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\vglue1.truecm
{\bf\\Renormalization group and the Fermi surface in the Luttinger model.}
\vglue1.truecm
{G. Benfatto\footnote{${}^1$}
{\arm Dipartimento di Matematica, II Universit\`a di
Roma, 00173 Roma, Italia},}
{G. Gallavotti\footnote{${}^2$}{\vtop{\hsize=15.9truecm\arm
\\Dipartimento di Fisica, Universit\`a di Roma,
P. Moro 5,00185 Roma, Italia; and Rutgers
University,\hfill\break\baselineskip=12.truept
Mathematics Dept., Hill Center, New Brunswick, N.J. 08903, USA.}},}
{V. Mastropietro\footnote{${}^3$}
{\arm Dipartimento di Fisica, Universit\`a di Roma,
P. Moro 5,00185 Roma, Italia}}
\vglue1.truecm
{\baselineskip=12.truept plus0.1truept minus0.1truept
{\it Abstract}:\ \arm the exactly soluble Luttinger model can be also
analyzed from the point of view of the renormalization group. A
perturbation theory of the beta function of the model is derived.
We argue that the
main terms of the beta function vanish identically if the anomalous
dimension is properly treated and if suitable properties of the exact
solution are taken into account.
Our treatment is purely perturbative and we do not
discuss the problems of convergence of the formal series defining
the beta function: however
the property that the series defining it is convergent has
been recently established.}
\vglue1.5truecm
{\it\S1 The Luttinger model}
\vglue1.truecm\numsec=1\numfor=1\pgn=1
The recent interest on the Luttinger model, [A], motivates our
discussion of its properties in a formalism which admits extensions to
higher dimensions, developed in [BG].
The model [L] describes two spinless fermions labeled by $\oo=\pm 1$, with a
one dimensional hamiltonian:
%
$$\eqalign{
&H = [T_0] + \{H_I\} = \Bigl[ \sum_{\oo=\pm 1} \ii_0^L d\xx \pst{+}{\xx,\oo}
\b_0(i\oo\Dp- p_F)\pst{-}{\xx,\oo} \Bigr]+\cr
&+ \Bigl\{\l\ii_0^L d\xx d\yy v(\xx-\yy) (\sum_{\oo=\pm 1}
q_{1,\oo}\pst{+}{\xx,\oo}
\pst{-}{\xx,\oo})(\sum_{\oo=\pm 1} q_{2,\oo} \pst{+}{\yy,\oo}
\pst{-}{\yy,\oo})+\cr
&+\n \sum_{\oo=\pm 1}[q_{1,\oo}+q_{2,\oo}] \ii_0^L d\xx
\pst{+}{\xx,\oo}\pst{-}{\xx,\oo} + \s\ii_0^L d\xx \Bigr\} \cr} \Eq(1.1)$$
%
where $\pst{\pm}{}$ are creation and annihilation field operators, $\xx$
and $\yy$ are position variables in the interval $[0,L]$ considered with
periodic boundary conditions, $p_F={2\p\over L}(n_F+1/2)$ is the {\it Fermi
momentum} ($n_F$ is an integer depending on $L$ so that $p_F$ is independent
of $L$ up to terms of order $1/L$),
and $\b_0$ is the {\it velocity at the Fermi surface}; $\l v({\V r})$ is the
interaction potential, which will be supposed with short range, equal to a
fixed length $p_0^{-1}$, and even as a function of ${\V r}$; the {\it
charges} $q_{1,\oo}$ and $q_{2,\oo}$ are arbitrary constants. Finally $\n$
and $\s$ are {\it counterterms}, necessary to balance the ultraviolet
divergences due to the unrealistic linear dispersion relation in the
(kinetic energy $-$ chemical potential) term $T_0$; in fact a fermion field
of type $\oo$ and momentum $\kk$ has energy $\b_0\oo\kk$. We fix units so
that the Fermi velocity is $\b_0=1$.
The case considered by Luttinger was $q_{1,+}=1$, $q_{1,-}=q_{2,+}=0$,
$q_{2,-}=1$. The model was solved in [ML], but the exact solution applies to
the general choice of $q_i$; the case $q_{i,\pm}=1/2$ is explicitly
treated in [ML] and extended to a simple spinning model in [M].
The values of $\n$ and $\s$ have to be computed by introducing a
ultraviolet cut-off in \equ(1.1) (which otherwise does not have a well
defined meaning) and, subsequently, by imposing that the Schwinger
functions of the model are well defined uniformly in the cut-off. Their
values depend upon the way the ultraviolet regularization is introduced and
can be altered by an arbitrary finite constant (possibly affecting the
physical value of the Fermi momentum or the Fermi velocity).
The regularization which is implicit in the exact theory of the ground state
seems to be simply the suppression of the modes with $\kk < -2^U p_0$ for
the $\oo=+1$ fermions and $\kk > 2^U p_0$ for the $\oo=-1$ fermions, where
$p_0$ is an arbitrary (for the time being) momentum scale and $U$ is a cut-off
parameter to be let to $\i$ eventually. It is natural and convenient to fix
$p_0{}^{-1}$ equal to the range of the interaction potential, supposed of
finite range.
Since the momenta $\pm p_F$ play a special role for the two fermions, it is
convenient to measure the momenta of the $\oo$-type fermions from $p_F\oo$. If
we call $\a^{\pm}_{\kk,\oo}$ the creation and annihilation operators of the
two fermions, we introduce the following field operators:
$$\eqalign{
\pst{\pm}{\xx,t,\oo} & \equiv e^{tT_0}\pst{\pm}{\xx,\oo}e^{-tT_0} =
{1\over \sqrt{L}} \sum_{\kk} e^{\pm [i\kk\xx +(\oo\kk-p_F)t]}
\a^{\pm}_{\kk,\oo} = \cr
& = e^{\pm ip_F\oo\xx} \ps{\pm}{\xx,t,\oo} \cr
\ps{\pm}{\xx,t,\oo} & \equiv e^{tT_0}\ps{\pm}{\xx,\oo}e^{-tT_0} =
{1\over \sqrt{L}} \sum_{\kk} e^{\pm (i\kk\xx +t\oo\kk)} a^{\pm}_{\kk,\oo}\cr
a^{\pm}_{\kk,\oo} & \equiv \a^{\pm}_{\kk+p_F\oo,\oo} \cr } \Eq(1.2)$$
The following hamiltonian operators, necessary to establish contact with the
existing literature, will also be introduced, following [ML]:
%
$$\eqalignno{
T_0' & = \sum_\oo\sum_{\kk>0} \kk (a^+_{\oo\kk,\oo} a^-_{\oo\kk,\oo} +
a^-_{-\oo\kk,\oo} a^+_{-\oo\kk,\oo} )&\eq(1.3) \cr
H_I' & = L^{-1}\l\sum_{\pp >0} \vh(\pp)[R_1(\pp)R_2(-\pp) +
R_1(-\pp)R_2(\pp)] + \cr
& + L^{-1}\l\vh(0) \left[ \sum_\oo q_{1,\oo} \NN_\oo \right] \,
\left[ \sum_\oo q_{2,\oo} \NN_\oo \right]\cr }$$
%
where:
%
$$\eqalign{
R_i(\pp) & = \sum_\oo q_{i,\oo} \r_\oo(\pp),\qquad
\r_\oo(\pp)=\sum_\kk a^+_{\kk+\pp,\oo}a^-_{\kk,\oo}\cr
\NN_\oo & = \sum_{\kk>0} (a^+_{\oo\kk,\oo} a^-_{\oo\kk,\oo} -
a^-_{-\oo\kk,\oo} a^+_{-\oo\kk,\oo} ) \cr } \Eq(1.4)$$
%
note that $T'_0$ is equal to $\sum_{\oo,\kk} \oo\kk
a^+_{\kk,\oo}a^-_{\kk,\oo}$ up to a constant; but the constant, see below,
is infinite, hence this simpler form for $T'_0$ is not defined (although it
can be very useful for heuristic purposes).
One can check that the operators \equ(1.3), \equ(1.4) can be regarded as
operators acting on a Hilbert space $\HH$ constructed as follows. Let:
%
$$\ket0= \prod_{\kk\le
0} a^+_{\kk,+1}a^+_{-\kk,-1} \ket{\rm vacuum} \Eq(1.5)$$
%
be an abstract vector, formally in Fock space. Let $\HH_0$ be the abstract
linear span of the formal vectors obtained by applying finitely many creation
and annihilation operators to $\ket0$. We get an abstract linear space on which
we introduce the scalar product between two vectors by computing it in the
obvious way {\it as if} they were Fock space vectors (no problem arises
because we only deal with vectors obtained by acting finitely many times on
$\ket0$ with the basic operators); then we define $\HH$ to be the completion of
$\HH_0 $ in the just introduced scalar product.
With such definitions it is easy to check the following basic commutation
relations:
%
$$\eqalign{
&[\r_\oo(\e\pp),\r_{\oo'}(-\e\pp^{\,'})]=-\e\oo\pp\d_{\oo,\oo'}\d_{\pp,\pp'}
{L\over2\p}, \qquad [\r_\oo(\e\pp),T'_0]=-\e\oo\pp \r_\oo(\e\pp)\cr
&[\r_\oo(\e\pp),\sum_{\pp>0,\oo'}
\r_{\oo'}(\oo'\pp)\r_{\oo'}(-\oo'\pp)]=-\e\oo\pp {L\over2\p}\r_\oo(\e \pp),
\qquad\r_\oo(-\oo\pp)\ket0\equiv0\cr}\Eq(1.4*1)$$
%
for $\pp>0,\e=\pm1$.
Furthermore the operators \equ(1.3), \equ(1.4), regarded as operators on
$\HH$ with domain $\HH_0$, are essentially selfadjoint. A simple
calculation shows that, if (setting $q_\e=\sum_\oo q_{\e\oo})$:
%
$$\eqalign{
\n &= -\l\vh(0) (2^Up_0+p_F)/2\p \cr\s &= q_+ q_-
\l\vh(0) (2^Up_0+p_F)^2/(4\pi^2) -L^{-1}\bra0 T_0\ket0 \cr }\Eq(1.6)$$
%
then one has $T_0'+H_I' =T_0+H_I$, in a formal sense as the $T_0+H_I$ is
defined using a ultraviolet cut off $2^Up_0$. The latter relation becomes
an identity in the limit $U\to\infty$.
Moreover one can also write:
%
$$\eqalign{
T_0'+H_I' &= \sum_\oo \ii d\xx\,:\ps{+}{\xx,\oo} (i\oo\Dp)\ps{-}{\xx,\oo}:+\cr
&+ \l\ii d\xx d\yy\, v(\xx-\yy) :(\sum_\oo q_{1,\oo}\ps{+}{\xx,\oo}
\ps{-}{\xx,\oo}):\, :(\sum_\oo q_{2,\oo} \ps{+}{\yy,\oo} \ps{-}{\yy,\oo}):\cr }
\Eq(1.7)$$
%
where $:\,\,:$ denotes the Wick ordering with respect to the vacuum $|0>$ of
$\HH$ (\ie the Wick ordering of a product of creation and annihilation
operators is obtained by rearranging the order so that $a^+_{-\kk,+},
a^-_{\kk,+}, a^+_{\kk,-}, a^-_{-\kk,-},\kk>0$ are always to the right of the
other operators, and the new product is multiplied by the parity sign of the
permutation necessary to produce it).
We adopt the choice \equ(1.6) of the counterterms because it allows the
simple interpretation \equ(1.7) of the hamiltonian in terms of Wick
ordering. However the model thus obtained is not, strictly speaking,
identical to the model of Luttinger as solved by Mattis-Lieb [ML]. They in
fact add to \equ(1.1) an extra term so that the operator $H_I'$ in
\equ(1.3) is just given by the first line, \ie $\vh(0)$ is in some sense
forced to vanish, without requiring $\vh(\pp)$ to vanish continously when
$\pp\to 0$. But the second line in the definition \equ(1.3) of $H_I'$ is an
operator $C'_I$ commuting with $T_0'$, as well as with all the operators
$\r_\oo(\pp)$ and this, of course, implies that the model \equ(1.1) with
the choices \equ(1.6) of $\n$, $\s$, \ie \equ(1.3) or \equ(1.7), is exactly
soluble in the same sense of the Luttinger model and the two hamiltonians
are defined on the same Hilbert space and have the same eigenvectors.
The only problem is that $C'_I$ is not bounded below; however it is easy to
see that $T''_0\equiv T'_0+C'_I$ is still bounded below if $\l\hat v(0)$
satisfies the stability condition:
%
$$\eqalign{
(\l\hat v(0)P)^2 &\le (2\p+2\l\hat v(0)q_{1+}q_{2+})
(2\p+2\l\hat v(0)q_{1-}q_{2-}) \cr
P & =q_{1+}q_{2-}+q_{1-}q_{2+}\cr} \Eq(1.8)$$
In fact, if we consider the action of $T''_0$ on the states with
$n_1$ {\it excitations} (\ie
number of particles minus number of holes) of tipe $\oo=+$ and $n_2$ of type
$\oo=-$, we have:
$$T''_0 \ge {\p\over L}(n_1^2+n_2^2) + {\l\hat v(0)\over L}(q_{1+}q_{2+}n_1^2
+ q_{1-}q_{2-}n_2^2)-|{\l\hat v(0)\over L}P||n_1||n_2| \Eq(1.8a)$$
and the r.h.s. is bounded below if and only if \equ(1.8) is satisfied.
As we shall see below, the condition \equ(1.8) is implied by the solubility
condition of the model in the case considered in [ML] ($q_{i,\oo}=1/2$), but
this is not true in general. However \equ(1.8) is always implied by the
stability condition for the full hamiltonian, if $\hat v(p)$ is a continuous
function, as we shall suppose.
Let us now define $H''_I\equiv H'_I-C'_I$, so that $T'_0+H'_I=T''_0+H''_I$.
The basic remark of [ML] is that the commutation relations in \equ(1.4*1)
imply:
%
$$[\r_\oo(\pm\pp),T-T''_0]\equiv0,\qquad{\rm for\ }\pp>0\Eq(1.9)$$
%
if $T\equiv(2\p/L)\sum_{\pp>0,\oo}\r_\oo(\oo\pp)\r_\oo(-\oo\pp)$. Hence
$T''_0-T$ commutes with all the operators $\r_\oo(\pp)$, and, therefore,
with $H''_I+T$. In this way we realize $T''_0+H''_I$ as the sum of two
commuting operators, the second of which is a sum of easily diagonalizable
commuting operators and this leads to the exact solubility of the model,
see [ML]. This is done by determining an even function $\f(p)$ such that
setting $S=2\p L^{-1}\sum_{all\ p\ne0}\f(p)p^{-1}\r_+(p)\r_-(-p)$ then the
operator $e^{iS}(H''_I+T)e^{-iS}$ does not contain {\it mixed terms}, \ie it
can be written, if $E_0(\l)$ is a suitable constant, in the form:
%
$${2\p \over L}
\sum_{p>0}[\e_+(p)\r_+(p)\r_+(-p)+\e_-(p)\r_-(-p)\r_-(p)]+E_0(\l)\Eq(1.10)$$
%
and one checks that this is achieved by taking:
%
$$\tanh2\f(p)=-{\l\hat v(p) P\over 2\p+\l\hat v(p)Q} \quad,\qquad
\cases{P=q_{1+}q_{2-}+q_{1-}p_{2+}\cr
Q=q_{1+}q_{2+}+q_{1-}q_{2-}\cr} \Eq(1.11)$$
%
and:
%
$$\eqalign{
\e_+(p)=&\ c(p)^2(1+{\l\hat v(p)\over \p}q_{1+}q_{2+}) +
s(p)^2(1+{\l\hat v(p)\over \p}q_{1-}q_{2-}) +\cr
+&\ {\l\hat v(p)\over \p}c(p)s(p)P \cr
\e_-(p)=&\ s(p)^2(1+{\l\hat v(p)\over \p}q_{1+}q_{2+}) +
c(p)^2(1+{\l\hat v(p)\over \p}q_{1-}q_{2-}) +\cr
+&\ {\l\hat v(p)\over \p}c(p)s(p)P \cr }\Eq(1.11a)$$
%
where $c(p)=\cosh \f(p)$, $s(p)=\sinh \f(p)$. Of course one needs that the
r.h.s. of the definition \equ(1.11) of the hyperbolic tangent be $< 1$ in
absolute value: we shall call this the "solubility condition".
Moreover \equ(1.10) and \equ(1.11a)
imply that the hamiltonian is bounded below if and only if:
%
$$(\l\hat v(p)P)^2 \le (2\p+2\l\hat v(p)q_{1+}q_{2+})
(2\p+2\l\hat v(p)q_{1-}q_{2-}) \Eq(1.11b)$$
%
This stability condition is a consequence of the solubility condition only if
$q_{1+}q_{2+}=q_{1-}q_{2-}$, as is the case considered in [ML] or in the
original Luttinger model. In general only the converse is true,
\ie the stability condition \equ(1.11b) implies that the r.h.s. of
\equ(1.11) is $< 1$ in absolute value, so that one
should always assume the stability condition \equ(1.11b).
In the rest of this paper we shall consider, as in [ML], the case
$q_{i\oo}=1/2$, $i=1,2$; then:
%
$$\e_+(p)=\e_-(p)=e^{-2\f(p)}=\bigl(1+{\l \hat v(p)\over2\p}\bigr)^{1/2}
\Eq(1.12)$$
%
and the ground state energy is: $E_0(\l)=\sum_{p>0}p(e^{-2\f(p)}-1)$.
Let us remark that the operator $T''_0-T$ can be explicitly computed and
it is a constant in every linear space containing a given number of
excitations
(this is non trivial and is implicit in [ML], as pointed out in [O]).
The constant can be computed in a state with $n_1$ excitations
of tipe $\oo=+$ and $n_2$ of type
$\oo=-$, simply by evaluating the expectation value of $T''_0-T$ on the
ground state with the same number of excitations, namely the state with the
first $n_1$ levels of type $\oo=+$ occupied and the first $n_2$ of type
$\oo=-$ occupied (if $n_i<0$ then one means, of course, holes created).
And the problem is solved by the remark that the commutation rules
\equ(1.4*1) imply that $e^{iS}\r_+(p)e^{-iS}$, $e^{iS}\r_-(-p)e^{-iS}$ are
bosonic creation operators
while $e^{iS}\r_+(-p)e^{-iS}$, $e^{iS}\r_-(p)e^{-iS}$ are bosonic
destruction operators annihilating the
new ground state which is: $\ket\O=e^{iS}\ket0$ as well as all
the similar ground states in the spaces with given numbers of excitations.
For completeness we give the argument (see [H]) showing that $T''_0-T$ is
constant on the space with a fixed number of excitations. Since $C'_I$ is
clearly constant on this space, it is sufficient to consider the case
$\l=0$, so that $T''_0=T'_0$. It is an immediate consequence of \equ(1.9)
that, if $E_j(n_1,n_2)$ are the eigenvalues of $T'_0-T$ in the space
with excitations numbers $n_1,n_2$, then each of the corresponding eigenstates
$\ket{n_1,n_2,j}$ generates a family of eigenvectors with the same
eigenvalue simply by applying the operators $\r_+(p)$ and $\r_-(-p)$ an
arbitrary number of times. Such vectors are all pairwise orthogonal and
non zero. Furthermore the eigenvector $\ket{n_1,n_2,j}$ with eigenvalue
$E_j(n_1,n_2)$ can be so chosen that $\r_+(-p)$ and $\r_-(p)$ annihilate
it. Then we see that by applying the operators $\r_+(p)$ and $\r_-(-p)$ an
arbitrary number of times to $\ket{n_1,n_2,j}$ one gets a family of vectors
with the property that $(T'_0-T)$ has eigenvalue $E(n_1,n_2,j)$ on each of
them while $T$ has eigenvalue $\sum_{p>0} p(n_+(p)+n_-(p))$, where
$n_+(p),n_-(p)$ are the number of times the operators $\r_+(p),\r_-(-p)$
are applied. Clearly the partition function for $T'_0$ at positive
temperature $\b^{-1}$ can be computed in two ways: one is by observing that
it is the partition function of a free Fermi gas with two particles with
dispersion relation $\oo(k-p_F)$, which is obviously:
%
$$Z=[\prod_{n>0}(1+z^{2n-1})]^4\Eq(1.17)$$
%
where $z=e^{-\b\p/L}$ (recall that $p_F=2\p/L(n_F+1/2)$).
Another way is to note that the above basis of
vectors $\ket{n_1,n_2,j,\{n_+(p)\},\{n_-(p)\}}$ is obviously complete and the
operator $T_0'\equiv (T'_0-T)+T$ has on it eigenvalues
$E(n_1,n_2,j)+\sum_{p>0} p(n_+(p)+n_-(p))$, so that the partition
function is:
%
$$Z=(\sum_{j,n_1} e^{-\b E(n_1,0,j)})^2\Bigl(\prod_{n>0}
(1-z^{2n})^{-1}\Bigr)^2\Eq(1.18)$$
%
where the independence of the two species of
fermions with $\oo=\pm1$ produces the squaring of the partition functions
and the identity $E(j,n_1,n_2)=E(j,n_1,0)+E(j,0,n_2)$.
Note that, as remarked above, we know explicitly at least one among
the eigenvectors
$\ket{j,n_1,n_2}$ of $T'_0-T$, namely the one in which all the levels are
filled up to the level $n_1$ (above $k=p_F$) with fermions of type $+$ and
down to the level $n_2 $ with fermions of type $-$. Furthermore on such
states it is easy to see that $T$ has eigenvalue $0$ while $T'_0$ has value
$(n_1^2+n_2^2)\p/L$. We see that if, {\it and only if}, such states were the
only ones with $n_1,n_2$ excitations it would follow that the
$(\sum_{j,n_1} e^{-\b E(n_1,0,j)})^2$ would have to be the sum
$(\sum_{k\in Z} z^{k^2})^2$. But
$(\sum_{j,n_1} e^{-\b E(n_1,0,j)})^2$
can be obviously ([ML])
computed by remarking that the two above methods of
computing the partition function of the free gas
must yield the same result (\ie
\equ(1.17) equals \equ(1.18)): so the property that
there is only one eigenstate of $T'_0-T$ which has the quantum numbers
$n_1,n_2$ and which is annihiilated by
$\r_+(-p),\r_-(p)$ is equivalent to the validity of the following identity
among power series:
%
$$\sum_{k=-\infty}^{+\infty}
z^{k^2}=\prod_{k=1}^\infty(1+z^{2k-1})^2(1-z^{2k})\Eq(12.a)$$
%
which is a well known identity about theta functions (see [GR], 8.180, 8.181).
Had we taken the Fermi
momentum to be $p_F=2\p n_F L^{-1}$ (instead of $p_F=2\p(n_F+1/2)L^{-1}$)
and performed consistently the above analysis, we would have found instead
of \equ(12.a) another remarkable identity:
%
$$\sum_{k=0}^\io z^{k(k+1)/2}\equiv \prod_{k=1}^\io(1+z^{k})^2(1-z^k).
\Eq(12.b)$$
%
\pagina\pgn=1
\vskip1.5truecm
{\it\S2 Schwinger functions}
\vglue1truecm\numsec=2\numfor=1
By repeating the classical analysis of [LW], one finds expressions for the
Schwinger functions of the Gibbs state at inverse temperature $\b$ for
the system confined in a box $[0,L]$ with periodic boundary conditions.
If $x\equiv (\xx,t)$, $\b>t_i>0$, $t_i\not=t_j$ if $i\not=j$, $\e_i=\pm 1$ and
$\{\pi(1),\ldots,\pi(n)\}$ is the permutation of $\{1,\ldots,n\}$ (with parity
$\s_\pi$) such that $\pi(1)>\pi(2)>\ldots>\pi(n)$, then:
$$\eqalign{
& S^{L,\b}(x_1,\oo_1,\e_1;\ldots;x_n,\oo_n,\e_n)=(-1)^{\s_\pi}
\left[ \hbox{\rm Tr}\,e^{-\b (H-E_0)} \right]^{-1}\cdot\cr
&\cdot\hbox{\rm Tr}\,e^{-(\b-t_{\pi(1)})(H-E_0)}
\ps{\e_{\pi(1)}}{\xx_{\pi(1)},\oo_{\pi(1)}}e^{-(t_{\pi(1)}-t_{\pi(2)})(H-E_0)}
\ldots \ps{\e_{\pi(n)}}{\xx_{\pi(n)},\oo_{\pi(n)}} e^{-t_{\pi(n)}(H-E_0)}
\cr } \Eq(2.1)$$
is the standard definition of the Schwinger functions, where $E_0$ is the
ground state energy.
Therefore, if $\ket{\O}$ denotes the ground state of $H$, it is:
%
$$\eqalign{
\lim_{\b\to\i} & S^{L,\b}(x_1,\oo_1,\e_1;\ldots;x_n,\oo_n,\e_n) \equiv
S_n^{L} =\cr
&=(-1)^{\s_\pi}\bra\O\ps{\e_{\pi(1)}}{\xx_{\pi(1)},\oo_{\pi(1)}}
e^{-(t_{\pi(1)}-t_{\pi(2)})(H-E_0)}
\ldots \ps{\e_{\pi(n)}}{\xx_{\pi(n)},\oo_{\pi(n)}}\ket\O\cr} \Eq(2.2)$$
The mean number of particles with momentum $\kk+p_F\oo$ and type $\oo$ can be,
consequently, evaluated as:
%
$$n_{\kk,\oo} = [\lim_{L\to\i} \, \lim_{\b\to\i} \, {1\over L}\ii d\xx d\yy
e^{i\kk(\xx-\yy)} S^{L,\b}((\xx,0^+),+,\oo);(\yy,0),-,\oo) ]\Eq(2.3)$$
%
where $0^+$ means that $0^+$ should be replaced by $t>0$ and then the limit of
the square bracket as $t\to 0$ has to be considered.
The r.h.s. of \equ(2.2) can be explicitely evaluated;
for example, for the model with $\r_i(\oo)=1/2$, $i=1,2$, one can show,
[Ma], that:
%
$$ S_n^{L} = e^{-Q_n^L} S_{0,n}^{L}\Eq(2.4)$$
%
where $S_{0,n}^L$ are the free Schwinger functions and:
%
$$\eqalign{
&Q_n^L (x_1,\oo_1,\e_1;\ldots;x_n,\oo_n,\e_n) =\cr
&\,\,= {2\pi\over L} \sum_{\pp>0}{1\over\pp} \sum_{\oo=\pm 1} \Bigl\{
s(\pp)^2 [{n\over 2}+2\sum_{i,j\in I_\oo \atop i -2\pi \Eq(2.7)$$
%
that we shall suppose satisfied in the following. We have seen that the
physical meaning of \equ(2.7) is simply that of the stability of the model,
(\ie boundedness from below of the energy spectrum, proportionally to the
number of particles and holes).
Denoting $S(x,\oo)\equiv \lim_{L\to\i}\lim_{\b\to\i}S^{L,\b}(x,\oo,-;
0,\oo,+)$ and $S_0(x,\oo)$ the corresponding free function, we find:
%
$$S(x,\oo)=S_0(x,\oo) e^{-Q(x)-R(x)-i\oo{t\over |t|}I(x)} \Eq(2.8)$$
%
with, see [Ma] for details:
%
$$\eqalign{
Q(x) &= \int_0^\i d\pp {2s(\pp)^2\over \pp}(1-e^{-\pp|t|/c_2(\pp)}
\cos\pp\xx) \cr
R(x) &= \int_0^\i d\pp {\cos\pp\xx\over \pp}
(e^{-\pp|t|}-e^{-\pp|t|/c_2(\pp)}) \cr
I(x) &= -\int_0^\i d\pp {\sin\pp\xx\over \pp}
(e^{-\pp|t|}-e^{-\pp|t|/c_2(\pp)}) \cr}\Eq(2.9)$$
and:
%
$$S_0(x,\oo)= {1\over (2\pi)^2} \int dk_0 d\kk
{e^{-i(k_0t+\kk\xx)} \over -ik_0+\oo\kk}
\,={1\over 2\pi}\,{1\over i\oo\xx+t}\Eq(2.10)$$
The \equ(2.9) and \equ(2.6) imply that $R(\xx,0)=I(\xx,0)=0$ and that $Q(\xx,0)\to
+\i$ as $|\xx|\to\i$ like $2\h \log|\xx|$, with, if $\l_1\equiv \l\vh(0)$:
%
$$\eqalign{
2\h &= 2[\sinh \f(0)]^2 =\cr
&= [(1+{\l_1\over 2\pi})^{1/2} + (1+{\l_1\over 2\pi})^{-1/2} -2]/2 =
{1\over 8}({\l_1\over 2\pi})^2 + \cdots \cr} \Eq(2.11)$$
%
This shows, using \equ(2.3) and \equ(2.8), that the occupation number
$n_{\kk,\oo}$ behaves, near $\kk=0$, \ie near the Fermi surface, as
$a-\e(\kk)\oo b|\kk|^ {\min\{2\h,1\}}$, with $\e(\kk)={\rm sign\,}\kk$
and $a,b$ two suitable positive numbers; hence we have no discontinuity
at the Fermi surface, if $\l_1\not= 0$, but just a singularity in the
derivatives of sufficiently high order, depending on the value of $\h$
(the first order if $2\h<1$). Note also that the stability condition
enters naturally in the solubility restriction \equ(2.7).
The \equ(2.9) and \equ(2.6) also imply that $R(\xx,t)+i\oo(t/|t|)I(\xx,t)$ and
$Q(\xx,t)$ behave respectively as $\log [(i\oo\xx+c_2(0)^{-1}t)/(i\oo\xx+t)]$
and $\h\log (\xx^2+c_2(0)^{-2}t^2)$, for $\xx^2+t^2 \to\i$, so that the
asymptotic behavior of $S(x,\oo)$ is:
$${1\over 2\pi}\,{1\over i\oo\xx+c_2(0)^{-1}t}\,
{A(\l_1)\over (\xx^2+c_2(0)^{-2}t^2)^\h} \Eq(2.12)$$
%
where $A(\l_1)$ is a constant such that $A(\l_1)\to 1$ as $\l_1\to 0$.
Formula \equ(2.12) can be written:
%
$$\hat S(p,\oo) \simeq B(\l_1) {|q|^{2\h} \over -i q_0 + \oo {\V q}}
\qquad\hbox{\it for\ } |p|\to 0 \Eq(2.12a) $$
%
where $q_0=p_0$ and ${\V q}= c_2(0)^{-1}\pp$. This implies that there is no
discontinuity at the Fermi surface and that the Fermi velocity is equal to
$c_2(0)^{-1}$, which goes to $1$ as $\l\to 0$. It is however possible to
consider a variation of the model \equ(1.1), such that the Fermi velocity
stays equal to $1$ for any $\l$. In fact, if we add a term $\bar\d\, T_0$ to
the Hamiltonian, the model is still exactly soluble and the Schwinger
functions are obtained from \equ(2.4),\equ(2.5) by the replacements [Ma]:
%
$$\eqalign{ & t \to (1+\bar\d)t \cr & c_2(\pp) \to
(1+{\l \hat{v}(\pp)\over 2\pi (1+\bar\d) })^{-1/2} \cr }\Eq(2.12b)$$
%
It is possible to choose $\bar\d$ so that:
%
$$\hat S(p,\oo) \simeq B(\l_1)|p|^{2\h}\hat S_0(p,\oo)
\qquad\hbox{\it for\ } |p|\to 0 \Eq(2.12c)$$
%
and $\bar\d$ is given by the condition [Ma]:
%
$$c_2(0)^{-1}(1+\bar\d)=1 \Eq(2.12d)$$
The exact solution \equ(2.4) allows us to deduce all the properties of the
Luttinger model. It is however interesting to investigate another
approach to the theory of the ground state, which does not rest in
principle on the solvability of the model and can then be extended to
more realistic examples.
Starting from the expression \equ(1.7) and going trough the well known
pattern of deductions used to set up the theory of the ground state as a
problem of the analysis of a suitable functional integral, one can easily
find a functional integral formulation of the Luttinger model.
If we introduce a family of grassmanian fields $\ps{\pm}{x,\oo}$, which we
denote with the same symbols already used for the Fermi field operators
(following a common practice, source of a lot of confusion), the \equ(2.1)
can be rewritten:
%
$$\eqalign{
S^{L,\b}(x_1,\oo_1,\e_1;\ldots;x_n,\oo_n,\e_n) & =
\X^{-1} \ii P_g^{L,\b}(d\psi) \ps{\e_1}{x_1,\oo_1} \ldots \ps{\e_n}{x_n,\oo_n}
e^{-V(\psi)} \cr \X & \equiv \ii P_g^{L,\b}(d\psi) e^{-V(\psi)} \cr
}\Eq(2.13)$$
%
where $V(\psi)$ is:
%
$$ \l\ii d\xx d\yy dt\; v(\xx-\yy) :(\sum_\oo q_{1,\oo}\ps{+}{\xx,t,\oo}
\ps{-}{\xx,t,\oo}):\,:(\sum_\oo q_{2,\oo} \ps{+}{\yy,t,\oo} \ps{-}{\yy,t,\oo}):
\Eq(2.14)$$
%
which is an element of the grassmanian algebra generated by
$\ps{\pm}{x,\oo}$ (hence it is not an operator), and the integrals
over $\psi$ in \equ(2.13) are defined by expanding $\exp[-V(\psi)]$ in powers of
$V(\psi)$, hence of $\psi$, and evaluating the integrals using the Wick rule
with field propagator vanishing for all the pairings except for those between
a $\ps{-}{}$ field and a $\ps{+}{}$ field; in the latter case the propagator
has the value:
$$\ii P_g^{L,\b}(d\psi) \ps{-}{x,\oo}\ps{+}{0,\oo}
= {\d_{\oo\oo'} \over (2\pi)^2} {\ii}^{(L,\b)} dk_0 d\kk
{e^{-i(k_0t+\kk\xx)} \over -ik_0+\oo\kk} \equiv \d_{\oo\oo'}
S_0^{L,\b}(x,\oo) \Eq(2.15)$$
%
where $(2\pi)^{-2}{\ii}^{(L,\b)}$ means $\sum_\kk\sum_{k_0}(L\b)^{-1}$, with
the sums running over the values $\kk=2\pi L^{-1}n$, $k_0=2\pi \b^{-1}
(m+1/2)$, $m$ and $n$ integers.
The latter structure of $k\equiv (\kk,k_0)$ means that
one should regard the inverse temperature interval $[0,\b]$ with antiperiodic
boundary conditions: $\ps{\pm}{\xx,0,\oo} = -\ps{\pm}{\xx,\b,\oo}$.
In the following sections we shall study, instead of the functions \equ(2.13),
the {\it truncated} Schwinger functions, which are simply related to them
and can be derived by a well known procedure from the {\it generating
function} $\SS^T(\f)$ in the following way (we suppress the indices $L,\b$):
%
$$\eqalign{
&S^T_{2n} (x_1,\oo_1;\ldots;x_n,\oo_n;y_1,\oo_1';\ldots;y_n,\oo_n')=
{\d^{2n} \SS^T(\f)\over \d\f^+_{x_1\oo_1} \ldots \d\f^-_{y_n\oo_n'}}
\Big|_{\f=0}\cr
&\SS^T(\f) \equiv \log\int
P_g(d\psi)e^{-V(\psi)+ (\f^+,\psi^-) + (\psi^+,\f^-)}
\cr} \Eq(3.19)$$
where $\f^\pm_{x\oo}$ are auxiliary grassmanian variables, anticommuting
also with the $\ps\pm{x\oo}$ fields, $\d$ denotes the formal functional
derivative which, togheter with the logarithm and exponential, is defined in
the sense of formal power series, and
$$(\f^+,\psi^-) \equiv \sum_\oo\ii dx \f^+_{x\oo}\ps-{x\oo} \Eq(3.19a)$$
The truncated Schwinger functions are
then constructed as power series in $\l$, whose terms are represented by
suitable Feynman graphs. As long as $L,\b<\i$, the series can be shown to be
convergent. One could then try to collect terms of the series so that the
limits $L,\b\to\i$ can be taken, using renormalization group techniques.
This is what we shall do, using however a different
infrared cutoff, which has a meaning only with respect to the representation
\equ(3.19) of the Schwinger functions ( it seems that our method does
not work with the original cutoff). In order to solve the problem we use
essential information from the fact that the model is exactly soluble.
However the results that we obtain can be easily extended to the {\it more
realistic} system of spinless electrons interacting with a symmetric
potential (which is not soluble), see [BG]. Furthermore there is an
intrinsic interest in the technique that we shall explain in the following
sections, because of the {\it anomalous scaling} \equ(2.12), which can be
completely understood in this model from the point of view of the
renormalization group.
\pagina\pgn=1
\fiat
\vskip1.5truecm
{\it\S3 Anomalous scaling and running couplings.}
\vglue1truecm\numsec=3\numfor=1
We consider the grassmanian integration with propagator:
%
$$\d_{\oo\oo'} {\ii} {dk_0 d\kk \over (2\pi)^2}
{e^{-i(k_0t+\kk\xx)} \over -ik_0+\oo\kk} \equiv \d_{\oo\oo'} g(x,\oo)
\Eq(3.1)$$
%
which is the limit of \equ(2.15) when $L,\b\to\i$.
The expression \equ(3.1) must be handled with care; in fact one can see that
the perturbative expansion of the Schwinger functions, expressed in terms of
Feynman graphs, can agree with the exact expression \equ(2.4) (in the
limit $L\to\i$) only if one
calculates each contribution with an ultraviolet cutoff on the space momentum,
$|\kk| \le 2^U p_0$, and then takes the limit $U\to\i$.
We now consider the {\it scaling decomposition}:
%
$$\eqalign{
g(x,\oo) &= \sum_{n=-\infty}^{1} g^{(n)}(x,\oo) \cr
g^{(n)}(x,\oo) &= \int_{p_0^{-2}2^{-2n-2}}^{p_0^{-2}2^{-2n}}d\a
\int {dk_0d^d\kk\over(2\p)^2}
e^{-i(k_0t+\kk\xx)-\a(k_0^2+\kk^2)}(ik_0+ \oo\kk),
\;\quad \; {\rm if}\, n\le0 \cr}\Eq(3.2)$$
%
while $g^{(1)}$ is given by the same integral over $\a$ with a different
domain, namely $\a\in[0,p_0^{-2}/4]$. Of course the remark which follows
\equ(3.1) affects only $g^{(1)}$, which represents the ultraviolet part of the
propagator.
We introduce, in correspondence with \equ(3.2), a sequence of grassmanian
fields $\ps{(n)\pm}{x,\oo}$ with propagators $\d_{n,n'}\d_{\oo\oo'}
g^{(n)}(x,\oo)$. The reason for introducing them, as well as the related
fields:
%
$$\ps{(\le h)\pm}{x,\oo} = \sum_{n=-\i}^h \ps{(n)\pm}{x,\oo}
\Eq(3.3)$$
%
is to define a recursive method to study the functional integrals
in \equ(2.13).
The {\it normal scaling approach} would simply be to use that
$P_g(d\psi) = \prod_{n=-\i}^1 P(d\ps{(n)}{})$, in the sense that the
integral of a function of $\psi$ is the same regarding $\psi$ as a
field with propagator $g$ or regarding it as $\psi=\ps{(\le 1)}{}$ via
\equ(3.3) and integrating over the various fields $\ps{(n)}{}$.The
integration should be done recursively over $\ps{(1)}{}, \ps{(0)}{},
\ps{(-1)}{}, \ldots$, trying to find recursive estimates. It will be
clear, however, that such approach is bound to fail. Therefore we set
up an anomalous scaling approach, as it has been done in the theory of
scalar fields in $4-\e$ dimensions, where a normal scaling approach
could not have worked.
We write $\psi=\ps{(1)}{}+\ps{(\le 0)}{}$ and perform the integration over
$\ps{(1)}{}$ defining:
%
$$e^{-\bar V^{(0)}(\ps{(\le0)}{})} \equiv \ii P(d\ps{(1)}{})
e^{-V(\ps{(1)}{} + \ps{(\le0)}{})} \Eq(3.4)$$
%
This is a preliminary step dealing with the ultraviolet part of the
propagator; it is a step which has no relation with the long range slow
decay of the propagator $g$, which is the main difficulty. Heuristically,
one expects that $\bar V^{(0)}$ is not very different from the original $V$.
This can be checked by studying the perturbation series for $\bar V^{(0)}$ in
powers of $\l$.
>From the point of view of field theory, the evaluation of $\bar V^{(0)}$ is a
problem of renormalizable type, then it is not trivial. However one can
easily show that the theory is divergence free (once $\n,\s$ are properly
chosen as in \S2) and that, to all orders of perturbation theory, $V^{(0)}$ is
an interaction containing terms of arbitrary degree in the fields, but with
coefficients decaying exponentially fast on the scale $p_0^{-1}$. We think it
possible to show that, for $\l$ small enough, one can sum the terms of the
same degree in the fields (there is some preliminary result in this direction
[Ge]). We therefore proceed by assuming that $V^{(0)}$ has the form
of a short range potential with many-body components (\ie terms containing any
number of $\ps{\pm}{}$ fields), becoming very small as the number of bodies
increases.
It is important to realize why it is inconvenient to break also $g^{(1)}$
into scales ranging from $p_0^{-1}$ to $0$ (in geometric progression with
ratio $2$); in fact at first sight this seems to provide the possibility of
a symmetric treatment of the problem in its ultraviolet and infrared parts.
But this would be illusory, for the simple reason that the interaction can
be regarded as short ranged only on scales $p_0^{-1}$ or larger. In the
ultraviolet scales the interaction is very long ranged and we should rather
treat it as a mean field. The only case in which it would seem reasonable
not to distinguish between ultraviolet and infrared scales is the case of a
delta function interaction (which has no scales intrinsic to it); this case
is, however, well known to be pathological [ML] and in our formalism it is
not even allowed because we suppose $p_0^{-1} <\i$. In fact the model with
the $\d$ interaction is equivalent to the Thirring model for a quantum
relativistic field theory and requires wave function renormalization to
remove the ultraviolet divergences (absent if the range $p_0^{-1}$ of the
potential is positive), see [K, Ma].
To perform the integrals over $\ps{(\le 0)}{}$ using an anomalous scaling
method, we introduce a sequence $Z_0,Z_{-1},\ldots$ of constants. While $Z_0$
is fixed to be $Z_0=1$, the others are left free to be determined inductively.
The choice $Z_j=1$ would give back the normal scaling procedure, but it will
not be our choice, although most of what we do holds also for this choice (but
the results are not useful, as it will appear).
In order to proceed we need:
\item{1)} the notion of relevant terms,
\item{2)} a more flexible notation for grassmanian integration.
The second point is an easy one; we denote $P_Z^{(h)}(d\psi)$ or
$P_Z^{(\le h)}(d\psi)$ the grassmanian integrations with propagators:
%
$$Z^{-1} g^{(h)}\qquad {\rm or} \qquad Z^{-1} g^{(\le h)} \Eq(3.5)$$
%
If we introduce the convolution operator $C_h$ with Fourier transform:
%
$$C_h(k)=e^{(k_0^2+\kk^2)2^{-2h}p_0^{-2}/4} \Eq(3.6)$$
%
we can write, formally:
%
$$P_Z^{(\le h)}(d\psi) \propto e^{-Z\sum_\oo \ii dx
\ps{+}{x,\oo}(\dpr_t + i\oo\dpr_\xx)C_h(\dpr)\ps{-}{x,\oo}}d\psi \Eq(3.7)$$
Coming to the notion of relevant operators, we consider a general element of
the grassmanian algebra and we define the operation $\LL$, the {\it
localization operation}, as follows. $\LL$ is a linear operator which
annihilates all monomials in the field operators of degree $>4$. Its
definition on the monomials of degree $4$ or $2$ is simply:
%
$$\eqalign{
&\LL \ps+{x_1\oo_1}\ps-{x_2\oo_2}=\ps+{x_1\oo_1}
(\ps-{x_1\oo_2}+(x_2-x_1)\dpr\ps-{x_1\oo_2}) \cr
&\LL \ps+{x_1\oo_1}\ps+{x_2\oo_2}\ps-{x_3\oo_3}\ps-{x_4\oo_4}=
2^{-1}\sum_{j=1,2}
\ps+{x_j\oo_1}\ps+{x_j\oo_2}\ps-{x_j\oo_3}\ps-{x_j\oo_4}\cr }\Eq(3.8)$$
This implies that the action of $\LL$ on a $V$ of the form:
%
$$\eqalign{
V(\psi) =&\sum_n \sum_{\oo_1,\ldots,\oo_n \atop \oo_1',\ldots,\oo_n'}
\ii W_n(x_1,\oo_1;\ldots;x_n,\oo_n;y_1,\oo_1';\ldots;y_n,\oo_n') \cr
& \ps{+}{x_1,\oo_1} \cdots \ps{+}{x_n,\oo_n} \ps{-}{y_1,\oo_1'} \cdots
\ps{-}{y_n,\oo_n'}\,dx_1\ldots dy_n \cr} \Eq(3.9)$$
%
gives a result which can be written, by collecting similar terms:
%
$$\eqalign{
\LL V(\psi) &= \l'\ii dx \ps{+}{x,+} \ps{+}{x,-} \ps{-}{x,-} \ps{-}{x,+}
+ \n'\sum_\oo \ii dx \ps{+}{x,\oo}\ps{-}{x,\oo} +\cr
& + \z'\sum_\oo \ii dx \ps{+}{x,\oo} \dpr_t \ps{-}{x,\oo}+
i\a'\sum_\oo \ii dx \ps{+}{x,\oo} \oo \dpr_\xx
\ps{-}{x,\oo}\cr}\Eq(3.10)$$
%
provided the $W$'s in \equ(3.9) are not too singular distributions.
To make precise what we mean by {\it not too singular} we introduce the
following fields:
$$\eqalign{
\ps{\pm}{x,\oo}\;,\quad \dpr \ps{\pm}{x,\oo} \quad & \cr
D^{\pm}_{x,y,\oo} = \ps{\pm}{x,\oo} - \ps{\pm}{y,\oo}\;,\quad &
S^1_{x,y,\oo} = \ps{-}{x,\oo}-\ps{-}{y,\oo} -(x-y)\dpr \ps{-}{y,\oo} \cr
S^2_{x,y,\oo} = \dpr \ps{-}{x,\oo} - \dpr \ps{-}{y,\oo}\;, \quad &
S^3_{x_1,x_2,x_3,x_4,\oo} = (x_3-x_4)S^1_{x_1,x_2,\oo} \cr
K^{(h)}_{x,\oo} = (\dpr_t + i\oo\dpr_\xx)&(1-C_h(\dpr))\ps{-}{x,\oo} \cr
}\Eq(3.11)$$
where $C_h$ is the operator in \equ(3.6).
We shall only consider $V$'s of the form \equ(3.9), which can be rewritten
as:
%
$$V(\psi)=\LL V(\psi)+\sum_n \ii d\xi_1 \ldots d\xi_n \tilde
W_n(\xi_1,\ldots,\xi_n) \F_{\xi_1}\ldots\F_{\xi_n} \Eq(3.12)$$
%
where $\F_\xi$ denotes one of the fields in \equ(3.11) and $\xi$ is
$(x,\oo)$ or $(x,y,\oo)$ or $(x_1,x_2,x_3,x_4,\oo)$ and $d\xi$ means
integration over the $x,y,\ldots$ coordinates and summation over the $\oo$
coordinates; furthermare the $\tilde W$ are products of ordinary smooth
kernels by suitable time delta functions.
We shall write the function $\bar V^{(0)}$ in \equ(3.4) as:
%
$$\bar
V^{(0)}(\psi)= \bar\z(\psi^+,(\dpr_t+i\oo\cdot\dpr_\xx)\,C_0(\dpr)\psi^-)+
V^{(0)}(\sqrt{Z_0}\psi)\Eq(3.12.1)$$
%
where $\bar\z$ is the coefficient of $(\psi^+,\dpr_t\psi^-)$ in the
expansion of $\bar V^{(0)}(\psi)$, and we set:
%
$$Z_0\equiv 1+\bar\z \Eq(3.12.2)$$
%
We can now set up a recursive procedure for the analysis of the integral
(which coincides with the $\X$ in \equ(2.13), because of
\equ(3.4) and the
last two definitions):
%
$$\ii P_{Z_0}^{(\le 0)}(d\psi) e^{-V^{(0)}(\sqrt{Z_0} \psi)} \Eq(3.13)$$
%
by writing $P_{Z_0}^{(\le 0)}(d\psi) = P_{Z_0}^{(0)}(d\bar\psi)
P_{Z_0}^{(\le -1)}(d\tilde\psi), \psi=\bar\psi + \tilde\psi$. Integrating over
$\bar\psi$ and using \equ(3.7), we write \equ(3.13) as:
%
$$\eqalign{
& \ii P_{Z_0}^{(\le -1)}(d\psi) e^{-\tilde V^{(-1)}(\sqrt{Z_0} \psi)}=\cr
& = {\rm const} \ii d\psi e^{-Z_0 \sum_\oo \ii dx \ps{+}{x,\oo}
(\dpr_t + i\oo\dpr_\xx) C_{-1}(\dpr) \ps{-}{x,\oo}}\;\cdot \cr
& e^{-\LL \tilde V^{(-1)}(\sqrt{Z_0} \psi) - (1-\LL) \tilde
V^{(-1)}(\sqrt{Z_0} \psi)}\cr} \Eq(3.14)$$
%
which we rewrite, using \equ(3.7), as:
%
$$\eqalign{
{\it const}& \ii P_{Z_{-1}}^{(\le -1)}(d\psi) e^{-(Z_0-Z_{-1}) \sum_\oo \ii dx
\ps{+}{x,\oo} (\dpr_t + i\oo\dpr_\xx) C_{-1}(\dpr) \ps{-}{x,\oo}} \;\cdot\cr
&\cdot e^{-\z' \sum_\oo \ii dx
\ps{+}{x,\oo} (\dpr_t + i\oo\dpr_\xx) C_{-1}(\dpr) \ps{-}{x,\oo} +\hbox{other
relevant terms} } \;\cdot \cr
&\cdot e^{-(1-\LL)\tilde V -\z'\sum_\oo \ii dx \ps{+}{x,\oo} K^{(0)}_{x,\oo}}
\cr }\EQ(3.15)$$
%
where {\it const\ } is a formally infinite but trivial constant, which we shall
neglect in the following, together with similar ones.
In the anomalous scaling procedure one chooses $Z_{-1}$ so that $Z_0-Z_{-1}
+\z'=0$, \ie the coefficient of
$\ii dx \ps{+}{x,\oo} (\dpr_t + i\oo\dpr_\xx) C_{-1}(\dpr) \ps{-}{x,\oo}$
vanishes and \equ(3.15) becomes (thus defining $V^{(-1)}$):
%
$$\ii P_{Z_{-1}}^{(\le -1)}(d\psi) e^{-V^{(-1)}(\sqrt{Z_{-1}}
\psi)}\Eq(3.16)$$
%
where $V^{(-1)}$ can be expressed in terms of the fields \equ(3.11) as in
\equ(3.12), with $h\ge-1$,
if $V^{(0)}$ was expressible in terms of them as in \equ(3.12). The latter
property is seen to hold order by order of perturbation theory.
%???? invece si?? (and no $K$-fields are needed in $V^{(0)}$, in fact).
The iteration produces a sequence $Z_0,Z_{-1},\ldots$, as well as a sequence
of potentials $V^{(h)}$ such that, up to a trivial constant: $$\ii
P_{Z_0}^{(\le 0)}(d\psi) e^{-V^{(0)}(\sqrt{Z_0} \psi)} = \ii P_{Z_{h}}^{(\le
h)}(d\psi) e^{-V^{(h)}(\sqrt{Z_{h}}\psi)}\EQ(3.16a)$$ and a sequence of
coefficients $\rr_h=(\n_h,\d_h,\l_h)$, called {\it running couplings}, which
are defined by writing:
%
$$\eqalign{
\LL V^{(h)} &= Z_h^2\l_h \sum_\oo\ii dx \ps{+}{x,+}\ps{+}{x,-}
\ps{-}{x,-}\ps{-}{x,+} +\cr
&+ Z_h i\d_h \sum_\oo\ii dx \ps{+}{x,\oo}\oo\dpr_\xx \ps{-}{x,\oo}
+ Z_h 2^h \n_h\sum_\oo\ii dx \ps{+}{x,\oo}\ps{-}{x,\oo}\cr} \Eq(3.17)$$
Furthermore the $\tilde W_h$-functions, appearing in the expansion of $(1-
\LL)V^{(h)}$ in powers of the fields, are also produced as formal power
series in $\rr_{h+1},\ldots,\rr_0$. No term proportional to $\ii dx\,
\ps{+}{x,\oo} \dpr_t \ps{-}{x,\oo}$ appears in \equ(3.17), because of our
definition of the sequence $Z_h$. Finally, using the oddness of the
propagator, it is easy to see that, for each $h$:
%
$$\n_h=0 \EQ(3.17a)$$
%
This property of the potential is related to the fact that, in the Luttinger
model, the interaction does not modify the position of the Fermi surface, so
that we have effectively only two running couplings.
Of course one could envisage other prescriptions to construct the sequence
$Z_j$, but it will appear that only one of them has the possibility of being
applicable to our problem, namely the just illustrated anomalous scaling
choice.
On heuristic grounds we expect that an asymptotic behaviour of the running
couplings like:
%
$$a)\,\cases{Z_h=z 2^{-2\h h}\cr\n_h\to0\cr\d_h\to0\cr\l_h\to\l_{-\io}\cr}
\qquad b)\,\cases{|\n_h|,|\d_h|,\,|\l_h|\,< C_0 |\l_0|\cr
e^{-q \e|\l_0|}<|{Z_{h+q}\over Z_h}|0$ and independent of $h$, and if $\lim_{h\to -\infty}
|h|^{-1}\log Z_h =\tilde \h >0$, then the asymptotic behaviour, for $h\to
-\infty$, of $\hat S_2(k,\oo)$ is of the type: $$\hat S_2(k,\oo)\simeq
{|k|^{\tilde\h} \over (-ik_0+\oo\kk)}\left\{b(\bar k)+ {\oo\kk \over
-ik_0+\oo\kk}[\d_h + a(\bar k)]\right\}
\Eq(3.30)$$
We can not compare \equ(3.30) with the asymptotic behaviour of the pair
correlation \equ(2.12a), because our renormalization procedure
fixed the Fermi velocity to $1$, which is not the value in the model
\equ(1.1).
Instead of modifying the renormalization procedure, we choose to study the new
model discussed after \equ(2.12a), obtained by adding a term $\bar\d T$ to the
Hamiltonian, so that the Fermi velocity is fixed to $1$, independently of
$\l$. Then, by comparing \equ(2.12c) and \equ(3.30), we see that
$b(\bar k)$ is a function of $|\bar k|$, $a(\bar k)=0$ (it should be
possible to derive directly these two results, but we did not do that) and
%
$$\d_h \to 0 \qquad {\rm as\ \ }h\to -\i \Eq(3.30a)$$
$$\tilde\h = 2\h = 2[\sinh \f(0)]^2 \Eq(3.30b)$$
%
A similar discussion for the four fields Schwinger function yields a similar
result, that is:
%
$$\eqalign{
&\hat S^T_4(p_1,+,p_2,-;p_3,+,p_4,-) = -\d(p_1+p_2-p_3-p_4)\cdot\cr
&\cdot g^{(>h)}(p_1,+) g^{(>h)}(p_2,-) g^{(>h)}(p_3,+) g^{(>h)}(p_4,-)
{1\over Z_h^2} \hat W_4^{(h)}(p_1,p_2,p_3) \cr}\Eq(3.31)$$
%
where, for momenta of order $2^h$:
$$\hat W_4^{(h)}(p_1,p_2,p_3) = \l_h + \bar W_4^{(h)}
(2^{-h}p_1,2^{-h}p_2,2^{-h}p_3) \Eq(3.32)$$
with $\bar W_4^{(h)}$ having a smooth limit as $h\to -\i$ and being of the
second order in the running couplings.
The asymptotic behaviour of the l.h.s. in \equ(3.31) can be calculated from
the exact solution and one can see [Ma] that it is compatible with \equ(3.31)
and \equ(3.32) only if $\l_h$ has a finite limit as $h\to-\i$:
$$\l_h\to \l_\i(\l_1) \Eq(3.34)$$.
\pagina\pgn=1
\vskip1.5truecm
{\it\S5 The beta function.}
\vglue1truecm\numsec=5\numfor=1
The above analysis not only permits us to define the running couplings
$\rr_h$ and the scalings $Z_h$, \equ(3.26),
but also to find an expression of $\rr_{h-1}, Z_{h-1}/Z_h$
in terms of $\rr_{\ge h}, Z_{\ge h}/Z_h$.
The latter can be studied from the explicit expressions of $\rr_h$ in terms
of the Feynman graphs of the model, which are constructed from the formal
integration formula:
%
$$ e^{\bar V(\psi)} \equiv \ii P(d\bar\psi) e^{V(\psi+\bar\psi)} =
\exp \sum_{n=1}^\i {1\over n!} \ET (V;\ldots;V) \Eq(4.1)$$
%
where $\ET$ denotes the truncated expectation with respect to the
integration over $\bar\psi$. The latter is defined simply by imposing that,
in the evaluation of the integrals $\E(V^n)$ with the Wick rule, only some
terms are to be retained. Namely, if we think all the fields appearing in a
monomial in one of the $V$ factors as lines and we represent a Wick
contraction by suitably joining together pairs of lines, then we only
retain terms corresponding to Wick contractions generating a connected
graph of lines.
The theory of the estimates of the series expansion of $\rr_{h-1},
Z_{h-1}/Z_h$ in powers of $\rr_{\ge h}, Z_{\ge h}/Z_h$ is technically
involved, see [BG], where the main result is that the formal power series
has coefficients of order $n$ bounded uniformly in the scale parameter $h$,
provided for some $\x>0$:
%
$$e^{-q\x}<|Z_{h+q}/Z_h|0\Eq(4.1.1)$$
%
by the bound:
%
$$D_\x C_\x^{n-1}(n-1)!\Eq(4.2)$$
%
for some $C_\x,D_\x$ and all $h\le0$.
In fact the model is technically very similar to the one flavour
Gross-Neveu model and it seems reasonable to us that one could improve
\equ(4.2) by taking out the $n!$ from the bounds.
It is in fact possible to prove [GS],
[BGPS], that there is a convergent power series expansion:
%
$$\eqalign{
\rr_{h-1}=&\L \rr_h+B'_h(\rr_h,Z_{h+1}/Z_h,\rr_{h+1},\ldots,Z_0/Z_h\rr_0)\cr
{Z_{h-1}\over Z_h}=&
(1+B"_h(\rr_h,Z_{h+1}/Z_h,\rr_{h+1},\ldots,Z_0/Z_h\rr_0)\cr
}\Eq(4.2*1)$$
%
where $\L$ is a $3\times3$ diagonal matrix, with elements $1,1,2$, see below,
and with the functions $B^\s_h$ being holomorphic when all their arguments
$\rr$-arguments are in
a small enough disk with $h$-independent radius while the arguments
$Z_{h+q}/Z_h,\; q\ge0$ vary in an annulus like \equ(4.1.1) for some $\x$
small enough. Furthermore the limit
$\lim_{h\to\io}B^s_h$ converging to a holomorphic function of infinitely many
variables $B(\V z^{1},\th_1,\V z^{2},\th_2,\ldots)$, holomorphic
in a disk of some radius $\r>0$ for the $\V z$ variables and in an annulus
like \equ(4.1.1) for the $\th_q$ variables with some $\x>0$. So
that if $\lim\rr^{h}=\rr^{-\io}$ and $\lim Z_{h+q}/Z_h=\th_q$
exist then $\rr^{-\io}$ is a fixed point of the
relation $\rr^{-\io}=\L\rr^{-\io}+B_\io(\rr^{-\io})$
where $B_\io(\rr)$ is defined
by setting $B_\io(\rr)=B'(\rr,\th_1,\rr,\th_2,\rr,\ldots)$.
For this reason we shall call {\it beta function} the function of three
complex variables $\L\V z+B_\io(\V z)$, while we call {\it beta functional}
the functions $B^s_h$ in \equ(4.2*1), depending on $h$ arguments. The beta
function in the above sense is a function whose fixed points are the limit
values of the running couplings $\rr_h$ of our model.
In the literature one also considers often the function relating
$\rr_{h-1}$ to $\rr_h$: it follows from the above that the latter also has
a well defined expansion around $\rr_h=\V0$ but
its coefficients grow as $n!$ with the order, hence it is not a priori well
defined, and it seems to us that even if it is well defined it will be such
because of non generic cancellations, absent in the case of spin non zero,
for instance. The proof of the above convergence properties can be found in
[GeS],[BGSP]. Hence we shall assume them and study their implications.
We stress, before continuing, that the above results would also hold if one
used the normal scaling procedure. The bounds \equ(4.2) and our convergence
conjecture hold also in the normal scaling approach. The importance of the
scaling does not come in at this point, yet.
If $\l_{\ge h}$ denotes the sequence $\l_h,\l_{h+1},\ldots,\l_0$ and a
similar notation is adopted for $\d_{\ge h},\n_{\ge h}$, then the
computation, via the Feynman graphs, of the running couplings leads to the
following:
%
$$\eqalignno{ \l_{h-1}=&({Z_h/ Z_{h-1}})^2
\bigl[\l_h+\l_h^3B_1(\l_{\ge h})+\d_h\l_h^3B_2(\l_{\ge h},\d_{\ge h})+\cr&
+\l_h^2\n_h^2B_3(\l_{\ge h},\d_{\ge h},\n_{\ge h}) +2^h\bar R_1(\l_{\ge
h},\d_{\ge h},\n_{\ge h},2^h)\bigr]\cr \d_{h-1}=&(Z_h/
Z_{h-1})[\d_h+\l_h^2\d_hB_4(\l_{\ge h})+\n_h^2{B_5}(\l_{\ge h},\d_{\ge
h},\n_{\ge h})+\cr& 2^h\bar R_2(\l_{\ge h},\d_{\ge h},\n_{\ge
h},2^h)\bigr]\cr \n_{h-1}=&2(Z_h/ Z_{h-1})[\n_h+\n_h\l_h^2B_6(\l_{\ge
h},\d_{\ge h},\n_{\ge h})+&\eq(4.3)\cr &+2^{h}\bar R_3(\l_{\ge h},\d_{\ge
h},\n_{\ge h},2^h)\bigr]\cr 1=&(Z_h/ Z_{h-1})[1+\l_h^2B_{8}(\l_{\ge h})+\cr
&+\d_h\l_h^2B_{9}(\l_{\ge h},\d_{\ge h})+ \l_h^2\n_h^2 B_{10}(\l_{\ge
h},\d_{\ge h},\n_{\ge h})+2^h \bar R_4(\l_{\ge h},\d_{\ge h},\n_{\ge
h},2^h)\bigr]\cr}$$
%
where all the $B_j$ functions do depend also on the ratios $Z_{h+q}/Z_h,\,
q\ge0$, as discussed above, but such dependence is noty explicitly indicated
to simplify the notation. Furthermore we have computed a little more
carefully the lowest terms to find out the minimal power to which each
running constant is raised; in particular we have used the following facts:
\item{a)} the graphs containing two $\l_h$-vertices and any number of
$\d_h$-vertices cancel out; \item{b)} since the propagator is an odd
function of $x$, in the equation for $\n_{h-1}$ there is no contribution due
to graphs containing only $\l_h$-vertices (and therefore an odd number of
innner lines) or containing only $\l_h$- and $\d_h$-vertices (a
$\d_h$-vertex does not change the parity of the graph).
As we have stressed in the previous section, $\n_h$ is exactly zero in the
model \equ(1.1), so we could cancel out the third equation in \equ(4.3).
However we prefer to study the complete set of equations \equ(4.3), since they
are valid also in the model with an ordinary kinetic energy, where $\n_h$ is
not zero, see [BG].
As a consequence of the discussion preceding \equ(4.3), the functions
$B_j,\bar R_j$ should be analytic in their arguments $\l_{\ge h},
\d_{\ge h},\n_{\ge h}$ (with a suitably small radius $M$ of convergence)
and in $Z_{h+q}/Z_h,\, q>0$ (in a suitably thin annulus around
the unit circle, see \equ(4.1.1)).
Furthermore the $B_j$ can be shown to have a limit as $j\to-\io$ while the
$\bar R_j$ terms disappear in this limit. The $\bar R_j$ vanish to second
order in $\l_h,\d_h,\n_h$, see [BGPS].
Had we used the normal scaling approach, we would have found an equation
like \equ(4.3) with $\d_h$ (or $\d_{\ge h}$) replaced by a pair
$(\a_h,\z_h)$ of constants (or by $(\a_{\ge h},\z_{\ge h}$), representing
the coefficients of $\ii dx \ps{+}{x,\oo} i\dpr_\xx \ps{-}{x,\oo}$ and
$\ii dx \ps{+}{x,\oo}\dpr_t \ps{-}{x,\oo}$, and each of the two new
relations would have had a non vanishing term proportional to $\l_h^2$. The
reason why such term is missing in \equ(4.3) is precisely due to our
definition of anomalous scaling combined with the symmetry in the
propagator between $\xx$ and $t$, which makes identical the contributions
to the variations of $\a_h$ and $\z_h$ due to graphs only involving
$\l_h$-vertices, hence it makes idenically identically zero the
contributions to $\d_h$ of the same graphs.
It is convenient to eliminate completely the factors $Z_h/ Z_{h-1}$ from
\equ(4.3), using the last of \equ(4.3) and expanding the denominators in
power series:
%
$$\eqalignno{ \l_{h-1} = & \l_h+\l_h^3G_1(\l_{\ge h})+
\d_h \l_h^3G_2(\l_{\ge h},\d_{\ge h})+\n_h^2\l_h^2 G_3(\l_{\ge h},\d_{\ge
h},\n_{\ge h})+\cr
+& t_h R_1(\l_{\ge h},\d_{\ge h},\n_{\ge h},t_h)\cr
\d_{h-1}= & \d_h+\l_h^2\d_h G_4(\l_{\ge h},\d_{\ge h})+
\n_h^2G_5(\l_{\ge h},\d_{\ge h},\n_{\ge h})+t_h R_2(\l_{\ge h},\d_{\ge
h},\n_{\ge h},t_h)\cr
\n_{h-1}= & 2\n_h+
\n_h\l_h^2G_6(\l_{\ge h},\d_{\ge h},\n_{\ge h}) +\cr
& + t_hR_3(\l_{\ge h},\d_{\ge h},\n_{\ge h},t_h) & \eq(4.4)\cr
t_{h-1}=&2^{-1}t_h\cr}$$
%
having set $t_h=2^h$, and not having once more written explicitly the
dependence of the $G_j$ on the variables $\th_{q,h}=Z_{h+q}/Z_h,\, q>0$.
The relation \equ(4.4), defining the beta functional, does not permit us to
infer much about the properties of the model; but we can derive extra
information about the $G_j$ functions from the fact that the model is
exactly soluble.
Let us assume:
\item{1) } that the flow \equ(4.4) admits, for each $\l_0\not=0$ small
enough, initial values $\d_0(\l_0)$, $\n_0(\l_0)$ such that:
%
$$\eqalign{
&\d_h,\n_h \to 0 \quad,\quad \l_h\to \l_\i(\l_0) \cr &Z_h\simeq2^{-\h h}
\quad,\quad \h=O(\l_0^2)>0 \cr &|\rr_h| \le C_0|\l_0| \cr} \Eq(4.5)$$
%
where $\simeq$ means that the logarithms of both sides, divided by $|h|$,
have the same limit (see \equ(3.22)), and $\l_{-\io}(\l),\h(\l)$ being
analytic near $\l=0$.
Call $\bar G_1(\l,\h)\equiv\lim_{h\to-\io} G_1(\l_{\ge h})$,
with $\l_j\equiv\l$ and $\th_{q,j}=2^{-2\h q}$,
and let us suppose that:
\item{2) } the function $G_1$ in the first equation of \equ(4.4)
is analytic and not identically zero.
The assumption \equ(4.5) immediately
implies that $\bar G_1(\l_\i,\h)=0$; then $\l_\i$ is independent of $\l_0$, as
a consequence of the analyticity hypothesis. But the hypotheses \equ(4.5) and
imply that $|\l_\i|\le C_0|\l_0|$ has
to hold for all $\l_0$ small enough, so $\l_\i=0$ and the fourth of
\equ(4.3) tells us that $Z_h/Z_{h-1} \to 1$, which is incompatible with
$\h>0$.
In conclusion, if the assumptions 1), 2) above are satisfied:
%
$$\bar G_1(\l)=0 \Eq(4.6)$$
This makes more precise the argument leading to the conjecture given in [BG].
A similar property has been proposed in [DM], supported by a symmetry
argument.
We now observe that 1), \ie \equ(4.5), should be deducible from
the exact solution of the model, using \equ(2.11), \equ(3.23a), \equ(3.25a).
Moreover $\l_\i=\l_0+O(\l_0^3)$, so that, if
$\l_0$ is small, also $\l_\i$ is small.
Also 2) should be provable by known techniques, as discussed above.
Then, by the previous discussion,
our basic result is that the main term in the beta function
not only is zero to second order, where it is easily calculated, but
vanishes to all orders. We have checked by explicit calculation that
\equ(4.6) is verified also to third order [Ma].
It is remarkable that \equ(4.6) holds: in fact as it can be used in other
models which are not exactly soluble, but which can be shown to have the
same $G_i$ functions. One case, see [BG], is the model of one spinless
species of fermions interacting via a short range interaction and with an
ordinary kinetic energy (namely $(\kk^2-p_F^2)/2m$).
\vskip3truecm
{\it References.}
\vglue1.truecm
\halign{[#]& \vtop{\hsize=14.truecm\\#}\cr
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Ma&{Mastropietro, V.: tesi di laurea in Fisica, 1990 and preprint.}\cr
ML&{Mattis, D., Lieb, E.: {\it Exact solution of a many fermion system
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}
\ciao
ENDBODY