!\! \ts {\lambda^r\over \sqrt{ L^d}^r} \, G_k(p_2)\cdots G_{kp_2\cdots p_{r-1}}(p_r)\, v_{k-p_2} \cdots v_{p_r-k} \eqno (2.29)$$ We cutoff the $r$-sum in (2.29) at some arbitrary but fixed order $\ell p_i\ne k} \!\!\!\! G(p_2)\cdots G(p_r)\, v_{k-p_2} \cdots v_{p_r-k}\eqno (2.31)$$ Consider first two strings $s_k^{r_1}$, $s_k^{r_2}$ where $$ s_k^r(v)={\ts {\lambda^r\over \sqrt{ L^d}^r}} \!\!\!\! \sum_{p_2\cdots p_{r} \atop p_i\ne p_j,\> p_i\ne k} \!\!\!\! c^r_{kp_2\cdots p_r}\, v_{k-p_2} \cdots v_{p_r-k}\eqno (2.32)$$ and the $c^r_{kp_2\cdots p_r}$ are some numbers. Then in the infinite volume limit $$\la s_k^{r_1} s_k^{r_2} \ra= \la s_k^{r_1}\ra\, \la s_k^{r_2} \ra \eqno (2.33)$$ because all pairings which connect the two strings have an extra volume factor $1/L^d$. Namely, if the two strings are disconnected, there are $(r_1+r_2)/2$ loops and a volume factor of $1/\sqrt{L^d}^{(r_1+r_2)}$ giving $(r_1+r_2)/2$ Riemannian sums. If the two strings are connected, there are only $(r_1+r_2-2)/2$ loops leaving an extra factor of $1/L^d$. By the same argument one has in the infinite volume limit $$\la (s_k^{r_1})^{n_1}\cdots (s_k^{r_m})^{n_m}\ra = \la s_k^{r_1} \ra^{n_1}\cdots \la s_k^{r_m}\ra^{n_m}\eqno (2.34)$$ which results in $$\la G\ra(k)= {1\over \ds a_k -\sum_{r=2}^\ell {\ts {(-\lambda)^r\over \sqrt{ L^d}^{\,r}}}\!\!\!\! \sum_{p_2\cdots p_{r} \atop p_i\ne p_j,\> p_i\ne k} \!\!\!\! \la G\ra(p_2)\cdots \la G\ra (p_r)\, \la v_{k-p_2} \cdots v_{p_r-k} \ra} \eqno (2.35) $$ The condition $p_2,\cdots,p_r\ne k$ and $p_i\ne p_j$ means exactly that two-legged subgraphs are forbidden. Namely, for a two-legged subdiagram as in (c) in figure 3, the incomming and outgoing momenta $p$, $p'$ (to which are assigned propagators $\la G\ra(p)$, $\la G\ra(p')$) must be equal which is forbidden by the condidtion $p_i\ne p_j$ in (2.35). However, we cannot take the limit $\ell\to\infty$ in (2.35) since the series in the denominator of (2.35) is only an asymptotic one. To see this a bit more clearly suppose for the moment that there were no restrictions on the momentum sums. Then, if $V=({\lambda\over\sqrt{L^d}}v_{k-p})_{k,p}$ and $\la G\ra=(\la G\ra (k)\,\delta_{k,p})_{k,p}$, $${\ts {\lambda^r\over \sqrt{ L^d}^r}} \sum_{p_2\cdots p_r}\la G\ra(p_2)\cdots \la G\ra(p_r)\, \la v_{k-p_2} \cdots v_{p_r-k} \ra =\left\la (V[\la G\ra V]^{r-1})_{kk} \right\ra \eqno (2.36)$$ and for $\ell \to \infty$ we would get $$\la G\ra (k)={1\over a_k-\la (V {\la G\ra V\over Id +\la G\ra V} )_{kk} \ra } = {1\over a_k-\la (V {1\over \la G\ra^{-1} + V} V )_{kk} \ra } \eqno (2.37)$$ That is, the factorials produced by the number of diagrams in the denominator of (2.35) are basically the same as those in the expansion $$ \int_{\Bbb R}{\ts {x^2\over z+\lambda x}}\, e^{-{x^2\over 2}} {\ts {dx\over\sqrt{2\pi}}}\> =\>\sum_{r=0}^\ell \ts {\lambda^{2r}\over z^{2r+1}}\,(2r+1)!!\> +R_{\ell+1}(\lambda)\, \eqno (2.38)$$ where the remainder satisfies the bound $|R_{\ell+1}(\lambda)|\le \ell!\,const_{\! z}^\ell\, \lambda^\ell $. \bigskip We close this section with two further remarks. So far the computations were done in momentum space. One may wonder what one gets if the inversion formula (1.3) is applied to $[-\Delta+z+\lambda V]^{-1}$ in coordinate space. Whereas a geometric series expansion of $[-\Delta+z+\lambda V]^{-1}$ gives a representation in terms of the simple random walk, application of (1.3) results in a representation in terms of the self avoiding walk: $$[-\Delta+z+\lambda V]_{0,x}^{-1}=\sum_{\gamma:0\to x\atop \gamma\;{\rm self\;avoiding}} {\det\left[ (-\Delta+z+\lambda V)_{y,y'\in\Gamma\setminus\gamma} \right] \over\det\left[ (-\Delta+z+\lambda V)_{y,y'\in\Gamma} \right] } \eqno (2.39)$$ where $\Gamma$ is the lattice in coordinate space. Namely, if $|x|>1$, the inversion formula (1.4) for the off-diagonal elements gives \beq \lefteqn{[-\Delta+\lambda V]_{0,x}^{-1}=\sum_{r=3}^{L^d}(-1)^{r+1} \!\!\!\!\!\!\!\!\! \sum_{x_3\cdots x_r\in\Gamma\setminus\{0,x\}\atop x_i\ne x_j} \!\!\!\!\!\!\!\! G(0)G_0(x_3)\cdots G_{0x_3\cdots x_r}(x) \,(-\Delta)_{0 x_3} \cdots (-\Delta)_{x_r x} } \\ &&=\sum_{r=3}^{L^d} \sum_{x_2=0,x_3,\cdots ,x_r,x_{r+1}=x\in\Gamma\atop |x_i-x_{i+1}|=1\; \forall i=2\cdots r} \!\!\!\! {\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma\setminus\{0\}} \right] \over\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma} \right] } \cdots {\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma\setminus\{0,x_3\cdots x_r,x\}} \right] \over\det\left[ (-\Delta+\lambda V)_{y,y'\in\Gamma\setminus\{0,x_3\cdots x_r\}} \right] } \eeq which coincides with (2.39). Finally we remark that, while the argument following (2.32) leads to a factorization property for on-diagonal elements in momentum space, $\la G(k)\,G(p)\ra=\la G(k)\ra\,\la G(p)\ra$, there is no such property for products of off-diagonal elements which appear in a quantity like $$\Lambda(q)= {\ts {1\over L^d}}\sum_{k,p}\ts \left\la \left[ a_k\delta_{k,p}+{\lambda\over \sqrt L^{\,d}} v_{k-p} \right]^{-1}_{k,p} \left[ \bar a_k\delta_{k,p}+{\lambda\over \sqrt L^{\,d}} \bar v_{k-p} \right]^{-1}_{k-q,p-q} \right\ra \eqno (2.40) $$ which is the Fourier transform of $\bigl\la \left| [-\Delta+z+\lambda V]^{-1}_{x,y} \right|^2\bigr\ra$. (Each off-diagonal inverse matrix element is proportional to $1/\sqrt{L^d}$, therefore the prefactor of $1/L^d$ in (2.40) is correct.) \bigskip \bigskip % % % S E C T I O N I I I % % \section{Proof of the Inversion Formula} %\setcounter{equation}{5} \label{sec3} \bigskip \noindent{\bf Lemma 3.1:} Let $B\in\Bbb C^{k\times n}$, $C\in\Bbb C^{n\times k}$ and let $Id_k$ denote the identity in $\Bbb C^{k\times k}$. Then: \bitem \item[{\bf (i)}] $Id_k-BC\;\;{\rm invertible}\;\;\; \Leftrightarrow\;\;\;Id_n-CB\;\;{\rm invertible}$. \item[{\bf (ii)}] If the left or the right hand side of (i) fullfilled, then $\ts C\,{1\over Id_k-BC} =\ts {1\over Id_n-CB}\,C $. \eitem \bigskip \noindent{\bf Proof:} Let $$B=\left(\bmatr{ccc} - & \vec b_1 & - \\ & \vdots & \\ - & \vec b_k & - \ematr\right),\;\;\;\; C=\left(\bmatr{ccc} |& &|\\ \vec c_1&\cdots &\vec c_k\\ |& &| \ematr\right) $$ where the $\vec b_j$ are $n$-component row vectors and the $\vec c_j$ are $n$-component column vectors. Let $\vec x\in {\rm Kern}(Id-CB)$. Then $\vec x=CB \vec x=\sum_j \lambda_j \vec c_j$ if we define $\lambda_j:= (\vec b_j,\vec x)$. Let $\vec \lambda=(\lambda_j)_{1\le j\le k}$. Then $[(Id-BC)\vec\lambda]_i =\lambda_i-\sum_j(\vec b_i,\vec c_j)\lambda_j =(\vec b_i,\vec x)-\sum_j(\vec b_i,\vec c_j)\lambda_j=0$ since $\vec x=\sum_j \lambda_j \vec c_j$, thus $\vec \lambda \in {\rm Kern}(Id-BC)$. On the other hand, if some $\vec\lambda\in {\rm Kern}(Id-BC)$, then $\vec x:=\sum_j\lambda_j \vec c_j\in {\rm Kern}(Id-CB)$ which proves (i). Part (ii) then follows from $ C\ts={1\over Id_n-CB}\,(Id_n-CB)C ={1\over Id_n-CB}\,C (Id_k-BC)$ $\blacksquare$ \bigskip \bigskip The inversion formula (1.3,4) is obtained by iterative application of the next lemma, which states the Feshbach formula for finite dimensional matrices. For a more general version one may look in [BFS], Theorem 2.1. \bigskip \bigskip \noindent{\bf Lemma 3.2:} Let $\ts h=\left({A\atop C}\;{B\atop D}\right)\in\Bbb C^{n\times n}$ where $A\in \Bbb C^{k\times k}$, $D\in \Bbb C^{(n-k)\times (n-k)}$ are invertible and $B\in \Bbb C^{k\times (n-k)}$, $C\in\Bbb C^{(n-k)\times k}$. Then $$ h\;\;{\rm invertible}\;\;\;\; \Leftrightarrow\;\;\;\;A-BD^{-1}C\;\;{\rm invertible}\;\;\;\; \Leftrightarrow\;\;\;\; D-CA^{-1}B\;\;{\rm invertible} \eqno (3.1)$$ and if one of the conditions in (3.1) is fullfilled, one has $h^{-1} = \left({E\atop G}\;{F\atop H}\right)$ where \setcounter{equation}{1} \beqn \ts E={1\over A-BD^{-1}C}\>,&\hskip 0.7cm&\ts H={1\over D-CA^{-1}B}\>, \\ F=-EBD^{-1}=-A^{-1}BH\>,&\hskip 0.7cm& G=-HCA^{-1}=-D^{-1}CE\>. \eeqn \bigskip \noindent{\bf Proof:} We have, using Lemma 3.1 in the second line, \beq A-BD^{-1}C\;\;{\rm inv.}&\Leftrightarrow & Id_k-A^{-1}BD^{-1}C\;\;{\rm inv.} \\ & \Leftrightarrow & Id_{n-k}-D^{-1}CA^{-1}B\;\;{\rm inv.} \\ & \Leftrightarrow & D-CA^{-1}B\;\;{\rm inv.} \eeq Furthermore, again by Lemma 3.1, $$\ts D^{-1}C\,{1\over Id-A^{-1}BD^{-1}C}= {1\over Id-D^{-1}CA^{-1}B}\,D^{-1}C= {1\over D-CA^{-1}B}\,C=HC$$ and $$\ts A^{-1}B\,{1\over Id-D^{-1}CA^{-1}B}= {1\over Id-A^{-1}BD^{-1}C}\,A^{-1}B= {1\over A-BD^{-1}C}\,B=EB$$ which proves the last equalities in (3.3), $HCA^{-1}=D^{-1}CE$ and $EBD^{-1}= A^{-1}BH $. Using these equations and the definition of $E,F,G$ and $H$ one computes $$\ts \left({A\atop C}\;{B\atop D}\right) \left({E\atop G}\;{F\atop H}\right) = \left({E\atop G}\;{F\atop H}\right) \left({A\atop C}\;{B\atop D}\right) =\left({Id\atop 0}\;{0\atop Id}\right)$$ It remains to show that the invertibility of $h$ implies the invertibility of $A-BD^{-1}C$. To this end let $P= \left({Id\atop \phantom{C}}\;{\phantom{B}\atop 0}\right)$, $\bar P= \left({0\atop \phantom{C}}\;{\phantom{B}\atop Id}\right)$ such that $A-BD^{-1}C=PhP-Ph\bar P(\bar P h\bar P)^{-1} \bar P h P$. Then \beq (A-BD^{-1}C)Ph^{-1}P&=&PhPh^{-1}P- Ph\bar P(\bar P h\bar P)^{-1} \bar P h Ph^{-1}P \\ &=&Ph(1-\bar P)h^{-1}P- Ph\bar P(\bar P h\bar P)^{-1} \bar P h(1- \bar P)h^{-1}P \\ &=&P-Ph\bar P h^{-1} P +Ph \bar P h^{-1} P=P \eeq and similarly $Ph^{-1}P(A-BD^{-1}C)=P$ which proves the invertibility of $A-BD^{-1}C$ $\blacksquare$ \bigskip \bigskip \noindent{\bf Theorem 3.3:} Let $B\in \Bbb C^{nN\times nN}$ be given by $B=(B_{kp})_{k,p\in\M}$, $\M$ some index set, $|\M|=N$, and $B_{kp}= (B_{k\sigma,p\tau})_{\sigma,\tau\in I}\in \Bbb C^{n\times n}$ where $I$ is another index set, $|I|=n$. Suppose that $B$ and, for any $\N\subset \M$, the submatrix $(B_{kp})_{k,p\in\N}$ is invertible. For $k\in \M$ let $$G(k):=\left[B^{-1}\right]_{kk}\in \Bbb C^{n\times n}\eqno (3.4)$$ and, if $\N\subset\M$, $k\notin \N$, $$G_{\N}(k):= \left[\{(B_{st})_{s,t\in\M\setminus\N}\}^{-1}\right]_{kk} \in \Bbb C^{n\times n} \eqno (3.5)$$ Then one has \bitem \item[\bf (i)] The on-diagonal block matrices of $B^{-1}$ are given by $$G(k)={1\over\ds B_{kk}-\sum_{r=2}^N (-1)^{r} \!\!\!\!\!\!\!\!\!\!\! \sum_{p_2\cdots p_{r}\in\M\setminus\{k\}\atop p_i\ne p_j} \!\!\!\!\!\!\! \!\! B_{kp_2}G_k(p_2)B_{p_2 p_3}\cdots B_{p_{r-1} p_r} G_{kp_2\cdots p_{r-1}}(p_r)B_{p_r k} } \eqno (3.6)$$ where $1/\;\cdot\;$ is inversion of $n\times n$ matrices. \item[\bf (ii)] Let $k,p\in\M$, $k\ne p$. Then the off-diagonal block matrices of $B^{-1}$ can be expressed in terms of the $G_{\N}(s)$ and the $B_{st}$, $$[B^{-1}]_{kp}=-G(k) B_{kp} G_k(p)- \sum_{r=3}^N(-1)^{r} \!\!\!\!\!\!\! \!\!\! \sum_{t_3\cdots t_r\in\M\setminus\{k,p\}\atop t_i\ne t_j} \!\!\!\!\!\! G(k) B_{kt_3} G_k(t_3) B_{t_3t_4} \cdots B_{t_rp}G_{kt_3\cdots t_r}(p) \eqno (3.7) $$ \eitem \bigskip \noindent{\bf Proof:} Let $k$ be fixed and let $p,p'\in\M\setminus\{k\}$ below label columns and rows. By Lemma 3.2 we have $$ \left(\bmatr{cccc} G(k) & && \\ & && \\ &&*& \\ &&& \ematr\right)\;=\;\left( \bmatr{cccc} B_{kk} & \;-&B_{kp}&-\; \\ | & && \cr B_{p'k}&&B_{p'p}& \\ |&&& \ematr\right)^{-1} =\;\; \left(\bmatr{cccc} E &\;- &F&-\; \\ |& && \\ G&&H& \\ |&&& \ematr\right) $$ where \setcounter{equation}{7} \beqn G(k)=E& =&{1\over\ds B_{kk}-\sum_{p,p'\ne k}B_{kp}\left[\{(B_{p' p})_{ p',p\in\M\setminus\{k\}}\}^{-1}\right]_{pp'} B_{p'k} } \\ &=&{1\over \ds B_{kk}-\sum_{p\ne k}B_{kp} G_k(p) B_{pk}- \sum_{p,p'\ne k\atop p\ne p'} B_{kp}\left[\{ (B_{p' p})_{ p',p\in\M\setminus\{k\}}\}^{-1}\right]_{pp'} B_{p'k} }\;\; \nonumber \eeqn and \beqn F_{kp}=\left[ B^{-1}\right]_{kp}&=& -G(k)\sum_{t\ne k} B_{kt} \left[\{ (B_{p' p})_{p',p\in\M\setminus\{k\}}\}^{-1}\right]_{tp} \\ &=&-G(k)B_{kp}G_k(p)-G(k)\sum_{t\ne k,p} B_{kt} \left[\{ (B_{p' p})_{p',p\in\M\setminus\{k\}}\}^{-1}\right]_{tp}\nonumber \eeqn Apply Lemma 3.2 now to the matrix $\{ (B_{p' p})_{p',p\in\M\setminus\{k\}}\}^{-1}$ and proceed by induction to obtain after $\ell$ steps \beqn G(k)&=&{1\over\ds B_{kk}-\sum_{r=2}^\ell (-1)^{r} \!\!\!\!\!\!\!\!\!\!\! \sum_{p_2\cdots p_{r}\in\M\setminus\{k\}\atop p_i\ne p_j} \!\!\!\!\!\!\! \!\! B_{kp_2}G_k(p_2)B_{p_2 p_3}\cdots B_{p_{r-1} p_r} G_{kp_2\cdots p_{r-1}}(p_r)B_{p_r k} -R_{\ell+1} } \\ F_{kp}&=&-G(k) B_{kp} G_k(p)- \sum_{r=3}^\ell (-1)^{r} \!\!\!\!\!\!\! \!\!\! \sum_{t_3\cdots t_r\in\M\setminus\{k,p\}\atop t_i\ne t_j} \!\!\!\!\!\! G(k) B_{kt_3} G_k(t_3) B_{t_3t_4} \cdots B_{t_rp}G_{kt_3\cdots t_r}(p) -\tilde R_{\ell+1} \nonumber \\ && \eeqn where \beqn R_{\ell+1}&=&(-1)^\ell \!\!\!\!\!\!\!\!\!\!\! \sum_{p_2\cdots p_{\ell+1}\in\M\setminus\{k\}\atop p_i\ne p_j} \!\!\!\!\!\!\! \!\! B_{kp_2}G_k(p_2)\cdots B_{p_{\ell-1} p_\ell} \left[\{ (B_{p' p})_{ p',p\in\M\setminus\{kp_2\cdots p_\ell\}}\}^{-1}\right]_{p_\ell p_{\ell+1}} B_{p_{\ell+1} k}\nonumber\\ &&\\ &&\nonumber\\ \tilde R_{\ell+1}&=& (-1)^\ell \!\!\!\!\!\!\! \!\!\! \sum_{t_3\cdots t_{\ell+1}\in\M\setminus\{k,p\}\atop t_i\ne t_j} \!\!\!\!\!\! G(k) B_{kt_3} \cdots G_{kt_3\cdots t_{\ell-1}}(t_\ell)B_{t_\ell t_{\ell+1}} \left[\{ (B_{p' p})_{p',p\in\M\setminus\{k t_3\cdots t_\ell\}}\}^{-1} \right]_{t_{\ell+1} p} \nonumber\\ && \eeqn Since $R_{N+1}=\tilde R_{N+1}=0$ the theorem follows $\blacksquare$ \goodbreak % % % S E C T I O N I V % % \bigskip \bigskip \bigskip \section{Application to the Many-Electron System and to the $\varphi^4$-Model} \bigskip \bigskip \subsection{\bf The Many-Electron System} \bigskip \setcounter{equation}{4} \label{sec4} We consider the many-electron system in the grand canonical ensemble in finite volume $[0,L]^d$ and at some small but positive temperature $T=1/\beta>0$ with attractive delta-interaction given by the Hamiltonian $$H=H_0-\lambda H_{\rm int}={\ts {1\over L^d}}\sum_{\bk\sigma}({\ts {\bk^2\over 2m}}-\mu)a_{\bk\sigma}^+ a_{\bk\sigma}-{\ts {\lambda\over L^{3d}}} \sum_{\bk\p\q} a_{\bk\up}^+ a_{\q-\bk\down}^+ a_{\q-\p\down} a_{\p\up} \eqno (4.1)$$ Our normalization conventions concerning the volume factors are such that the canonical anticommutation relations read $\{a_{\bk\sigma}, a_{\p\tau}^+\}=L^d\,\delta_{\bk,\p}\delta_{\sigma,\tau}$. The momentum sums range over some subset of $\left({2\pi\over L}\Bbb Z\right)^d$, say ${\cal M}=\bigl\{\bk\in \left({2\pi\over L}\Bbb Z\right)^d\,\bigr|\> |e_\bk|\le 1\bigr\}$, $e_\bk=\bk^2/2m-\mu$, and $\q\in\{\bk-\p\,|\bk,\p\in\M\}$. We are interested in the momentum distribution $$\la a_{\bk\sigma}^+a_{\bk\sigma}\ra= Tr[e^{-\beta H}a_{\bk\sigma}^+ a_{\bk\sigma}]\bigr/ Tr\,e^{-\beta H} \eqno (4.2)$$ and in the expectation value of the energy $$\la H_{\rm int}\ra=\sum_{\q} \Lambda(\q) \eqno (4.3)$$ where $$\Lambda(\q)={\ts {\lambda\over L^{3d}}}\sum_{\bk,\p} Tr[e^{-\beta H} a_{\bk\up}^+ a_{\q-\bk\down}^+ a_{\q-\p\down} a_{\p\up}] \bigr/ Tr\,e^{-\beta H} \eqno (4.4)$$ By writing down the perturbation series for the partition function, rewriting it as a Grassmann integral \beqn {\ts {Tr\,e^{-\beta( H_0-\lambda H_{\rm int})} \over Tr\, e^{-\beta H_0}}}&=&\int e^{ {\lambda\over (\beta L^d)^3}\sum_{kpq}\bar\psi_{k\up}\bar\psi_{q-k\down} \psi_{q-p\down}\psi_{p\up} } d\mu_C(\psi,\bar\psi)\\ d\mu_C&=&\pro_{k\sigma}{\ts {\beta L^d\over ik_0-e_\bk}}\> e^{-{1\over \beta L^d}\sum_{k\sigma}(ik_0-e_\bk)\bar\psi_{k\sigma}\psi_{k\sigma}} \pro_{k\sigma} d\psi_{k\sigma} d\bar\psi_{k\sigma} \>, \nonumber \eeqn performing a Hubbard-Stratonovich transformation ($\phi_q=u_q+iv_q$, $d\phi_q d\bar\phi_q:=du_q dv_q$) $$ e^{-\sum_q a_q b_q}= \int e^{i\sum_q(a_q\phi_q+b_q\bar\phi_q)} e^{-\sum_q|\phi_q|^2}\pro_q\ts {d\phi_q d\bar\phi_q\over \pi} \eqno (4.6)$$ with $$ a_q={\ts {\lambda^{1\over 2}\over (\beta L^d)^{3\over2}}}\sum_k \bar\psi_{k\up} \bar\psi_{q-k\down},\;\;\; b_q={\ts {\lambda^{1\over 2}\over (\beta L^d)^{3\over2}}} \sum_p \psi_{p\up} \psi_{q-p\down} \eqno (4.7)$$ and then integrating out the $\psi,\bar\psi$ variables, one arrives at the following representation which is the starting point for our analysis (for more details, see [FKT] or [L1]): $$ {\ts {1\over L^d}}\la a_{\bk\sigma}^+a_{\bk\sigma}\ra= {\ts {1\over \beta L^d}\>{1\over \beta}}\!\!\!\!\sum_{k_0\in {\pi\over\beta} (2\Bbb Z+1)} \!\!\la \psi_{\bk k_0\sigma}^+\psi_{\bk k_0\sigma}\ra \eqno (4.8)$$ where, abbreviating $k=(\bk,k_0)$, $\kappa=\beta L^d$, $a_k=ik_0-e_\bk$, $g=\lambda^{1\over2}$, $${\ts {1\over\kappa}}\,\la \bar\psi_{t\sigma}\psi_{t\sigma}\ra=\int \left[ \bmatr{cc} a_{k}\delta_{k,p} & {ig\over \sqrt{\kappa}}\, \bar\phi_{p-k} \\ {ig\over \sqrt{\kappa}}\, \phi_{k-p} & a_{-k}\delta_{k,p} \ematr \right]_{t\sigma,t\sigma}^{-1} \!\!\! dP(\phi) \eqno(4.9)$$ and $dP(\phi)$ is the probability measure $$dP(\phi)={\ts {1\over Z}} \>\det\left[ \bmatr{cc} a_{k}\delta_{k,p} & {ig\over \sqrt{\kappa}}\, \bar\phi_{p-k} \\ {ig\over \sqrt{\kappa}}\, \phi_{k-p} & a_{-k}\delta_{k,p} \ematr \right] e^{-\sum_q|\phi_q|^2} \pro_q {\ts {d\phi_q d\bar\phi_q}} \eqno (4.10)$$ Furthermore $$\Lambda(\q)={\ts {1\over\beta}}\sum_{q_0\in {2\pi\over\beta}\Bbb Z} \Lambda(\q,q_0) \eqno (4.11)$$ where \setcounter{equation}{11} \beqn \Lambda(q)&=&{\ts {\lambda\over (\beta L^d)^3}}\sum_{k,p} \la \bar\psi_{k\up}\bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up}\ra \nonumber \\ &=& \la |\phi_q|^2\ra -1 \eeqn and the expectation in the last line is integration with respect to $dP(\phi)$. The expectation on the $\psi$ variables $\la \bar\psi_{k\sigma}\psi_{k\sigma}\ra={\ts {1\over{\cal Z}}} \int \bar\psi_{k\sigma}\psi_{k\sigma}\, e^{ {\lambda\over \kappa^3}\sum_{k,p,q}\bar\psi_{k\up} \bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up} } d\mu_C$ is Grassmann integration, but these representations are not used in the following. The matrix and the integral in (4.9) become finite dimensional if we choose some cutoff on the $k_0$ variables which is removed in the end. The set $\M$ for the spatial momenta is already finite since we have chosen a fixed UV-cuttoff $|e_\bk|=|\bk^2/2m-\mu|\le 1$ which will not be removed in the end since we are interested in the infrared properties at $\bk^2/2m=\mu$. Our goal is to apply the inversion formula to the inverse matrix element in (4.9). Instead of writing the matrix in terms of four $N\times N$ blocks $(a_k\delta_{k,p})_{k,p}$, $(\bar\phi_{p-k})_{k,p}$, $(\phi_{k-p})_{k,p}$ and $(a_{-k}\delta_{k,p})_{k,p}$ where $N$ is the number of the $d+1$-dimensional momenta $k,p$, we interchange rows and columns to rewrite it in terms of $N$ blocks of size $2\times 2$ (the matrix $U$ in the next line interchanges the rows and columns): $$U\left[ \bmatr{cc} a_{k}\delta_{k,p} & {ig\over \sqrt{\kappa}}\, \bar\phi_{p-k} \\ {ig\over \sqrt{\kappa}}\, \phi_{k-p} & a_{-k}\delta_{k,p} \ematr \right]U^{-1}=B=(B_{kp})_{k,p}$$ where the $2\times 2$ blocks $B_{kp}$ are given by $$ B_{kk}=\left( \bmatr{cc} a_{k} & {ig\over \sqrt{\kappa}}\, \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & a_{-k} \ematr \right),\;\;\;B_{kp}={\ts {ig\over \sqrt{\kappa}}} \left( \bmatr{cc} 0 &\bar\phi_{p-k} \\ \phi_{k-p} & 0 \ematr \right)\;\;{\rm if}\;k\ne p\,. \eqno (4.13) $$ We want to compute the $2\times 2$ matrix $$ \la G\ra(k)=\int G(k)\,dP(\phi) \eqno (4.14)$$ where $$G(k)=[B^{-1}]_{kk} \eqno (4.15)$$ \bigskip We start again with the two loop approximation which retains only the $r=2$ term in the denominator of (1.3). The result will be equation (4.20) below where the quantities $\la\sigma_k\ra$ and $\la|\phi_0|^2\ra$ appearing in (4.20) have to satisfy the equations (4.21) and (4.24) which have to be solved in conjunction with (4.26). The solution to these equations is discussed below (4.27) and leads to (4.32). \par We first derive (4.20). In the two loop approximation, \setcounter{equation}{15} \beqn G(k)&\approx& \biggl[ B_{kk}-\sum_{p\ne k} B_{kp}\, G_k(p)\, B_{pk} \biggr]^{-1} \nonumber \\ &=&\biggl[\left(\bmatr{cc} a_{k} & {ig\over \sqrt{\kappa}}\, \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & a_{-k} \ematr\right) +{\ts {\lambda\over\kappa}} \sum_{p\ne k} \left(\bmatr{cc} & \bar\phi_{p-k} \\ \phi_{k-p} & \ematr\right) \,G_k(p)\, \left(\bmatr{cc} & \bar\phi_{k-p} \\ \phi_{p-k} & \ematr\right) \biggr]^{-1}\nonumber \\ &=:& \biggl[\left(\bmatr{cc} a_{k} & {ig\over \sqrt{\kappa}}\, \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & \bar a_{k} \ematr\right) + \Sigma(k) \biggr]^{-1} \eeqn where, substituting again $G_k(p)$ by $G(p)$ in the infinite volume limit, $$\Sigma(k)={\ts {\lambda\over\kappa}} \sum_{p\ne k} \left(\bmatr{cc} & \bar\phi_{p-k} \\ \phi_{k-p} & \ematr\right)\, \biggl[ \left(\bmatr{cc} a_{p} & {ig\over \sqrt{\kappa}}\, \bar\phi_{0} \\ {ig\over \sqrt{\kappa}}\, \phi_{0} & \bar a_{p} \ematr\right)+ \Sigma(p) \biggr]^{-1}\, \left(\bmatr{cc} & \bar\phi_{k-p} \\ \phi_{p-k} & \ematr\right) \eqno (4.17)$$ Anticipating the fact that the off-diagonal elements of $\la\Sigma\ra(k)$ will be zero (for `zero external field'), we make the Ansatz $$\Sigma(k)=\left(\bmatr{cc} \sigma_k& \\ & \bar\sigma_k\ematr\right) \eqno (4.18) $$ and obtain $$ \left(\bmatr{cc} \sigma_k& \\ & \bar\sigma_k\ematr\right)= {\ts {\lambda\over\kappa}} \sum_{p\ne k} \ts {1\over (a_p+\sigma_p)(\bar a_p+\bar\sigma_p) +{\lambda\over\kappa}|\phi_0|^2} \left(\bmatr{cc} (a_k+\sigma_k)|\phi_{p-k}|^2 & -{ig\over \sqrt{\kappa}}\, \phi_{0}\bar\phi_{k-p}\bar\phi_{p-k}\\ -{ig\over \sqrt{\kappa}}\, \bar\phi_{0} \phi_{k-p}\phi_{p-k} & (\bar a_k+\bar\sigma_k)|\phi_{k-p}|^2\ematr\right) \eqno(4.19)$$ As for the Anderson model, we perform the functional integral by substituting the quantities $|\phi_q|^2$ by their expectation values $\la|\phi_q|^2\ra$. Apparently this is less obvious in this case since $dP(\phi)$ is no longer Gaussian and the $|\phi_q|^2$ are no longer identically, independently distributed. We will comment on this after (4.37) below and at the end of the next section by reinterpreting this procedure as a resummation of diagrams. For now, we simply continue in this way. Then $$\la G\ra(k)=\ts {1\over |a_k+\la\sigma_k\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra}\left(\bmatr{cc} \bar a_k +\la\bar\sigma_k\ra & \ts -{ig\over \sqrt\kappa}\,\la\bar\phi_0\ra \\ -{ig\over \sqrt\kappa} \,\la\phi_0\ra & a_k+\la\sigma_k\ra \ematr\right) \eqno (4.20)$$ where the quantity $\la\sigma_k\ra$ has to satisfy the equation $$\la\sigma_k\ra= {\ts {\lambda\over\kappa}} \sum_{p\ne k} \ts {\bar a_p+\la\bar\sigma_p\ra \over |a_p+\la\sigma_p\ra|^2 +{\lambda\over\kappa}\la |\phi_0|^2\ra }\, \la |\phi_{p-k}|^2\ra \eqno (4.21) $$ \medskip Since $dP(\phi)$ is not Gaussian, we do not know the expectations $\la|\phi_q|^2\ra$. However, by partial integration, we obtain $$ \la |\phi_q|^2\ra =1+ {\ts {ig\over \sqrt\kappa}} \sum_p \int \phi_q\, [B^{-1}(\phi)]_{p\up,p+q\down}\, dP(\phi) \eqno(4.22) $$ Namely, \beq \la |\phi_q|^2\ra&=&{\ts {1\over Z}}\int \phi_q\bar\phi_q\, \det\left[ \{B_{kp}(\phi)\}_{k,p}\right]\, e^{-\sum_q |\phi_q|^2} d\phi_qd\bar\phi_q \\ &=&1+{\ts {1\over Z}}\int \phi_q\,\ts \Bigl( {\pt\over \pt\phi_q} \det\left[ \{B_{kp}(\phi)\}_{k,p}\right]\Bigr)\, e^{-\sum_q |\phi_q|^2} d\phi_qd\bar\phi_q \\ &=&1+{\ts {1\over Z}}\int \phi_q\,\sum_{p,\tau} \ts \det\left[\bmatr{ccc} |&|&|\\ B_{k\sigma,p'\tau'}& \ts {\pt B_{k\sigma,p\tau}\over \pt\phi_q}& B_{k\sigma,p''\tau''} \\ |&|&|\ematr \right] e^{-\sum_q |\phi_q|^2} d\phi_qd\bar\phi_q \eeq Since $$\ts {\pt \over \pt\phi_q}B_{kp} ={ig\over \sqrt\kappa}\left(\bmatr{cc} 0&0 \cr1&0 \ematr\right) \,\delta_{k-p,q}$$ we have $$ {\det\left[\bmatr{ccc} |&|&|\\ B_{k\sigma,p'\tau'}& \ts {\pt B_{k\sigma,p\tau}\over \pt\phi_q}& B_{k\sigma,p''\tau''} \\ |&|&|\ematr \right]\Bigr/ \det\left[ \{B_{kp}\}_{k,p}\right]} =\left\{\begin{array}{ll} 0& {\rm if}\; \tau=\down\\ & \\ \ts {ig\over \sqrt\kappa}\, [B^{-1}]_{p\up,p+q\down}& {\rm if}\; \tau=\up\end{array}\right. $$ which results in (4.22). The inverse matrix element in (4.22) we compute again with (1.3,4) in the two loop approximation. Consider first the case $q=0$. Then one gets \setcounter{equation}{22} \beqn \la |\phi_0|^2\ra&=&1+ {\ts {ig\over \sqrt\kappa}} \sum_p \int \phi_0\, G(p)_{\up\down}\, dP(\phi) \nonumber\\ &=&1+ {\ts {ig\over \sqrt\kappa}} \sum_p \int \phi_0\, \ts {1\over |a_p+\sigma_p|^2 +{\lambda\over\kappa}|\phi_0|^2}\left(\bmatr{cc}\bar a_p +\bar\sigma_p & \ts -{ig\over \sqrt\kappa}\,\bar\phi_0 \\ -{ig\over \sqrt\kappa} \,\phi_0 & a_p+\sigma_p\ematr\right)_{\up\down} \, dP(\phi) \nonumber\\ &=&1+ {\ts {\lambda\over \kappa}} \sum_p \int \phi_0\, \ts {\bar\phi_0\over |a_p+\sigma_p|^2 +{\lambda\over\kappa}|\phi_0|^2} \, dP(\phi) \eeqn Performing the functional integral by substitution of expectation values gives $$\la |\phi_0|^2\ra=1+ {\ts {\lambda\over \kappa}} \sum_p \la |\phi_0|^2\ra\, \ts {1\over |a_p+\la\sigma_p\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra } $$ or $$\la |\phi_0|^2\ra={1\over 1-{\ts {\lambda\over \kappa}} {\ds \sum_p^{\phantom{,}}} \ts {1\over |a_p+\la\sigma_p\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra } } \eqno (4.24) $$ Before we discuss (4.24), we write down the equation for $q\ne 0$. In that case we use (1.4) to compute $[B^{-1}(\phi)]_{p\up,p+q\down}$ in the two loop approximation. We get \setcounter{equation}{24} \beqn \lefteqn{ [B^{-1}(\phi)]_{p\up,p+q\down} \approx -\left[G(p)B_{p,p+q}G(p+q)\right]_{\up\down} }\nonumber \\ & &\nonumber \\ \lefteqn{ =- \ts {1\over |a_p+\sigma_p|^2 +{\lambda\over\kappa}|\phi_0|^2} \, \ts {1\over |a_{p+q}+\sigma_{p+q}|^2 +{\lambda\over\kappa}|\phi_0|^2}\,{ig\over\sqrt\kappa}\times }\nonumber \\ & & \nonumber \\ & & \left(\bmatr{cc} -{ig\over \sqrt\kappa}[ (\bar a+\bar \sigma)_{p+q}\bar\phi_0\phi_{-q}+(\bar a+\bar \sigma)_{p} \phi_0\bar\phi_q]& (\bar a+\bar \sigma)_{p}(a+\sigma)_{p+q}\bar\phi_q -{\lambda\over\kappa} \bar\phi_0^2\phi_{-q}\\ ( a+ \sigma)_{p}(\bar a+\bar\sigma)_{p+q}\phi_{-q} -{\lambda\over\kappa} \phi_0^2\bar\phi_{q} & -{ig\over \sqrt\kappa}[ ( a+ \sigma)_{p+q}\phi_0\bar\phi_{q}+( a+ \sigma)_{p} \bar\phi_0\phi_{-q}] \ematr\right)_{\up\down} \nonumber \\ & & \nonumber \\ \lefteqn{ =- \ts{ig\over\sqrt\kappa} {(\bar a+\bar \sigma)_{p}(a+\sigma)_{p+q}\bar\phi_q -{\lambda\over\kappa} \bar\phi_0^2\phi_{-q} \over \left(|a_p+\sigma_p|^2 +{\lambda\over\kappa}|\phi_0|^2\right)\left( |a_{p+q}+\sigma_{p+q}|^2 +{\lambda\over\kappa}|\phi_0|^2\right)} } \eeqn which gives \beq \la |\phi_q|^2\ra&=&1+ {\ts {\lambda\over \kappa}} \sum_p \int \phi_q\, \ts {(\bar a+\bar \sigma)_{p}(a+\sigma)_{p+q}\bar\phi_q -{\lambda\over\kappa} \bar\phi_0^2\phi_{-q} \over \left(|a_p+\sigma_p|^2 +{\lambda\over\kappa}|\phi_0|^2\right)\left( |a_{p+q}+\sigma_{p+q}|^2 +{\lambda\over\kappa}|\phi_0|^2\right)} \, dP(\phi) \\ &=&1+ \la |\phi_q|^2\ra {\ts {\lambda\over \kappa}} \sum_p \ts {(\bar a_p+ \la\bar\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra) \over \left(|a_p+\la\sigma_p\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra\right) \left( |a_{p+q}+\la\sigma_{p+q}\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra \right)} \eeq or $$\la |\phi_q|^2\ra={1\over 1- {\ts {\lambda\over \kappa}} {\ds \sum_p^{\phantom{,}} } \ts {(\bar a_p+ \la\bar\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra) \over \left(|a_p+\la\sigma_p\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra\right) \left( |a_{p+q}+\la\sigma_{p+q}\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra \right)} }\>,\;\;\;q\ne 0 \eqno (4.26)$$ \bigskip We now discuss the solutions to (4.24) and (4.26). We assume that the solution $\la \sigma_k\ra$ of (4.21) is sufficiently small such that the BCS equation $$ {\ts {\lambda\over \kappa}} \sum_p^{\phantom{,}} { \ts {1\over |a_p+\la\sigma_p\ra|^2 + |\Delta|^2 }} = 1 \eqno (4.27)$$ has a nonzero solution $\Delta\ne 0$ (in particular this excludes Luttinger liquid behaviour, for $d=1$ one should make a seperate analysis), and make the Ansatz $$\lambda\la|\phi_0|^2\ra= \beta L^d\, |\Delta|^2 +\eta \eqno (4.28) $$ where $\eta$ is independent of the volume. Then \setcounter{equation}{28} \beqn {\ts {\lambda\over \kappa}} \sum_p^{\phantom{,}} \ts {1\over |a_p+\la\sigma_p\ra|^2 +{\lambda\over\kappa}\la|\phi_0|^2\ra } &= & {\ts {\lambda\over \kappa}} \sum_p^{\phantom{,}} \ts {1\over |a_p+\la\sigma_p\ra|^2 + |\Delta|^2+{\eta\over\kappa} } \nonumber\\ &=&{\ts {\lambda\over \kappa}} \sum_p^{\phantom{,}} {\ts {1\over |a_p+\la\sigma_p\ra|^2 + |\Delta|^2 }} - {\ts {\lambda\over \kappa}} \sum_p^{\phantom{,}} \ts {\eta/\kappa\over ( |a_p+\la\sigma_p\ra|^2 + |\Delta|^2)^2 } +O\left(({\eta\over\kappa})^2\right) \nonumber\\ &=&\ts 1 - c_\Delta\,{\eta\over\kappa}+O\left(({\eta\over\kappa})^2\right) \eeqn where we put $c_\Delta= {\ts {\lambda\over \kappa}} \sum_p \ts {1\over ( |a_p+\la\sigma_p\ra|^2 + |\Delta|^2)^2 }$ and used the BCS equation (4.27) in the last line. Equation (4.24) becomes \beqn \kappa\, |\Delta|^2 +\eta &=&{\lambda\over c_\Delta \,{\eta\over\kappa} +O\left(({\eta\over\kappa})^2\right) } =\kappa\,{\lambda\over c_\Delta\,\eta} +O(1) \eeqn and has a solution $\eta=\lambda/(c_\Delta|\Delta|^2)$. Now consider $\la|\phi_q|^2\ra$ for small but nonzero $q$. Substituting (4.28) and Taylor expanding around $q=0$ we get \beqn \la |\phi_q|^2\ra&=&{1\over 1- {\ts {\lambda\over \kappa}} {\ds \sum_p^{\phantom{,}} } \ts {(\bar a_p+ \la\bar\sigma_p\ra)(a_{p}+\la\sigma_{p}\ra) \over \left(|a_p+\la\sigma_p\ra|^2 +|\Delta|^2 \right) \left( |a_{p}+\la\sigma_{p}\ra|^2 +|\Delta|^2 \right)} +O(q) } \nonumber \\ &=&{1\over {\ts {\lambda\over \kappa}} {\ds \sum_p^{\phantom{,}} } \ts { |\Delta|^2 \over \left(|a_p+\la\sigma_p\ra|^2 +|\Delta|^2 \right)^2} +O(q) } \eeqn which is bounded for $q\to 0$. Thus, in the infinite volume limit and at zero temperature we find $$ \lambda\la|\phi_q|^2\ra=|\Delta|^2\,\delta(q)+{\rm regular} \eqno (4.32)$$ \par The function $\Lambda(q)={\ts {\lambda\over (\beta L^d)^3}}\sum_{k,p} \la \bar\psi_{k\up}\bar\psi_{q-k\down}\psi_{q-p\down}\psi_{p\up}\ra = \la |\phi_q|^2\ra -1$ or some closely related four-point functions have been discussed by some authors [FMRT,CFS,B]. The conclusion is $\Lambda(q)\sim 1/q^2$ for small $q$. Basically this is infered from the second order Taylor expansion of the effective potential $$V_{\rm eff}(\{\phi_q\})=\sum_q|\phi_q|^2-\log \det\left[ \bmatr{cc} \delta_{k,p} & {ig\over \sqrt{\kappa}}\, {\bar\phi_{p-k}\over a_k} \\ {ig\over \sqrt{\kappa}}\, {\phi_{k-p}\over a_{-k}} & \delta_{k,p} \ematr \right] \eqno (4.33) $$ around its global minimum [L2] $$\phi_q^{\rm min}=\sqrt{\beta L^d}\,\ts {|\Delta|\over \sqrt\lambda}\,\delta_{q,0}\, e^{i\theta_0} \eqno (4.34) $$ where the phase $\theta_0$ of $\phi_0$ is arbitrary. If one expands $V_{\rm eff}$ up to second order in $$ \xi_{q}=\phi_{q}-\delta_{q,0}\sqrt{\beta L^d}\, \ts {|\Delta|\over \sqrt\lambda}\, e^{i\theta_0} =\left\{ \begin{array}{ll} \left(\rho_0-\sqrt{\beta L^d}\,\ts {|\Delta|\over \sqrt\lambda}\right) \,e^{i\theta_0} &\mbox{for $q=0$ }\\ \>\rho_{q}\, e^{i\theta_{q}} &\mbox{for $q\ne 0$} \end{array}\right. \eqno(4.35) $$ one obtains [L2] \setcounter{equation}{35} \beqn V_{\rm eff}(\{\phi_q\})&=&V_{\rm min}+2\beta_0 \, (\rho_0-\sqrt{\beta L^d}\,{\ts {|\Delta|\over \sqrt\lambda}})^2 + \sum_{q\ne 0} (\alpha_q+i\gamma_q)\, \rho_q^2 \nonumber\\ &\phantom{m}& + {\ts{1\over2}}\sum_{q\ne 0} \beta_q \,|e^{-i\theta_0} \phi_q +e^{i\theta_0} \bar\phi_{-q}|^2 +O(\xi^3) \eeqn where for small $q$ one has $\alpha_q,\gamma_q\sim q^2$. Hence, if $V_{\rm eff}$ is substituted by the right hand side of (4.36) one obtains $\la |\phi_q|^2\ra\sim 1/q^2$. Thus, in view of the above computations, it is questionable to what extent one can draw any conclusions from the quadratic approximation. We believe that the probability of the configurations $\{\phi_q\}$ for which (4.36) is a good approximation , the small field region, is too small to give a significant contribution to the functional integral. \bigskip Finally we argue why it is reasonable to substitute $|\phi_0|^2$ by its expectation value while performing the functional integral. We may write the effective potential (4.33) as $$V_{\rm eff}(\{\phi_q\})=V_1(\phi_0)+ V_2(\{\phi_q\}) \eqno (4.37)$$ where \setcounter{equation}{37} \beqn V_1(\phi_0)&=&|\phi_0|^2- \sum_{k}\ts \log\left[ 1+ {\lambda\over\kappa} { |\phi_0|^2 \over k_0^2+e_\bk^2}\right] \nonumber \\ &=& \kappa \Bigl({\ts {|\phi_0|^2\over\kappa}- {1\over\kappa}}\sum_{k}\ts \log\left[ 1+ {\lambda {|\phi_0|^2\over\kappa}\over k_0^2+e_\bk^2}\right] \Bigr)\equiv \kappa\, V_{\rm BCS}\ts\left({|\phi_0|\over\sqrt\kappa}\right) \eeqn and $$ V_2(\{\phi_q\})= \sum_{q\ne 0}|\phi_q|^2-\log \det\left[ \left(\bmatr{cc} \delta_{k,p} & {ig\over \sqrt{\kappa}}\, {\bar\phi_{0}\over a_k}\delta_{k,p} \\ {ig\over \sqrt{\kappa}}\, {\phi_{0}\over a_{-k}}\delta_{k,p} & \delta_{k,p} \ematr\right)^{-1} \left(\bmatr{cc} \delta_{k,p} & {ig\over \sqrt{\kappa}}\, {\bar\phi_{p-k}\over a_k} \\ {ig\over \sqrt{\kappa}}\, {\phi_{k-p}\over a_{-k}} & \delta_{k,p} \ematr\right) \right] \eqno(4.39)$$ If we ignore the $\phi_0$-dependence of $V_2$, then the $\phi_0$-integral \setcounter{equation}{39} \beqn {\int F\left({1\over\kappa}|\phi_0|^2\right)\, e^{-V_{1}(\phi_0)} d\phi_0 d\bar\phi_0 \over \int e^{-V_{1}(\phi_0)} d\phi_0 d\bar\phi_0} &=& {\int F\left( \rho^2\right)\, e^{-\kappa V_{\rm BCS}(\rho)} \rho\,d\rho \over \int e^{-\kappa V_{\rm BCS}(\rho)} \rho\,d\rho} \buildrel \kappa\to\infty\over \to F(\rho_{\rm min}^2) =\ts F\left({1\over\kappa}\la|\phi_0|^2\ra\right)\nonumber \\ & & \eeqn simply puts $|\phi_0|^2$ at the global minimum of the (BCS) effective potential. \bigskip \bigskip \goodbreak % % % P H I 4 % % \subsection{The $\vp^4$-Model} \bigskip In this section we choose the $\varphi^4$-model as a typical bosonic model to demonstrate our method. As in section 2, we start in finite volume $[0,L]^d$ on a lattice with lattice spacing $1/M$. The two point function is given by $$S(x,y)=\la \vph_x\vph_y\ra:={ \int_{\Bbb R^{N^d}} \vph_x\vph_y\> e^{-{g^2\over 2}{1\over M^d} \sum_x \vph_x^4}\> e^{-{1\over M^{2d}}\sum_{x,y} (-\Delta+m^2)_{x,y}\vph_x\vph_y} \pro_x d\vph_x \over \int_{\Bbb R^{N^d}} e^{-{g^2\over 2}{1\over M^d} \sum_x \vph_x^4}\> e^{-{1\over M^{2d}}\sum_{x,y} (-\Delta+m^2)_{x,y}\vph_x\vph_y} \pro_x d\vph_x} \eqno (4.41)$$ where $$(-\Delta+m^2)_{x,y}=M^d\Bigl[ -M^2\sum_{i=1}^d(\delta_{x,y-e_i/M} +\delta_{x,y+e_i/M} -2\delta_{x,y} ) +m^2\delta_{x,y}\Bigr]\, \eqno (4.42) $$ First we have to bring this into the form $\int[P+Q]^{-1}_{x,y}d\mu$, $P$ diagonal in momentum space, $Q$ diagonal in coordinate space. This is done again by making a Hubbard Stratonovich transformation which in this case reads $$ e^{-{1\over 2} \sum_x a_x^2}=\int e^{i\sum_x a_x u_x} e^{-{1\over 2} \sum_x u_x^2}\pro_x {\ts {d u_x\over\sqrt{2 \pi}}} \eqno (4.43)$$ with $$a_x={\ts {g\over \sqrt{M^d}}}\vph_x^2 \eqno (4.44) $$ The result is Gaussian in the $\vp_x$-variables and the integral over these variables gives $$S(x,y)= \int_{\Bbb R^{N^d}}\ts \left[ {1\over M^{2d}} (-\Delta+m^2)_{x,y}-{ig\over \sqrt{M^d}} u_x\delta_{x,y} \right]_{x,y}^{-1} dP(u) \eqno (4.45)$$ where $$ dP(u)={\ts{1\over Z}} \, {\ts\det\left[ {1\over M^{2d}} (-\Delta+m^2)_{x,y}-{ig\over \sqrt{M^d}} u_x\delta_{x,y} \right]^{-{1\over2}} } e^{-{1\over 2}\sum_x u_x^2}\pro_x du_x \eqno (4.46) $$ Since we have bosons, the determinant comes with a power of $-1/2$ which is the only difference compared to a fermionic system. In momentum space this reads (compare equations (2.7-11)) $$S(x-y)={\ts {1\over L^d}}\sum_k e^{ik(x-y)} \la G\ra (k) \eqno (4.47)$$ where ($\gamma_q=v_q+iw_q$, $\gamma_{-q}=\bar\gamma_q$, $d\gamma_qd\bar\gamma_q:=dv_qdw_q$) $$\la G\ra (k)= \int_{\Bbb R^{N^d}}\ts \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-1}_{kk} dP(\gamma) \eqno (4.48)$$ and $$dP(\gamma)={\ts {1\over Z}}\, {\ts \det\left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-{1\over 2}}} e^{-{1\over 2}v_0^2}dv_0 \pro_{q\in \M^+} e^{-|\gamma_q|^2} {\ts {d\gamma_q d\bar\gamma_q}} \eqno (4.49)$$ and $\M^+$ again is a set such that either $q\in \M^+$ or $-q\in \M^+$. Furthermore $$a_k= 4M^2\sum_{i=1}^d\ts \sin^2\left[ {k_i\over 2M}\right] \;+\; m^2 \eqno (4.50)$$ \medskip Equation (4.48) is our starting point. We apply (1.3) to the inverse matrix element in (4.48). In the two loop approximation one obtains $(\gamma_0=v_0\in\Bbb R)$ $${\ts \left[a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-1}_{kk} }\approx {1\over a_k -{igv_0\over \sqrt{L^d}} +{g^2\over L^d}\sum_{p\ne k}G_k(p)|\gamma_{k-p}|^2 } =: {1\over a_k+\sigma_k} \eqno (4.51)$$ where $$\sigma_k =-{\ts {ig\over \sqrt{L^d}}}v_0+ {\ts {g^2\over L^d}}\sum_{p\ne k}{ |\gamma_{k-p}|^2 \over a_p-{igv_0\over \sqrt{L^d}} +\sigma_p} \eqno (4.52) $$ which results in $$\la G\ra(k)= {1\over a_k +\la\sigma_k\ra } \eqno (4.53)$$ where $\la\sigma_k\ra$ has to satisfy the equation \setcounter{equation}{53} \beqn \la\sigma_k\ra & =& -{\ts {ig\over \sqrt{L^d}}}\la v_0\ra +{\ts {g^2\over L^d}} \sum_{p\ne k}{ \la |\gamma_{k-p}|^2\ra \over a_p +\la\sigma_p\ra }\nonumber \\ &=& {\ts {g^2\over 2 L^d}}\sum_{p} \la G\ra(p) +{\ts {g^2\over L^d}} \sum_{p\ne k}{ \la |\gamma_{k-p}|^2\ra \over a_p +\la\sigma_p\ra } ={\ts {g^2\over L^d}} \sum_{p\ne k}{ \la |\gamma_{k-p}|^2\ra +{1\over 2} \over a_p +\la\sigma_p\ra } \eeqn where the last line is due to \beqn \la v_0\ra&=&{\ts {1\over Z}}\int v_0\> {\ts \det\left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-{1\over 2}}} e^{-{1\over 2}v_0^2}dv_0 \pro_{q\in \M^+} e^{-|\gamma_q|^2} {\ts {d\gamma_q d\bar\gamma_q}} \nonumber \\ &=&{\ts {1\over Z}}\int \Bigl\{ {\ts {\pt\over \pt v_0}} {\ts \det\left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-{1\over 2}}}\Bigr\} e^{-{1\over 2}v_0^2}dv_0 \pro_{q\in \M^+} e^{-|\gamma_q|^2} {\ts {d\gamma_q d\bar\gamma_q}} \nonumber\\ &=&{\ts -{1\over 2}}\sum_p {\ts (-{ig\over \sqrt{L^d}}) }\int \ts \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-1}_{pp} dP(\gamma) \eeqn \smallskip As for the Many-Electron system, we can derive an equation for $\la|\gamma_q|^2\ra$ by partial integration: \beqn \la|\gamma_q|^2\ra&=&{\ts {1\over Z}} \int \gamma_q \bar\gamma_q\> {\ts \det\left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p} } \right]^{-{1\over 2}}} e^{-{v_0^2\over 2}}dv_0 \pro_{q} e^{-|\gamma_q|^2} {\ts {d\gamma_q d\bar\gamma_q}} \nonumber \\ &=&\;1\;+\; {\ts {1\over Z}} \int \gamma_q {\ts{\partial\over \partial \gamma_q}}\biggl\{ {\ts \det\left[ a_k \delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]^{-{1\over 2}}} \biggr\} e^{-{v_0^2\over 2}}dv_0 \pro_{q} e^{-{1\over 2}|\gamma_q|^2} {\ts {d\gamma_q d\bar\gamma_q}} \nonumber\\ &=&\;1\;-\;{\ts{1\over2}}\> \int \gamma_q\,{ {\partial\over \partial \gamma_q} {\ts \det\left[ a_k \delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]} \over {\ts \det\left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right] } } \,dP(\gamma) \nonumber\\ &=&\;1\;-\;{\ts{1\over2}}\>\sum_p {\ts {-ig\over \sqrt{L^d}}} \int \ts \gamma_q\, \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p}} \right]_{p,p+q}^{-1} \,dP(\gamma) \eeqn Computing the inverse matrix element in (4.56) again in the two loop approximation, one arrives at $$\la |\gamma_q|^2\ra=1-\la |\gamma_q|^2\ra {\ts {g^2\over 2 L^d}} \sum_p \ts {1\over (a_p+\la\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra)}$$ or $$\la |\gamma_q|^2\ra={1\over 1+ {\ts {g^2\over 2}} \int_{[0,2\pi M]^d} {d^dp\over (2\pi)^d}\, {1\over (a_p+\la\sigma_p\ra)(a_{p+q}+\la\sigma_{p+q}\ra)} } \eqno (4.57)$$ which has to be solved in conjunction with $$\la\sigma_k\ra =\ts {g^2} \int_{[0,2\pi M]^d}\ts {d^dp\over (2\pi)^d}\, { \la |\gamma_{k-p}|^2\ra+{1\over2} \over a_p +\la\sigma_p\ra } \eqno (4.58) $$ Introducing the rescaled quantities $$\la \sigma_k\ra= M^2 s_{p\over M}\,,\;\;\;\; \la |\gamma_q|^2\ra=\lambda_{q\over M}\,,\;\;\;\; a_k=M^2 \vep_{k\over M}\,,\;\;\vep_k=\sum_{i=1}^d \sin^2\ts{k_i\over2} +{m^2\over M^2}\eqno (4.60)$$ (4.57,58) read \setcounter{equation}{60} \beqn s_k & =&\ts M^{d-4} {g^2} \int_{[0,2\pi ]^d}\ts {d^dp\over (2\pi)^d}\, { \lambda_{k-p}+{1\over2} \over \vep_p + s_p} \\ \lambda_q&=&{1\over 1+ {\ts M^{d-4}{g^2\over 2}} \int_{[0,2\pi ]^d} {d^dp\over (2\pi)^d}\, {1\over (\vep_p+ s_p)(\vep_{p+q}+s_{p+q})} } \eeqn Unfortunately we cannot check this result with the rigorously proven triviality theorem since $\la\sigma_k\ra$ and $\la|\gamma_q|^2\ra$ only give information on the 2-point function $S(x,y)$, (4.41), and on ${g^2\over M^d}\sum_x \la \vp(x)^4\ra=\sum_q\Lambda(q)$ where $\Lambda(q)=\la|\gamma_q|^2\ra-1$. However, the triviality theorem [F,FFS] makes a statement on the connected 4-point function $S_{4,c}(x_1,x_2,x_3,x_4)$ at noncoinciding arguments, namely that this function vanishes in the continuum limit in dimension $d>4$. \bigskip We now include the higher loop terms of (1.3,4) and give an interpretation in terms of diagrams. The exact equations for $\la G\ra(k)$ and $\la |\gamma_q|^2\ra$ are \setcounter{equation}{62} \beqn \la G\ra (k)&=& \int \ts \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p} } \right]^{-1}_{kk} dP(\gamma) \;=\;\left\la {1\over a_k+\sigma_k} \right\ra \\ \sigma_k&=& -{\ts {ig\over \sqrt{L^d}}}\, v_0 +\sum_{r=2}^{N^d} {\ts \left({ig\over \sqrt{L^d}}\right)^r}\!\!\!\! \sum_{p_2\cdots p_r\ne k\atop p_i\ne p_j} G_k(p_2)\cdots G_{kp_2\cdots p_{r-1}}(p_r)\,\gamma_{k-p_2}\gamma_{p_2-p_3}\cdots \gamma_{p_r-k} \nonumber \eeqn and \beqn \la |\gamma_q|^2\ra&=&1+{\ts {ig\over 2\sqrt{L^d}}}\sum_p \int \ts \gamma_q \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p} } \right]^{-1}_{p,p+q} dP(\gamma) \\ &\buildrel p\to p_2\over=& 1+{\ts {1\over 2}} \sum_{r=2}^{N^d} \left( {\ts {ig\over\sqrt{L^d}}}\right)^r \!\!\!\!\!\sum_{p_2\cdots p_r\ne p_2+q\atop p_i\ne p_j} \Bigl\la G(p_2)G_{p_2}(p_3) \cdots G_{p_2\cdots p_{r-1}}(p_r) G_{p_2\cdots p_r}(p_2+q)\times \nonumber \\ & &\phantom{mmmmmmmmmmmmmmmm } \gamma_{p_2-p_3}\cdots \gamma_{p_{r-1}-p_r}\gamma_{p_r-p_2-q}\gamma_{p_2+q-p_2} \Bigr\ra \nonumber \eeqn For $r>2$, we obtain terms $\la \gamma_{k_1}\cdots\gamma_{k_r}\ra$ whose connected contributions \begin{figure}[thb] \centerline{\epsfbox{bild4.eps}} \caption{three and higher loop contributions} \label{Figure 4} \end{figure} \noindent are, in terms of the electron or $\vp^4$-lines, are at least six-legged. Since for the many-electron system and for the $\vp^4$-model (for $d=4$) the relevant diagrams are two- and four-legged [FT,R], one may start with an approximation which ignores the connected $r$-loop contributions for $r>2$. This is obtained by writing $$\bigl\la\gamma_{k_1}\cdots \gamma_{k_{n}} \bigr\ra \approx \bigl\la\gamma_{k_1}\cdots \gamma_{k_{n}} \bigr\ra_2 \eqno (4.65)$$ where (the index `2' for `retaining only two-loop contributions') $$\bigl\la\gamma_{k_1}\cdots \gamma_{k_{2n}} \bigr\ra_2 :=\sum_{{\rm pairings}\;\sigma} \la\gamma_{k_{\sigma 1}} \gamma_{k_{\sigma 2}} \ra\cdots \la\gamma_{k_{\sigma (2n-1)}} \gamma_{k_{\sigma 2n}} \ra =\int \gamma_{k_1}\cdots \gamma_{k_{2n}} \, dP_2(\gamma) \eqno (4.66) $$ if we define $$dP_2(\gamma):= e^{-\sum_q{|\gamma_q|^2\over \la |\gamma_q|^2\ra} } \pro_q \ts {d\gamma_q d\bar\gamma_q\over \pi\, \la |\gamma_q|^2\ra } \eqno (4.67)$$ Substituting $dP$ by $dP_2$ in (4.63,64), we obtain a model which differs from the original model only by irrelevant contributions and for which we are able to write down a closed set of equations for the two-legged particle correlation function $\la G\ra (k)$ and the two-legged squiggle correlation function $\la |\gamma_q|^2\ra$ by resumming all two-legged (particle and squiggle) subdiagrams. The exact equations of this model are \setcounter{equation}{67} \beqn \la G\ra (k)&=& \int \ts \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p} } \right]^{-1}_{kk} dP_2(\gamma) \\ \la |\gamma_q|^2\ra&=&1+{\ts {ig\over 2\sqrt{L^d}}}\sum_p \int \ts \gamma_q \left[ a_k\delta_{k,p} -{ig\over \sqrt{L^d}} {\gamma_{k-p} } \right]^{-1}_{p,p+q} dP_2(\gamma) \eeqn and the resummation of the two-legged particle and squiggle subdiagrams is obtained by applying the inversion formula (1.3,4) to the inverse matrix elements in (4.68,69). A discussion similar to those of section 2 gives the following closed set of equations for the quantities $\la G\ra(k)$ and $\la |\gamma_q|^2\ra$: $$\la G\ra(k)= {1\over a_k+\la \sigma_k\ra}\,,\;\;\;\;\;\; \la |\gamma_q|^2\ra={1\over 1+\la\pi_q\ra} \eqno (4.70)$$ where \setcounter{equation}{70} \beqn \la \sigma_k\ra&=&{\ts {g^2\over 2L^d}}\sum_p\la G\ra(p) +\sum_{r=2}^{\ell} {\ts \left({ig\over \sqrt{L^d}}\right)^r}\!\!\!\! \sum_{p_2\cdots p_r\ne k\atop p_i\ne p_j} \la G\ra(p_2)\cdots \la G\ra (p_r)\, \la\gamma_{k-p_2}\gamma_{p_2-p_3}\cdots \gamma_{p_r-k}\ra_2 \nonumber\\ & & \\ \la\pi_q\ra&=&-{\ts {1\over 2}}\sum_{r=2}^\ell {\ts \left({ig\over \sqrt{L^d}}\right)^r} \sum_{s=3}^{r-1} \sum_{p_2\cdots p_r\ne p_2+q\atop p_i\ne p_j} \!\!\! \Bigl( \delta_{q,p_{s+1}-p_s}\, \la G\ra(p_2)\cdots \la G\ra(p_r)\, \la G\ra(p_2+q)\times \nonumber \\ & &\phantom{mmmmmmmmmmmmmmmmm } \la \gamma_{p_2-p_3}\cdots \widehat{\gamma}_{p_s-p_{s+1}} \cdots \gamma_{p_{r-1}-p_r} \gamma_{p_r-p_2-q}\ra_2 \Bigr) \nonumber\\ & & \eeqn In the last line we used that $\gamma_q$ in (4.64) cannot contract to $\gamma_{p_2-p_3}$ or to $\gamma_{p_r-p_2-q}$. If the expectations of the $\gamma$-fields on the right hand side of (4.71,72) are computed according to (4.66), one obtains the expansion into diagrams. The graphs contributing to $\la \sigma_k\ra$ have exactly one string of particle lines, each line having $\la G\ra$ as propagator, and no particle loops (up to the tadpole diagram). Each squiggle corresponds to a factor $\la |\gamma|^2\ra$. The diagrams contributing to $\la \pi\ra$ have exactly one particle loop, the propagators being again the interacting two point functions, $\la G\ra$ for the particle lines and $\la|\gamma|^2\ra$ for the squiggles. In both cases there are no two-legged subdiagrams. However, although the equation $\la |\gamma_q|^2\ra={1\over 1+\la\pi_q\ra}$ resums ladder or bubble diagrams (which is apparent from (4.57) or (4.26)) and more general four-legged particle subdiagrams if the terms for $r\ge 4$ in (4.72) are taken into account, the right hand side of (4.71,72) still contains diagrams with four-legged particle subdiagrams. Thus, the resummation of four-legged particle subdiagrams is only partially through the complete resummation of two-legged squiggle diagrams. Finally observe that, in going from (4.68,69) to (4.70-72), we cut off the $r$-sum at some fixed order $\ell$ independent of the volume since we can only expect that the expansions are asymptotic ones, compare the discussion in section 2. \bigskip \bigskip \bigskip \noindent{\Mittel References} \bigskip \bitem \item[{[AG]}] M. Aizenman, G.M. Graf, {\it Localization Bounds for an Electron Gas}, Journ. Phys. A, Bd. 31, S. 6783, 1998. \item[{[B]}] N. N. Bogoliubov, {\it Lectures on Quantum Statistics}, Vol.2, Gordon and Breach, 1970 \item[{[BFS]}] V. Bach, J. Fr\"ohlich, and I. M. Sigal, {\it Renormalization group analysis of spectral problems in quantum field theory}, Adv. in Math. 137, 205-298, 1998. \item[{[CFS]}] T. Chen, J. Fr\"ohlich, M. Seifert, {\it Renormalization Group Methods: Landau-Fermi Liquid and BCS Superconductor}, Proceedings of the Les Houches session {\it Fluctuating Geometries in Statistical Mechanics and Field Theory}, eds. F. David, P. Ginsparg, J. Zinn-Justin, 1994. \item[{[F]}] J. Fr\"ohlich, {\it On the Triviality of $\lambda\vp_d^4$ Theories and the Approach to the Critical Point}, Nucl. Physics B200, 281-296, 1982. \item[{[FFS]}] R. Fernandez, J. Fr\"ohlich, A. Sokal, {\it Random Walks, Critical Phenomena and Triviality in Quantum Field Theory}, Texts and Mongraphs in Physics, Springer 1992. \item[{[FKT]}] J. Feldman, H. Kn\"orrer, E. Trubowitz, {\it Mathematical Methods of Many Body Quantum Field Theory}, Lecture Notes, ETH Z\"urich. \item[{[FMRT]}] J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, {\it Ward Identities and a Perturbative Analysis of a U(1) Goldstone Boson in a Many Fermion System}, Helvetia Physica Acta 66, 1993, 498-550. \item[{[FT]}] J. Feldman, E. Trubowitz, {\it Perturbation Theory for Many Fermion Systems}, Helvetia Physica Acta 63, p.156-260, 1990; {\it The Flow of an Electron-Phonon System to the Superconducting State}, Helv. Phys. Acta 64, p.214-357, 1991. \item[{[K]}] A. Klein, {\it The Supersymmetric Replica Trick and Smoothness of the Density of States for Random Schr\"odinger Operators}, Proceedings of Symposia in Pure Mathematics, vol. 51, part 1, 1990. \item[{[L1]}] D. Lehmann, {\it The Many-Electron System in the Forward, Exchange and BCS Approximation}, Comm. Math. Phys. 198, 427-468, 1998. \item[{[L2]}] D. Lehmann, {\it The Global Minimum of the Effective Potential of the Many-Electron System with Delta-Interaction}, to appear in Rev. Math. Phys. Vol. 12, No 9, Sept. 2000. \item[{[MPR]}] J. Magnen, G. Poirot, V. Rivasseau, {\it Ward Type Identities for the 2D Anderson Model at Weak Disorder}, cond-mat/9801217; {\it The Anderson Model as a Matrix Model}, Nucl. Phys. B (Proc. Suppl.) 58, S.149, 1997. \item[{[P]}] G. Poirot, {\it Mean Green's Function of the Anderson Model at Weak Disorder with an Infrared Cutoff}, cond-mat/9702111 \item[{[R]}] V. Rivasseau, {\it From Perturbative to Constructive Renormalization}, Princeton Univ. Press 1991. \item[{[W]}] Wei-Min Wang, {\it Supersymmetry, Witten Complex and Asymptotics for Directional Lyapunov Exponents in $\mathbb Z^d$}, mp-arc/99-355; {\it Localization and Universality of Poisson Statistics Near Anderson Transition}, mp-arc/99-473. \eitem \end{document} \begin{figure}[thb] \centerline{\epsfbox{bild4.eps}} \caption{Figure 4} \label{Figure 4} \end{figure} \noindent ---------------0003061656364 Content-Type: application/postscript; name="bild1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bild1.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Title: bild2.eps %%Creator: fig2dev Version 3.2 Patchlevel 1 %%CreationDate: Mon Feb 28 14:13:13 2000 %%For: root@seven (root) %%Orientation: Portrait %%BoundingBox: 0 0 363 67 %%Pages: 0 %%BeginSetup %%EndSetup %%Magnification: 0.5000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} 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{0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -88.0 159.0 translate 1 -1 scale .9 .9 scale % to make patterns same scale as in xfig % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % right30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {<00030003000c000c0030003000c000c0030003000c000c00 30003000c000c00000030003000c000c0030003000c000c0 030003000c000c0030003000c000c000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P2 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 6047 m -1000 -1000 l 12130 -1000 l 12130 6047 l cp clip 0.03150 0.03150 sc 15.000 slw % Ellipse n 4065 4515 450 450 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P2 [16 0 0 -8 241.00 271.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 8100 4515 456 456 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P2 [16 0 0 -8 509.60 270.60] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 10350 4515 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 10710 4515 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Ellipse n 11085 4515 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % Polyline [90] 0 sd n 2925 4515 m 3615 4515 l gs col0 s gr [] 0 sd % Polyline [90] 0 sd n 4350 4905 m 4995 4905 l gs col0 s gr [] 0 sd % Polyline [90] 0 sd n 7785 4185 m 7230 4170 l gs col0 s gr [] 0 sd % Polyline [90] 0 sd n 7845 4920 m 7200 4920 l gs col0 s gr [] 0 sd % Polyline [90] 0 sd n 8415 4185 m 9045 4185 l gs col0 s gr [] 0 sd % Polyline [90] 0 sd n 8385 4920 m 9030 4920 l gs col0 s gr [] 0 sd % Polyline n 2925 4530 m 2820 4665 l gs col0 s gr % Polyline n 2940 4530 m 2820 4365 l gs col0 s gr % Polyline [90] 0 sd n 4305 4125 m 4995 4125 l gs col0 s gr [] 0 sd % Polyline n 4935 4140 m 5055 4005 l gs col0 s gr % Polyline n 4965 4155 m 5040 4245 l gs col0 s gr % Polyline n 4980 4920 m 5070 4800 l gs col0 s gr % Polyline n 5063 4800 m 5078 4800 l gs col0 s gr % Polyline n 4995 4935 m 5085 5025 l gs col0 s gr % Polyline n 5078 5025 m 5093 5025 l gs col0 s gr % Polyline n 7245 4185 m 7140 4290 l gs col0 s gr % Polyline n 7245 4185 m 7140 4050 l gs col0 s gr % Polyline n 7215 4935 m 7095 5025 l gs col0 s gr % Polyline n 7200 4905 m 7095 4830 l gs col0 s gr % Polyline n 9045 4200 m 9165 4065 l gs col0 s gr % Polyline n 9075 4215 m 9165 4305 l gs col0 s gr % Polyline n 9030 4935 m 9135 4815 l gs col0 s gr % Polyline n 9045 4905 m 9135 4995 l gs col0 s gr $F2psEnd rs end ---------------0003061656364--