#  Vanishing of Drude weight in interacting fermions on $\hbox{\msytw Z}^{d}$
with quasi-periodic disorder

Vieri Mastropietro  University of Milano, Department of Mathematics “F.
Enriquez”, Via C. Saldini 50, 20133 Milano, Italy

###### Abstract

We consider a fermionic many body system in $\hbox{\msytw Z}^{d}$ with a short
range interaction and quasi-periodic disorder. In the strong disorder regime
and assuming a Diophantine condition on the frequencies and on the chemical
potential, we prove at $T=0$ the exponential decay of the correlations and the
vanishing of the Drude weight, signaling Anderson localization in the ground
state. The proof combines Ward Identities, Renormalization Group and KAM
Lindstedt series methods.

##  1 Introduction

The conductivity properties in fermionic systems, describing electrons in
metals, are strongly affected by the presence of disorder, which breaks the
perfect periodicity of an ideal lattice and is unavoidable in real systems.
Disorder can be represented either by a random variable or by a quasi-periodic
potential; the first description is more suitable for impurities in solids
while the second appears naturally in quasi-crystals or cold atoms
experiments. In absence of many body interaction disorder produces the
phenomenon of Anderson localization [1], consisting in an exponential decay of
all eigenstates and in an insulating behavior with vanishing conductivity.
Such a phenomenon relies on the properties of the single particle Schroedinger
equation and it has been the subject of a deep mathematical investigation.
With random disorder Anderson localization was established for strong disorder
in any dimension [2], [3] and in one dimension with any disorder. In the case
of quasi-periodic disorder localization in one dimension is present only for
large disorder [4], [5], while for weak disorder is absent; in higher
dimensions localization was proved for strong disorder in $d=2$ [6], [7] and
for any $d$ in [8].

The interplay between disorder and interaction has been deeply analyzed in the
physical literature soon after [1]. The presence of many body interaction
induces new processes which can indeed destroy localization. At zero
temperature $T=0$ with random disorder qualitative scaling arguments gave
evidence of persistence of localization in $d=3$ [9], [10] for short range
weak interaction; in $d=1$ a second order Renormalization Group analysis was
shown to produce a complex phase diagram [11]. The case of quasi-random
disorder has been less studied, with the exception of [12], [13] focusing on
the extended weak disorder regime at $T=0$. In more recent times the
properties at $T>0$ were analyzed in [14], where perturbative arguments for
the vanishing of conductivity up to a certain critical $T$ in any dimension
were given (many body localized phase). Subsequently numerical simulations
found localization in certain systems in all the spectrum and vanishing of
conductivity for any $T$, a phenomenon called many body localization, see [15]
for random and [16] for quasi-periodic disorder. If all states are localized
one expects, in a non-equilibrium setting, that interaction is unable to
produce thermalization in an isolated quantum system, a phenomenon that in
classical mechanics is due to closeness to an integrable system. Interacting
quantum systems with quasi-periodic disorder have been realized in cold atoms
experiments [17], [18],[19] ; quasi-periodic disorder with many body
interaction has been extensively numerically analyzed [20]-[28].

While the above works suggest that localization persists in presence of
interaction, results based on numerical or perturbative analysis cannot be
conclusive. In particular the presence of small divisors has the effect that
physical informations are difficult to be extracted by lower order analysis
but are typically encoded in convergence or divergence of the whole series.
This is a well known phenomenon in classical mechanics; the Birkoff series for
prime integrals in Hamiltonian systems are generically diverging while
Lindsdtet series for Kolomogorov-Arnold-Moser (KAM) tori converge, even if
both series are order by order finite and present similar small divisors.
Therefore, even if perturbative analysis in [14] or [29] get localization at
finite temperature and in any dimension, one cannot exclude that the series
are divergent and localization eventually disappear (this would say that
thermalization in experiments is eventually reached, even if at long times). A
non-perturbative proof of many body localization for all eigenstates has been
indeed finally obtained in $d=1$ with random disorder in [30] but the result
is based on a certain unproven assumption. A complete proof have been obtained
only with vanishing densities [31], [32]. Arguments for breaking of many body
localization in $d>1$ have been indeed presented in [33].

In order to get rigorous results as benchmark for conjectures and
approximations, a natural starting point is the zero temperature case in the
thermodynamic limit. Our approach is to compute thermodynamical correlations;
they not only provide physical observables at equilibrium but give also
information on the spectrum (so their computation is of interest even for
situation where equilibrium is not reached). In particular at zero temperature
they provide information of correlations over the ground state, while the
vanishing of conductivity at any temperature is a signal of many body
localization in all the spectrum. It has been proven in [34],[35],[36] for one
dimensional interacting fermions with strong quasi-periodic disorder the $T=0$
exponential decay of 2-point correlations, indicating persistence of
localization in the ground state. Aim of this paper is twofold. The first is
to investigate the $d>1$ case. We consider a disorder of the form
$f(\vec{\omega}\vec{x})$ with $f$ periodic, as the one considered in [6] for
the single particle Schroedinger equation ; more general forms of disorder are
however possible, as $f(\vec{\omega}_{1}\vec{x},\vec{\omega}_{2}\vec{x})$
considered in [6]. The second aim is to compute the $T=0$ conductivity
expressed by Kubo formula, whose properties can be analyzed via a combination
of information provided by Ward Identities with regularity properties of the
current correlations. The thermodynamical quantities are expressed by a series
expansion showing a peculiar combinations of properties appearing in classical
and quantum physics; they show a small divisor problem, as in the Lindstedt
series for KAM [37], but loop graphs appear in the expansion, a signature of
quantum physics totally absent in classical mechanics. In order to achieve
convergence and exclude non perturbative effects one has from one side to show
that divisors can be controlled by number theoretical conditions on
frequencies, and from the other that the huge number of loop graphs is
compensated by cancellations from the fermionic anticommutative nature of the
problem.

The paper is organized in the following way. In §2 the model is presented and
in §3 the main results, together with open problems, are presented. In §4 we
discuss the implications of Ward Identities and regularity bounds. In §5 we
introduce the Grassmann representation and in §6 we introduce the multiscale
analysis. In §7 we prove the convergence of series expansion and in §8 we get
the asymptotic decay of correlations.

##  2 Interacting fermions with quasi-periodic disorder

We introduce the Fock space $\mathcal{F}_{L}=\bigoplus_{N\geq
0}\mathfrak{h}_{L}^{\wedge N}$ where the $N$ particle Hilbert space
$\mathfrak{h}_{L}^{\wedge N}$ is the set of the totally antisymmetric square
integrable functions in
$\Lambda_{L}:=\\{\vec{x}\in\mathbb{Z}^{d}\mid\vec{x}=n_{1}\vec{e}_{1}+n_{2}\vec{e}_{2}+...\;,\quad-L/2\leq
n_{i}\leq L/2\;,\quad i=1,2,..,d\\}$ where $\vec{e}_{i}$ are unit vectors. The
$a^{\pm}_{\vec{x}}$ are fermionic creation or annihilation operators sending
an element of $\mathfrak{h}_{L}^{\wedge N}$ in $\mathfrak{h}_{L}^{\wedge N+1}$
(creation) or $\mathfrak{h}_{L}^{\wedge N-1}$ (annihilation) and
$\\{a^{+}_{\vec{x}}\,,a^{-}_{\vec{y}}\\}=\delta_{\vec{x},\vec{y}}$,
$\\{a^{+}_{\vec{x}}\,,a^{+}_{\vec{y}}\\}=\\{a^{-}_{\vec{x}}\,,a^{-}_{\vec{y}}\\}=0$.
The Hamiltonian is

$H=-{\varepsilon\over
2}\sum_{\vec{x}}\sum_{i=1}^{d}(a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}}+a^{+}_{\vec{x}}a^{-}_{\vec{x}+\vec{e}_{i}})+u\sum_{\vec{x}}\phi_{\vec{x}}a^{+}_{\vec{x}}a^{-}_{\vec{x}}+\lambda\sum_{\vec{x}}\sum_{i=1}^{d}a^{+}_{\vec{x}}a^{-}_{\vec{x}}a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}+\vec{e}_{i}}$
(1)

where $a^{+}_{\vec{x}}$ must be interpreted as zero for
$\vec{x}\not\in\Lambda_{L}$ and
$\phi_{\vec{x}}=\bar{\phi}(\vec{\omega}\vec{x})$ with
$\bar{\phi}(t):\hbox{\msytw T}\rightarrow\hbox{\msytw R}$ periodic of period
$1$. In order to describe a quasi-periodic disorder we impose that
$\vec{\omega}$ is rationally independent and ”badly” approximated by rationals
(Diophantine condition). The first term in (1) represents the kinetic energy
of the fermions hopping on a lattice, the second represents the interaction
with a quasi-periodic potential and the last term represents a 2 body
interaction.

There are several interesting limits; $\lambda=0$ is the non interacting
limit; $\lambda=u=0$ is the integrable limit;ùù $\lambda=\varepsilon=0$ is the
anti-integrable limit (the therminology was introduced in [38] ). We consider
the case in which $\lambda,\varepsilon$ are small with respect to $u$, and we
set $u=1$ for definiteness; that is we consider a perturbation of the anti-
integrable limit.

If $N=\sum_{\vec{x}}a^{+}_{\vec{x}}a^{-}_{\vec{x}}$ we define

$\langle\cdot\rangle_{\beta,L}=\frac{\mathrm{Tr}_{\mathcal{F}_{L}}\cdot
e^{-\beta(H-\mu
N)}}{\mathcal{Z}_{\beta,L}}\;,\qquad\mathcal{Z}_{\beta,L}=\mathrm{Tr}_{\mathcal{F}_{L}}e^{-\beta(H-\mu
N)}$ (2)

where $\mu$ is the chemical potential, which is fixed by the density in the
Grand-Canonical ensamble, and $\mathcal{Z}_{\beta,L}$ is the partition
function. In the limit $\beta\rightarrow\infty$ they provide information on
the ground states. We define

$\langle\cdot\rangle=\lim_{\beta\rightarrow\infty}\lim_{L\rightarrow\infty}\langle\cdot\rangle_{\beta,L}$
(3)

The imaginary-time (or Euclidean) evolution of the fermionic operators is

$a^{\pm}_{{\bf x}}=e^{x_{0}(H-\mu N)}a^{\pm}_{\vec{x}}e^{-x_{0}(H-\mu N)}$ (4)

with ${\bf x}=(x_{0},\vec{x})\quad\text{with}\quad x_{0}\in[0,\beta)$, The
2-point function is given by

$S_{\beta,L}({\bf x},{\bf y})=\left\langle Ta^{-}_{\bf x}a^{+}_{\bf
y}\right\rangle_{\beta,L}$ (5)

and $T$ is the time order product. We also consider the truncated expectations
$\left\langle TA;B\right\rangle_{\beta,L}=\left\langle
TAB\right\rangle_{\beta,L}-\left\langle TA\right\rangle_{\beta,L}\left\langle
TB\right\rangle_{\beta,L}$. The density and the current are given by

$\rho_{\vec{x}}=a^{+}_{\vec{x}}a^{-}_{\vec{x}}\quad\quad
j^{i}_{\vec{x}}={\varepsilon\over
2i}(a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}}-a^{+}_{\vec{x}}a^{-}_{\vec{x}+\vec{e}_{i}})$
(6)

The (Euclidean) conductivity density in the zero temperature limit is defined
by Kubo formula

$\sigma^{i}_{\vec{y}}=\lim_{p_{0}\rightarrow 0}{1\over
p_{0}}\lim_{\beta\rightarrow\infty}\lim_{L\rightarrow\infty}[\sum_{\vec{x}\in\Lambda_{L}}\int_{0}^{\beta}dx_{0}e^{ip_{0}x_{0}}\left\langle
Tj^{i}_{\vec{x},x_{0}};j^{i}_{\vec{y},0}\right\rangle_{\beta,L}+<\tau^{i}_{\vec{y}}>_{\beta,L}]$
(7)

where

$\tau^{i}_{\vec{y}}=-{\varepsilon\over
2}(a^{+}_{\vec{y}+\vec{e}_{i}}a^{-}_{\vec{y}}+a^{+}_{\vec{y}}a^{-}_{\vec{y}+\vec{e}_{i}})$
(8)

The conductivity can be equivalently expressed in terms of the Fourier
transform which is, in the $\beta\rightarrow\infty,L\rightarrow\infty$ limit ,
$i=1,,d$

$\widehat{H}_{ii}({\bf p},\vec{y})=\sum_{\vec{x}\in\Lambda}\int_{\hbox{\msytw
R}}dx_{0}e^{i{\bf p}{\bf x}}<Tj^{i}_{\vec{x},x_{0}0};j^{i}_{\vec{y},0}>$ (9)

and similarly we define $\widehat{H}_{\mu\nu}({\bf p},\vec{y})$, with
$\mu=0,1,...d$ ($\mu=0$ is the density and $\mu=1,...,d$ the current
component). We can rewrite (7) as

$\sigma^{i}_{\vec{y}}=\lim_{p_{0}\rightarrow 0}\lim_{\vec{p}\rightarrow
0}{1\over p_{0}}[\widehat{H}_{ii}({\bf p},\vec{y})+<\tau^{i}_{\vec{y}}>]$ (10)

Finally the (zero temperature) Drude weight, see eg [39], [40] , is defined as

$D^{i}_{\vec{y}}=\lim_{p_{0}\rightarrow 0}\lim_{\vec{p}\rightarrow
0}[\widehat{H}_{ii}({\bf p},\vec{y})+<\tau^{i}_{\vec{y}}>]$ (11)

In a perfect metal at equilibrium the Drude weight is non-vanishing implying
that the conductivity is infinite; a vanishing Drude weight signals a non-
metallic behavior.

In the above definitions of conductivity the order in which the limits are
taken is essential; already in the integrable limit $u=\lambda=0$ reversing
the order of the limits one obtains a zero result, while the Drude weight is
indeed non vanishing as a consequence of the non-continuity of the Fourier
transform of the current correlation.

##  3 Main result

In the anti-integrable limit $\lambda=\varepsilon=0$ the eigenvalues of the
Hamiltonian are, $\vec{x}\in\Lambda_{L}$

$H_{0}=\sum_{\vec{x}\in\Lambda_{L}}\bar{\phi}(\vec{\omega}\vec{x})n_{\vec{x}}\quad\quad
n_{\vec{x}}=0,1$ (12)

and the single particle eigenfunctions have the form of
$\delta_{\vec{x},\vec{y}}$. The 2-point function is given by

$g({\bf x},{\bf
y})=\delta_{\vec{x},\vec{y}}e^{(\phi_{\vec{x}}-\mu)(x_{0}-y_{0})}[\theta(x_{0}-y_{0}){1\over
1+e^{\beta(\phi_{\vec{x}}-\mu)}}-\theta(y_{0}-x_{0}){e^{\beta(\phi_{\vec{x}}-\mu)}\over
1+e^{\beta(\phi_{\vec{x}}-\mu)}}]$ (13)

which can be equivalently written as

$g({\bf x},{\bf
y})=\delta_{\vec{x},\vec{y}}{1\over\beta}\sum_{k_{0}={2\pi\over\beta}(n_{0}+{1\over
2})}e^{-ik_{0}(x_{0}-y_{0})}\widehat{g}(\vec{x},k_{0})=\delta_{\vec{x},\vec{y}}\bar{g}(\vec{x};x_{0}-y_{0})$
(14)

with

$\widehat{g}(\vec{x},k_{0})={1\over-ik_{0}+\phi_{\vec{x}}-\mu}$ (15)

We define

$\mu=\bar{\phi}(\alpha)$ (16)

and the occupation number on the ground state is
$\theta(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))$; the choice of
$\mu$ fixes the averaged density. The conductivity is exactly vanishing as the
is proportional to $\varepsilon$. The density correlation is

$<\rho_{\bf x};\rho_{\bf
y}>=\delta_{\vec{x},\vec{y}}\bar{g}(\vec{x};x_{0}-y_{0})\bar{g}(\vec{x};y_{0}-x_{0})$
(17)

We want to investigate what happens when we consider a non-vanishing hopping
$\varepsilon\not=0$ and interaction $\lambda\not=0$. As usual in small divisor
problems, we need to impose a Diophantine condition on the frequencies
$\vec{\omega}$ of the quasi-periodic disorder that is

$||(\vec{\omega}\vec{x})||_{\hbox{\msytw T}}\geq
C_{0}|\vec{x}|^{-\tau}\quad\quad\vec{x}\in\hbox{\msytw Z}^{d}/\vec{0}$ (18)

$||.||$ being the norm on the one dimensional torus with period $1$; we
require also a Diophantine condition on the chemical potential, that is

$||(\vec{\omega}\vec{x})\pm 2\alpha||_{\hbox{\msytw T}}\geq
C_{0}|\vec{x}|^{-\tau}\quad\quad\vec{x}\in\hbox{\msytw Z}^{d}/\vec{0}$ (19)

The complementary of the set of numbers $\omega,\alpha$ verifying the
diophantine conditions for some $C_{0}$ has measure $O(C_{0})$, see eg [41].

In general the value of the chemical potential is modified by the interaction;
in order to fix the interacting chemical potential to the value
$\bar{\phi}(\alpha)$ we choose the bare one to $\mu=\bar{\phi}(\alpha)+\nu$
with $\nu$ chosen properly.

Our main result is the following

######  Theorem 3.1.

Assume that $\mu=\bar{\phi}(\alpha)+\nu$ and
$\phi_{x}=\bar{\phi}(\vec{\omega}\vec{x})$ with
$\bar{\phi}:\mathbf{T}\rightarrow\hbox{\msytw R}$, even, differentiable and
such that $v_{0}=\partial\bar{\phi}(\alpha)\not=0$: in addition $\vec{\omega}$
verifies (18) and $\alpha$ verifies (19). There exists $\varepsilon_{0}$ and a
suitable choice of $\nu=O(\varepsilon_{0})$ such that, for
$|\lambda|\leq|\varepsilon|\leq\varepsilon_{0}$ in the zero temperature and
infinite volume limit

  1. 1.

The 2-point correlation verifies, for any $N$

$|S({\bf x},{\bf y})|\leq|\log\Delta_{\vec{x},\vec{y}}|C_{N}{e^{-{1\over
4}|\log|\varepsilon|||\vec{x}-\vec{y}|}\over
1+(\Delta_{\vec{x},\vec{y}}|x_{0}-y_{0}|)^{N}}$ (20)

with

$\Delta_{\vec{x},\vec{y}}=(1+\min(|\vec{x}|,|\vec{y}|))^{-\tau}$ (21)

  2. 2.

The density and current correlations verify

$|H_{\mu,\nu}({\bf x},{\bf
y})|\leq\Delta_{\vec{x},\vec{y}}^{-4}C_{N}{e^{-{1\over
4}|\log|\varepsilon|||\vec{x}-\vec{y}|}\over
1+(\Delta_{\vec{x},\vec{y}}|x_{0}-y_{0}|)^{N}}$ (22)

  3. 3.

The Drude weight is vanishing

$D^{i}_{\vec{x}}=0$ (23)

The above result says that there is exponential decay in the coordinate
difference in the fermionic and current correlations, signaling localization
in the ground state with quasi periodic potential of the form
$\bar{\phi}(\vec{\omega}\vec{x})$ in any dimension. Moreover the Drude weight
at $T=0$ is vanishing, implying a non-metallic behavior. This result is
obtained assuming a Diophantine condition on the frequencies and on the
chemical potential (or equivalently on the densities), see (19). As the
estimate of the radius of convergence $\varepsilon_{0}$ is proportional to
$C_{0}$ to some power, with fixed $\varepsilon,\lambda$ we get a large measure
set of densities for which localization is present (but not on an interval).

Information on the conductivity are obtained by combining the Ward Identities
following from the conservation of the current with regularity properties of
the Fourier transform of the correlations, which are related to the decay in
the coordinate space. In the case of non-interacting fermions, or for $1d$
interacting fermions without disorder, the slow power law decay of
correlations implies a non vanishing Drude weight, see [42]. In the present
case, the decay in space is exponentially fast but the decay in the imaginary
time has rate not uniform in $\vec{x},\vec{y}$, due to the lack of translation
invariance. As a consequence, we can deduce the vanishing of the Drude weight
but not of the conductivity.

The analysis is based on an extension of the Lindstedt series approach to KAM
tori with exact Renormalization Group methods for fermions. The correlations
are expressed by a series expansion showing a small divisor problem, as in the
Lindstedt series for KAM, in graphs with loops, which are a peculiarity of
quantum physics. Small divisors are controlled by the Diophantine conditions
and the huge number of loop graphs is compensated by cancellations due to
anticommutativity.

While we have proved here the vanishing of the Drude weight, it would be
interesting to understand if also the conductivity is vanishing or if a zero
result is found only by a suitable averaging over the phase, as is done in
numerical simulations [27].

The effective interaction is irrelevant in the Renormalization Group sense, as
consequence of Diophantine conditions and by cancellations due to
anticommutativity. The presence of spin [43] and an anisotropic hopping [44]
produce extra marginal couplings. They can in principle destroy the
convergence result of the present paper, and it is interesting to observe that
numerical [45] or cold atoms experiments [19] have found evidence of
delocalization is such cases. Another important point would be to extend the
analysis to a more general kind of disorder like
$f(\vec{\omega}_{1}\vec{x},\vec{\omega}_{2}\vec{x})$. The condition of strong
disorder is non technical; in the case of weak quasiperiodic disorder there is
no localization; in particular, this is the case of the interacting Aubry-
Andre’ model [46], of the bidimensional Hofstadter model [47] or of three
dimensional Weyl semimetals [48]. Finally, we stress that a rigorous
understanding of $T=0$ properties of interacting fermions with finite density
and random disorder is still unknown.

The main open problem if of course to extend the above result on transport
coefficients to finite temperature to get information on localization beyond
the ground state. While an extension of [39] allows to pass from Euclidean to
real time conductivity at $T=0$, this is expected to be a major difficulty for
$T>0$. Another difficulty is due to the fact that we do not get ground state
localization in an interval of densities, but only in a large measure set. The
absence of thermalization in the classical case is considered related to KAM
theorem; it is interesting to note that the persistence of localization in a
quantum system, which is considered an obstruction to thermalization, is also
obtained via the generalization of KAM methods in a quantum context.

##  4 Vanishing of Drude weight

We show that the vanishing of Drude weight (23) is consequence of the bound
(22) combined with Ward Identities. Note first that the Fourier transform in
the infinite volume limit is continuous as

$\displaystyle|\widehat{H}_{\mu,\nu}({\bf p},\vec{y})|\leq\sum_{\vec{x}}\int
dx_{0}|H_{\mu,\nu}({\bf x},{\bf y})|\leq\sum_{\vec{x}}\int
dx_{0}\Delta_{\vec{x},\vec{y}}^{-4}C_{N}{e^{-{1\over
4}|\log|\varepsilon||\vec{x}-\vec{y}|}\over
1+(\Delta_{\vec{x},\vec{y}}|x_{0}|)^{N}}\leq$ (24) $\displaystyle
C_{1}\sum_{\vec{x}}(|\vec{x}+\vec{y}|^{5\tau}+|\vec{y}|^{5\tau})e^{-{1\over
4}|\log|\varepsilon||\vec{x}||}\leq C_{2}\sum_{\vec{x}}e^{-{1\over
4}|\log|\varepsilon||\vec{x}||}(|\vec{x}|^{5\tau}+2|\vec{y}|^{5\tau})\leq
C_{3}|\vec{y}|^{5\tau}/(|\log|\varepsilon||)^{d+5\tau}$

Ward identities can be deduced from the continuity equation,

$\partial_{0}\rho_{\bf x}=[H,\rho_{\bf x}]=-i\sum_{i}(j^{i}_{\bf
x}-j^{i}_{{\bf x}-e_{i}})$ (25)

we get, setting $\partial_{i}j_{\bf x}\equiv j_{\bf x}-j_{{\bf x}-{\bf
e}_{i}}$ , $i=1,...,d$, ${\bf e}_{i}=(0,\vec{e}_{i})$

$\displaystyle\partial_{0}<T\rho_{\bf x};\rho_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj_{\bf x}^{i};\rho_{\bf
y}>+\delta(x_{0}-y_{0})<[\rho_{\bf x},\rho_{\bf y}]>$
$\displaystyle\partial_{0}<T\rho_{\bf x};j^{j}_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj^{i}_{\bf x};j^{j}_{\bf
y}>+\delta(x_{0}-y_{0})<[\rho_{\bf x},j^{j}_{\bf y}]>$ (26)

Note that $[\rho_{\vec{x},x_{0}},\rho_{\vec{y},x_{0}}]=0$ while

$[\rho_{\vec{x},x_{0}},j^{j}_{\vec{y},x_{0}}]=-i\delta_{\vec{x},\vec{y}}\tau^{j}_{\vec{x}}+i\delta_{\vec{x}-\vec{e}_{j},\vec{y}}\tau^{j}_{\vec{y}}$
(27)

so that, in the $L,\beta\rightarrow\infty$ limit

$\displaystyle\partial_{0}<T\rho_{\bf x};\rho_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj^{i}_{\bf x};\rho_{\bf y}>$ (28)
$\displaystyle\partial_{0}<T\rho_{\bf x};j^{j}_{\bf
y}>=-i\sum_{i}\partial_{i}<Tj^{i}_{\bf x};j^{j}_{\bf
y}>-i\delta(x_{0}-y_{0})(-\delta_{\vec{x},\vec{y}}<\tau^{j}_{\vec{y}}>+\delta_{\vec{x}-\vec{e}_{j},\vec{y}}<\tau^{j}_{\vec{y}}>)$

Taking the Fourier transform in ${\bf x}$ we get, using translation invariance
in time and setting $y_{0}=0$

$\sum_{\vec{x}}\int dx_{0}e^{i{\bf p}{\bf x}}(\partial_{0}<T\rho_{{\bf
x}};j^{j}_{\vec{y}}>+i\sum_{i}\partial_{i}<Tj^{i}_{\bf
x};j^{j}_{\vec{y}}>+i\delta(x_{0})(-\delta_{\vec{x},\vec{y}}<\tau^{j}_{\vec{y}}>+\delta_{\vec{x}-\vec{e}_{j},\vec{y}}<\tau^{j}_{\vec{y}}>)=0$
(29)

with $p_{0}\in\hbox{\msytw R}$ and $\vec{p}\in[-\pi,\pi)^{d}$ so that

$-ip_{0}\widehat{H}_{0,j}({\bf
p},\vec{y})+i\sum_{i}(1-e^{-ip_{i}})(\widehat{H}_{i,j}({\bf
p},\vec{y})+e^{-i\vec{p}\vec{y}}<\tau^{j}_{y,0}>)=0$ (30)

Setting $j=1$ for definiteness, we set $\bar{\vec{p}}=(p_{1},0,0)$ so that

$-ip_{0}\widehat{H}_{0,1}(\bar{\bf
p},\vec{y})+i(1-e^{-ip_{1}})(\widehat{H}_{1,1}(\bar{\bf
p},\vec{y})+e^{-ip_{1}y_{1}}<\tau^{1}_{y,y_{0}}>)=0$ (31)

so that

$\lim_{p_{1}\rightarrow
0}(\widehat{H}_{1,1}(0,p_{1},\vec{y})+e^{-ip_{1}y_{1}}<\tau^{1}_{y,y_{0}}>)=0$
(32)

but $\lim_{p_{1}\rightarrow 0}(e^{-ip_{1}y_{1}}-1)=0$. In conclusion

$\lim_{p_{1}\rightarrow
0}(\widehat{H}_{1,1}(0,p_{1},\vec{y})+<\tau^{1}_{y,y_{0}}>)=0$ (33)

Due to (4) $\widehat{H}_{1,1}({\bf p},\vec{y})$ is continuous in ${\bf p}$ so
that we can exchange the limits

$\lim_{p_{0}\rightarrow 0}\lim_{\vec{p}\rightarrow 0}(\widehat{H}_{1,1}({\bf
p},\vec{y})+<\tau^{1}_{y,y_{0}}>)=D^{1}_{\vec{x}}=0$ (34)

and this shows that the Drude weight is vanishing. Note the crucial role
played by continuity of the Fourier transform, following by the fast decay of
the correlations; without quasi-periodic disorder the Fourier transform is not
continuous due to its slow decay and the Drude weight is non vanishing.

##  5 Perturbation theory and Grassmann representation

The starting point of the analysis consists in expanding around the anti-
integrable limit (12); defining

$\displaystyle H-\mu N=H_{0}+V$ (35) $\displaystyle
H_{0}=\sum_{\vec{x}}(\phi_{\vec{x}}-\bar{\phi}(\alpha))a^{+}_{\vec{x}}a^{-}_{\vec{x}}$
$\displaystyle
V=\varepsilon\sum_{\vec{x},i}(a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}}+a^{+}_{\vec{x}}a^{-}_{\vec{x}+\vec{e}_{i}})+\lambda\sum_{\vec{x},i}a^{+}_{\vec{x}}a^{-}_{\vec{x}}a^{+}_{\vec{x}+\vec{e}_{i}}a^{-}_{\vec{x}+\vec{e}_{i}}+\nu\sum_{\vec{x}}a^{+}_{\vec{x}}a^{-}_{\vec{x}}$
(36)

and using the Trotter formula one can write the partition function and the
correlations as a power series expansion in $\lambda,\varepsilon$.

${\bf x}\pm{\bf e}_{i}$

${\bf x}\pm{\bf e}_{i}$

${\bf x}$

${\bf x}$

${\bf x}$

${\bf x}\pm{\bf e}_{i}$

${\bf x}$

${\bf x}$

$\nu$

$\varepsilon$

$\lambda$

Figure 1:  Graphical representation of the three terms in ${\cal V}(\psi)$
eq.(38)

The correlations can be equivalently written in terms of Grassmann integrals.
We can write

$e^{W(\eta,J)}=\int P(d\psi)e^{-{\cal V}(\psi)-{\cal B}(\psi,J,\eta)}$ (37)

with ${\bf e_{i}}=(0,\vec{e}_{i})$

${\cal V}(\psi)=\varepsilon\sum_{i}\int d{\bf x}(\psi^{+}_{{\bf x}+{\bf
e_{i}}}\psi^{-}_{{\bf x}}+\psi^{+}_{{\bf x}-{\bf e_{i}}}\psi^{-}_{{\bf
x}})+\lambda\int d{\bf x}\sum_{i}\psi^{+}_{{\bf x}}\psi^{-}_{{\bf
x}}\psi^{+}_{{\bf x}+{\bf e}_{i}}\psi^{-}_{{\bf x}+{\bf e_{i}}}+\nu\int d{\bf
x}\psi^{+}_{{\bf x}}\psi^{-}_{{\bf x}}$ (38)

where $\int d{\bf x}=\sum_{x\in\Lambda_{L}}\int_{-{\beta\over 2}}^{\beta\over
2}dx_{0}$ and $\psi^{\pm}_{\bf x}$ is vanishing outside $\Lambda_{L}$;
moreover

${\cal B}(\psi,J,\eta)=\int d{\bf x}[\eta^{+}_{{\bf x}}\psi^{-}_{{\bf
x}}+\psi^{+}_{{\bf x}}\eta^{-}_{{\bf x}}+\sum_{\mu=0}^{d}J_{\mu}({\bf
x})j_{\mu}({\bf x})]$ (39)

with

$\displaystyle j_{0}({\bf x})=\psi^{+}_{{\bf x}}\psi^{-}_{\bf x}\quad\quad
j_{i}({\bf x})=\varepsilon(\psi^{+}_{{\bf x}+{\bf e}_{i}}\psi^{-}_{\bf
x}-\psi^{+}_{{\bf x}}\psi^{-}_{{\bf x}+{\bf e_{i}}})$ (40)

The 2-point and the current correlations are given by

$S_{2}^{L,\beta}({\bf x},{\bf y})={\partial^{2}\over\partial\eta^{+}_{{\bf
x}}\partial\eta^{-}_{{\bf y}}}W(\eta,J)|_{0,0}\quad\quad H_{\mu,\nu}({\bf
x},{\bf y})={\partial^{2}\over\partial J_{\mu,{\bf x}}\partial J_{\nu,{\bf
y}}}W(\eta,J)|_{0,0}$ (41)

By expanding in $\lambda,\varepsilon,\nu$ one can write the correlations as a
series expansion, which can be expressed in terms of Feynman graphs obtained
contracting the half lines of vertices, see Fig. 1, and associating to each
line the propagator $g({\bf x},{\bf y})$. There is a basic difference between
the perturbative expansion in the non interacting case $\lambda=0$ and the
interacting case $\lambda\not=0$. In the first case there are only chain
graphs, while in the second there are also loops, producing further
combinatorial problems. One can verify that the perturbative expansions
obtained by Trotter formula for (2) and by the Grassmann generating functions
are the same (this is true up to the so called ”tadpoles” which can be easily
taken into account, see §1 D in [35]). The identity between (2) and (37) is
true in a rigorous sense provided that the Grassmann integral representation
is analytic in a disk uniformly in $L,\beta$, as proven in the following
sections. Indeed at finite $L,\beta$ the partition function in (2) is entire
and it coincides order by order with the Grassmann representation, which is
analytic in a disk independent on the volume, so they coincide. As the
denominator of the correlations is non vanishing in this finite disk and the
numerator is entire at finite $\beta,L$, also the correlations (2) is analytic
and coincide with the Grassmann representation, and the identity holds also in
the limit.

##  6 Multiscale decomposition and renormalization

The difficulty in controlling the perturbative expansion is due to a ”small
divisor problem” related to the size of the propagator; the denominator of
$\widehat{g}(\vec{x},k_{0})$ can be arbitrarily small if $\vec{\omega}\vec{x}$
is close to $\pm\alpha$, a fact which can produce in principle $O(n!)$-terms
which could destroy convergence. The starting point of the analysis is to
separate the propagator in two terms, one containing the quasi-singularity and
a regular part; we write

$g({\bf x},{\bf y})=g^{(1)}({\bf x},{\bf y})+\sum_{\rho=\pm}g_{\rho}^{(\leq
0)}({\bf x},{\bf y})$ (42)

where

$\displaystyle g^{(1)}({\bf x},{\bf
y})={\delta_{\vec{x},\vec{y}}\over\beta}\sum_{k_{0}}\chi^{(1)}(\vec{\omega}\vec{x},k_{0}){e^{-ik_{0}(x_{0}-y_{0})}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}=\delta_{\vec{x},\vec{y}}g^{(1)}(\vec{x},x_{0}-y_{0})$
$\displaystyle g^{(\leq 0)}_{\rho}({\bf x},{\bf
y})={\delta_{\vec{x},\vec{y}}\over\beta}\sum_{k_{0}}\chi^{(0)}_{\rho}(\vec{\omega}\vec{x},k_{0}){e^{-ik_{0}(x_{0}-y_{0})}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}=\delta_{\vec{x},\vec{y}}g^{(\leq
0)}_{\rho}(\vec{x},x_{0}-y_{0})$ (43)

with
$\chi^{(0)}_{\rho}(\vec{\omega}\vec{x},k_{0})={\widetilde{\theta}}_{\rho}(\vec{\omega}\vec{x})\bar{\chi}_{0}(\sqrt{k_{0}^{2}+(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))^{2}})$
with ${\widetilde{\theta}}_{\rho}$ is the periodic theta function
(${\widetilde{\theta}}_{\pm}=1$ if $\vec{\omega}\vec{x}$ mod. $1$ is
positive/negative and zero otherwise) and $\bar{\chi}_{0}$ such that
$C^{\infty}(\hbox{\msytw R}^{+})\rightarrow\hbox{\msytw R}$ such that
$\bar{\chi}_{0}(t)=1$ with $t\leq 1$ and $\bar{\chi}_{0}(t)=0$ for
$t\geq\gamma>1$; moreover $\chi^{(1)}+\sum_{\rho=\pm}\chi_{\rho}=1$. The
”infrared” propagator $g^{(\leq 0)}({\bf x},{\bf y})$ has denominator
arbitrarily small. We can further decompose the infrared propagator as sum of
propagators with smaller and smaller denominators

$g^{(\leq
0)}_{\rho}(\vec{x},x_{0}-y_{0})=\sum_{h=-\infty}^{0}g^{(h)}_{\rho}(\vec{x},x_{0}-y_{0})$
(44)

with $g^{(h)}_{\rho}$ similar $g^{(\leq 0)}_{\rho}$ witrh $f^{h}$ replacing
$\bar{\chi}_{0}$ with

$f^{h}=\bar{\chi}_{0}(\gamma^{h}\sqrt{k_{0}^{2}+(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))^{2}})-\bar{\chi}_{0}(\gamma^{h-1}\sqrt{k_{0}^{2}+(\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha))^{2}})$
(45)

For any integer $N$ one has

$|g^{(h)}_{\rho}(\vec{x},x_{0}-y_{0})|\leq{C_{N}\over
1+(\gamma^{h}|x_{0}-y_{0}|)^{N}}$ (46)

if $C_{N}$ is a suitable constant.

The integration of (37) is done iteratively by using two crucial properties of
Grassmann integrations. If $P(d\psi^{(1)})$ and $P(d\psi^{(\leq 0)})$ are
gaussian Grassmann integrations with propagators $g^{(1)}$ and $g^{(\leq 0)}$,
we can write $P(d\psi)=P(d\psi^{(1)})P(d\psi^{(\leq 0)})$ so that

$\displaystyle e^{W(\eta,J)}=\int P(d\psi^{(1)})P(d\psi^{(\leq 0)})e^{-{\cal
V}(\psi^{(1)}+\sum_{\rho=\pm}\psi^{(\leq 0)}_{\rho})-{\cal
B}(\psi^{(1)}+\sum_{\rho=\pm}\psi^{(\leq 0)}_{\rho},\eta,J)}=$
$\displaystyle\int P(d\psi^{(\leq 0)})e^{-{\cal V}^{(0)}(\psi^{(\leq
0)}_{\rho},\eta,J)}$ (47)

with

${\cal V}^{(0)}(\psi^{(\leq 0)}_{\rho},\eta,J)=\sum_{n=0}^{\infty}{1\over
n!}{\cal E}^{T}_{1}({\cal V}+{\cal B};n)$ (48)

and ${\cal E}^{T}_{1}$ are fermionic truncated expectations with propagator
$g^{(1)}$. By integrating $\psi^{(0)},\psi^{(-1)},..,\psi^{(h+1)}$ one obtains
a sequence of effective potentials ${\cal V}^{(h)}$, $h=0,-1,-2,..$. The way
in which we define the integration is dictated by the scaling dimension which
is, as we will see below, $D=1$; that is all terms are relevant in the
Renormalization Group sense. Remark Note that after the integration of
$\psi^{1}$ one gets a theory defined in terms of two fields
$\psi_{+},\psi_{-}$. This is due to the fact that
$\bar{\phi}(t)=\bar{\phi}(\alpha)$ in correspondence of two points
$\pm\alpha$. If we consider more general forms of quasi periodic disorder,
like $\bar{\phi}(t_{1},t_{2})$ as the one in [7] , then
$\bar{\phi}(t_{1},t_{2})-\mu=0$ in a set corresponding to a surface. In this
case one gets a description in terms of a field $\psi_{\rho}$, with $\rho$ a
parameter parametrizing this curve, a situation somewhat analogue to what
happens in interacting fermions with extended Fermi surface.

The multiscale integration is described iteratively in the following way.
Assume that we have already integrated the fields
$\psi^{(0)},\psi^{(-1)},..,\psi^{(h+1)}$ obtaining (we set $\eta=0$ for the
moment)

$e^{W(0,J)}=\int P(d\psi^{(\leq h)})e^{-{\cal V}^{(h)}(\psi^{(\leq h)},J)}$
(49)

where $P(d\psi^{(\leq h)}$ has propagator

$g^{(\leq h)}_{\rho}({\bf x},{\bf
y})={\delta_{\vec{x},\vec{y}}\over\beta}\sum_{k_{0}}\chi^{(h)}_{\rho}(k_{0},\vec{\omega}\vec{x}){e^{-ik_{0}(x_{0}-y_{0})}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}=\delta_{\vec{x},\vec{y}}g^{(\leq
0)}_{\rho}(\vec{x},x_{0}-y_{0})$ (50)

and

${\cal V}^{(h)}(\psi^{(\leq h)},J)=\sum_{l\geq 0,m\geq
0}\sum_{\underline{\varepsilon},\underline{\rho}}\int d{\bf x}_{1}...d{\bf
x}_{l}d{\bf y}_{1}...d{\bf y}_{m}H^{h}_{l,m}(\underline{{\bf
x}},\underline{{\bf y}})\prod_{i=1}^{l}\psi^{\varepsilon_{i}(\leq
h)}_{\rho_{i},{\bf x}_{i}}\prod_{i=l}^{m}J_{{\bf y}_{i}}$ (51)

If there is a subset of $\psi^{\varepsilon_{i}}_{\rho_{i},{\bf x}_{i}}$ with
the same $\varepsilon,\rho$ and $\vec{x}_{i}$, by the anticommuting properties
of Grassmann variables we can write, if $l>1$

$\prod_{i=1}^{l}\psi^{\varepsilon}_{\vec{x},x_{0,i}}=\psi^{\varepsilon}_{\vec{x},x_{0,1}}\prod_{i=2}^{l}D^{\varepsilon}_{\vec{x},x_{0,i},x_{0,1}}\quad\quad\quad
D^{\varepsilon}_{\vec{x},x_{0,i},x_{0,1}}=\psi^{\varepsilon}_{\vec{x},x_{0,i}}-\psi^{\varepsilon}_{\vec{x},x_{0,1}}$
(52)

We can therefore rewrite that effective potential in the following way

${\cal V}^{(h)}(\psi^{(\leq h)},J)=\sum_{l\geq 0,m\geq
0}\sum_{\underline{\varepsilon},\underline{\rho}}\int d{\bf x}_{1}...d{\bf
x}_{l}d{\bf y}_{1}...d{\bf y}_{m}H^{h}_{l,m}(\underline{{\bf
x}},\underline{{\bf
y}})\prod_{i=1}^{l}d^{\sigma_{i}}\psi^{\varepsilon_{i}}_{\rho_{i},{\bf
x}_{i}}\prod_{i=l}^{m}J_{{\bf y}_{i}}$ (53)

with $\sigma=0,1$ and $d^{0}\psi=\psi$ and $d^{1}\psi=D$.

We define resonant the terms with fields with the same coordinate $\vec{x}$,
that is ${\bf x}_{i}=(x_{0,i},\vec{x})$. Note that all the resonant terms with
$l\geq 4$ are such that there are at least two $D$ fields; the fields have the
same $\rho$ index as have the same $\vec{\omega}\vec{x}$.

We define a renormalization operation ${\cal R}$ in the following way

  1. 1.

If $l=2$, $m=0$

${\cal R}\sum_{\vec{x}}\int dx_{0,1}dx_{0,2}H_{2,0}^{(h)}\psi^{+(\leq
h)}_{\vec{x},x_{0,1},\rho}\psi^{-(\leq
h)}_{\vec{x},x_{0,2},\rho}=\sum_{\vec{x}}\int
dx_{0,1}dx_{0,2}H_{2,0}^{(h)}\psi^{+(\leq h)}_{\vec{x},x_{0,1},\rho}T^{-(\leq
h)}_{\vec{x},x_{0,1},x_{0,2}\rho}$ (54)

with

$T^{-(\leq h)}_{\vec{x},x_{0,1},x_{0,2}\rho}=\psi^{-(\leq
h)}_{\vec{x},x_{0,2},\rho}-\psi^{-(\leq
h)}_{\vec{x},x_{0,1},\rho}-(x_{0,1}-x_{0,2})\partial\psi^{-(\leq
h)}_{\vec{x},x_{0,1},\rho}$ (55)

  2. 2.

${\cal R}=0$ otherwise

We define ${\cal R}=1-{\cal L}$ and by definition ${\cal L}{\cal V}^{(h)}$ is
given by the following expression

${\cal L}{\cal
V}^{(h)}=\gamma^{h}F^{(h)}_{\nu}+F^{(h)}_{\zeta}+F^{(h)}_{\alpha}$ (56)

where, if
$H_{2,0}^{(h)}(\vec{x},x_{0}-y_{0})\equiv\bar{H}_{2,0}^{(h)}(\vec{\omega}\vec{x},x_{0}-y_{0})$
one has

$\nu_{h}=\int
dx_{0}\bar{H}_{2,0}^{(h)}(\rho\alpha,x_{0})\quad\xi_{h}(\vec{x})=\int
dx_{0}{\bar{H}_{2,0}^{(h)}(\vec{\omega}\vec{x},x_{0})-\bar{H}_{2,0}^{(h)}(\rho\alpha,x_{0})\over\vec{\omega}\vec{x}-\rho\alpha}$
(57)

and $\alpha_{h}(\vec{x})=\int
dx_{0}x_{0}\bar{H}_{2,0}^{(h)}(\vec{\omega}\vec{x},x_{0})$; moreover

$\displaystyle F^{(h)}_{\nu}=\sum_{\rho}\sum_{\vec{x}}\int
dx_{0}\nu_{h}\psi^{+(\leq h)}_{{\bf x},\rho}\psi^{-(\leq h)}_{{\bf
x},\rho}\quad\quad F^{(h)}_{\zeta}=\sum_{\rho}\sum_{\vec{x}}\int
dx_{0}((\vec{\omega}\vec{x})-\rho\alpha)\zeta_{h,\rho}(\vec{x})\psi^{+(\leq
h)}_{{\bf x},\rho}\psi^{-(\leq h)}_{{\bf x},\rho}$ $\displaystyle
F^{(h)}_{\alpha}=\sum_{\rho}\sum_{\vec{x}}\int
dx_{0}\alpha_{h,\rho}(\vec{x})\psi^{+(\leq h)}_{{\bf
x},\rho}\partial_{0}\psi^{-(\leq h)}_{{\bf x},\rho}\quad\quad$ (58)

The running coupling constants  $\vec{v}_{h}=(\nu_{h},\alpha_{h},\xi_{h})$ are
independent from $\rho$, as (37) is invariant under parity
$\vec{x}\rightarrow-\vec{x}$. Note also that
$(\widehat{g}^{(k)})^{*}(\vec{x},k_{0})=\widehat{g}^{(k)}(\vec{x},-k_{0})$ so
that
$(\widehat{H}^{(h)}_{2,\rho}(\vec{x},k_{0}))^{*}=\widehat{H}^{(h)}_{2,\rho}(\vec{x},-k_{0})$,
and this implies that $\nu_{h}$ is real.

Remark The ${\cal R}$ operation is defined in order to act non trivially on
the resonant terms with two fields and no $J$ fields; they are the only
resonant terms with no $D$ fields. This fact would be not true of there is the
spin or an extra degree of freedom, as in the case of lattice Weyl semimetals
[48]. In that case the local part of the effective potential would contain
also effective interactions. With the above definitions we can write (49)

$e^{W(0,J)}=\int P(d\psi^{(\leq h-1)})\int P(d\psi^{(h)})e^{-{\cal L}{\cal
V}^{(h)}(\psi^{(\leq h)},J)-{\cal R}{\cal V}^{(h)}(\psi^{(\leq h)},J)}=\int
P(d\psi^{(\leq h-1)})\ e^{-{\cal L}{\cal V}^{(h)}(\psi^{(\leq h-1)},J)}$ (59)

and the procedure can be iterated.

##  7 Convergence of series expansion

The effective potential can be written as a sum over Gallavotti trees $\tau$,
see Fig.2

${\cal V}^{(h)}(\psi^{(\leq h)},J)=\sum_{n=1}^{\infty}\sum_{\tau\in{\cal
T}_{h,n}}V^{(h)}(\tau,\psi^{(\leq h)})$ (60)

where $\tau$ are trees constructed adding labels to the unlabeled trees,
obtained by joining a point, the root, with an ordered set of $n\geq 1$
points, the endpoints, so that the root is not a branching point.

$v_{0}$

$v$

$v^{\prime}$

$h_{v}$

$1$

$2$

Figure 2:  A labeled tree

The set of labeled trees ${\cal T}_{h,n}$ is defined associating a label
$h\leq 0$ with the root and introducing a family of vertical lines, labeled by
an integer taking values in $[h,2]$ intersecting all the non-trivial vertices,
the endpoints and other points called trivial vertices.To a vertex $v$ is
associated $h_{v}$ and, if $v_{1}$ and $v_{2}$ are two vertices and
$v_{1}<v_{2}$, then $h_{v_{1}}<h_{v_{2}}$. Moreover, there is only one vertex
immediately following the root, which will be denoted $v_{0}$ and can not be
an endpoint; its scale is $h+1$. To the end-points are associated ${\cal
V}+{\cal B}$ , and in such a case the scale is $2$; or ${\cal L}{\cal
V}^{h_{v}-1}(\psi^{(\leq h_{v}-1)},J)$ and in this case the scale is
$h_{v}\leq 1$ and there is the constraint that $h_{v}=h_{\bar{v}}+1$, if
$\bar{v}$ is the first non trivial vertex immediately preceding $v$. The tree
structure induces a jerarchy of end-points which can be represented by
clusters, see Fig.3.

$1$

$2$

$3$

$4$

$5$

$\Longleftrightarrow$

$1$

$2$

$3$

$4$

$5$

Figure 3:  A tree of order 5 and the corresponding clusters.

If $v_{0}$ is the first vertex of $\tau$ and $\tau_{1},..,\tau_{s}$
($s=s_{v_{0}}$) are the subtrees of $\tau$ with root $v_{0}$,
$V^{(h)}(\tau,\psi^{(\leq h)})$ is defined inductively by the relation

$V^{(h)}(\tau,\psi)={(-1)^{s+1}\over s!}{\cal
E}^{T}_{h+1}[\bar{V}^{(h+1)}(\tau_{1},\psi^{(\leq
h+1)});..;\bar{V}^{(h+1)}(\tau_{s},\psi^{(\leq h+1)})]$ (61)

where $\bar{V}^{(h+1)}(\tau_{i},\psi^{(\leq h+1)})$ it is equal to ${\cal
R}{\cal V}^{(h+1)}(\tau_{i},\psi^{(\leq h+1)})$ if the subtree $\tau_{i}$ is
non trivial;if $\tau_{i}$ is trivial, it is equal to ${\cal L}{\cal
V}^{(h+1)}$. By iterating (61) we get a jerarchy of truncated expectations,
with a certain subset of fields contracted in each expectations. We can
therefore write $V^{(h)}(\tau,\psi^{(\leq h)})$ as sum over sets defined in
the following way. We call $I_{v}$ the set of $\psi$ associated to the end-
points following $v$ and $P_{v}$ is a subset of $I_{v}$ denoting the external
$\psi$. We denote by $Q_{v_{i}}$ the intersection of $P_{v}$ and $P_{v_{i}}$;
they are such that $P_{v}=\cup_{i}Q_{v_{i}}$ and the union ${\cal I}_{v}$ of
the subsets $P_{v_{i}}\setminus Q_{v_{i}}$ is, by definition, the set of the
internal fields of $v$, and is non empty if $S_{v}>1$. The effective potential
can be therefore written as

${\cal V}^{(h)}(\tau,\psi^{(\leq h)})=\sum_{{\bf P}\in{\cal P}_{\tau}}{\cal
V}^{(h)}(\tau,{\bf P})\quad\bar{\cal V}^{(h)}(\tau,{\bf P})=\int d{\bf
x}_{v_{0}}\widetilde{\psi}^{(\leq h)}(P_{v_{0}})K_{\tau,{\bf P}}^{(h+1)}({\bf
x}_{v_{0}})\;,$ (62)

where $\widetilde{\psi}^{(\leq h)}(P)=\prod_{f\in P}\psi_{{\bf x}(f)}$. If we
expand the truncated expectations by the Wick rule we get a sum of Feynman
graphs with an associated cluster structure; an example is in Fig.4.

Figure 4:  An example of graph with $\lambda$ and $\varepsilon$ vertices and
the associated cluster structure; the propagator in the cluster, represented
as a circle, has scale $h$ smaller than the scales of the propagators external
to the cluster.

The truncated expectations can be written by the Brydges-Battle-Federbush
formula

${\cal
E}^{T}_{h_{v}}({\widetilde{\psi}}^{(h_{v})}(P_{1}/Q_{1}),\cdots,{\widetilde{\psi}}^{(h_{v})}(P_{s}/Q_{s})))=\sum_{T_{v}}\prod_{l\in
T_{v}}\big{[}\delta_{\vec{x}_{l},\vec{y}_{l}}\bar{g}^{(h_{v})}(\vec{x}_{l},x_{0,l}-y_{0,l})\big{]}\,\int
dP_{T}({\bf t})\;{\rm det}\,G^{h_{v},T}({\bf t})\;,$ (63)

where $T_{v}$ is a set of lines forming an anchored tree graph between the
clusters of points ${\bf x}^{(i)}\cup{\bf y}^{(i)}$, that is $T_{v}$ is a set
of lines, which becomes a tree graph if one identifies all the points in the
same cluster. Moreover ${\bf t}=\\{t_{ii^{\prime}}\in[0,1],1\leq
i,i^{\prime}\leq s\\}$, $dP_{T_{v}}({\bf t})$ is a probability measure with
support on a set of ${\bf t}$ such that $t_{ii^{\prime}}={\bf u}_{i}\cdot{\bf
u}_{i^{\prime}}$ for some family of vectors ${\bf u}_{i}\in\hbox{\msytw
R}^{s}$ of unit norm.

$G^{h,T}_{ij,i^{\prime}j^{\prime}}=t_{ii^{\prime}}\delta_{\vec{x}_{ij},\vec{y}_{i^{\prime}j^{\prime}}}\bar{g}^{(h)}(\vec{x}_{ij},x_{0,ij}-y_{0,i^{\prime}j^{\prime}})\;,$
(64)

We define $\bar{T}_{v}=\bigcup_{w\geq v}T_{w}$ starting from $T_{v}$ and
attaching to it the trees $T_{v_{1}},..,T_{v_{S_{v}}}$ associated to the
vertices $v_{1},..,v_{S_{v}}$ following $v$ in $\tau$, and repeating this
operation until the end-points of $\tau$ are reached.

$w_{1}$

$w_{a}$

$w_{b}$

$w_{c}$

$w_{2}$

Figure 5:  A tree $\bar{T}_{v}$ with attached wiggly lines representing the
external lines $P_{v}$; the lines represent propagators with scale $\geq
h_{v}$ connecting $w_{1},w_{a},w_{b},w_{c},w_{2}$, representing the end-points
following $v$ in $\tau$.

The tree $\bar{T}_{v}$ connects the end-points $w$ of the tree $\tau$. To each
end-point $w$ we associate a factor $\vec{\delta}_{w}^{i_{w}}$, and a)
$\vec{\delta}^{i}_{w}=0$ if $w$ corresponds to a
$\nu_{h},\alpha_{h},\zeta_{h}$ end-point; b) $\vec{\delta}_{w}^{i}$ one among
$\pm\vec{e}_{i}$, $i=1,2,3$ if it corresponds to an $\varepsilon$ end-point;
c) $\delta^{i}_{w}$ one among $0,\pm\vec{e}_{i}$, $i=1,2,3$ if it corresponds
to a $\lambda$ end-point. If $\vec{x}_{w_{1}}$ and $\vec{x}_{w_{2}}$ are
coordinates of the external fields ${\widetilde{\psi}}(P_{v})$ we have, see
Fig.5

$\vec{x}_{w_{1}}-\vec{x}_{w_{2}}=\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}}$ (65)

where $c_{w_{1},w_{2}}$ is the set of endpoints in the path in $\bar{T}$
connecting $w_{1}$ and $w_{2}$. The above relation implies, in particular,
that the coordinates of the external fields ${\widetilde{\psi}}(P_{v})$ are
determined once that the choice of a single one of them and of
$\tau,\bar{T}_{v}$ and ${\bf P}$ is done. We can therefore write the effective
potential as sum over trees $T$, setting the Kronecker deltas in the
propagators in $l\in T$ equal to $1$

${\cal V}^{(h)}(\tau,\psi^{(\leq h)})=\sum_{{\bf P}\in{\cal
P}_{\tau}}\sum_{T}{\cal V}^{(h)}(\tau,{\bf P},T)\quad\bar{\cal
V}^{(h)}(\tau,{\bf P},T)=\sum_{\vec{x}}\int
dx_{0,v_{0}}\widetilde{\psi}^{(\leq h)}(P_{v_{0}})K_{\tau,{\bf
P},T}^{(h+1)}({\bf x}_{v_{0}})\;,$ (66)

where in $K_{\tau,{\bf P},T}^{(h+1)}$ the propagators in $T$ are
$g^{(h)}(\vec{x},x_{0}-y_{0})$ and the determinants are product of determinats
involving propagators with the same $\vec{x}$. We can bound the propagators in
$T$ by

$\int dx_{0}|g^{(h)}(\vec{x},x_{0}-y_{0})|\leq C\gamma^{-h}$ (67)

Moreover the determinants in the BFF formula can be bounded by the Gram-
Hadamard inequality . We introduce an Hilbert space ${\cal H}=\hbox{\msytw
R}^{s}\otimes L^{2}(\hbox{\msytw R}^{1})$ so that

${\widetilde{G}}^{h,T}_{ij,i^{\prime}j^{\prime}}=\Big{(}{\bf u}_{i}\otimes
A(x_{0,ij}-,x_{ij})\;,\ {\bf u}_{i^{\prime}}\otimes
B(y_{0,i^{\prime}j^{\prime}}-,x_{ij})\Big{)}\;,$ (68)

where ${\bf u}\in\hbox{\msytw R}^{s}$ are unit vectors
$(u_{i},u_{i})=t_{ii^{\prime}}$, and $A,B$

$(A,B)=\int dz_{0}A(\vec{x},x_{0}-z_{0})B^{*}(\vec{x},z_{0}-y_{0})$ (69)

given by

$A(\vec{x},x_{0}-z_{0})={1\over\beta}\sum_{k_{0}}e^{-ik_{0}(x_{0}-z_{0})}\sqrt{f_{h}}\quad\quad
B(\vec{x},y_{0}-z_{0})={1\over\beta}\sum_{k_{0}}{e^{-ik_{0}(y_{0}-z_{0})}\sqrt{f_{h}}\over-
ik_{0}+\bar{\phi}(\vec{\omega}\vec{x})-\bar{\phi}(\alpha)}$

Moreover $||A_{h}||^{2}=\int dz_{0}|A_{h}(x^{\prime},z_{0})|^{2}\leq
C\gamma^{h}$ and $||B_{h}||^{2}\leq C\gamma^{-h}$ so that By Gram-Hadamard
inequality we get:

$|{\rm det}{\widetilde{G}}^{h_{v},T_{v}}({\bf t}_{v})|\leq
C^{\sum_{i=1}^{S_{v}}|P_{v_{i}}|-|P_{v}|-2(S_{v}-1)}\;.$ (70)

One get therefore the bound, for $|\lambda|,|\vec{v}_{h}|\leq\varepsilon_{0}$,

$|K_{\tau,{\bf P},T}^{(h+1)}({\bf x}_{v_{0}})|\leq
C^{n}\varepsilon_{0}^{n}\prod_{v}{1\over S_{v}!}\gamma^{-h_{v}(S_{v}-1)}$ (71)

which is not suitable for summing over $\tau$ and $P$. In order to improve the
above bound we need to implement in the bounds some constraints which have
been neglected in the derivation of (71), and to take into account the effect
of the presence of the $D$ fields.

We define $V_{\chi}$ the set of non trivial vertices or the trivial ones with
non zero internal lines; we define $v^{\prime}$ the first vertex in $V_{\chi}$
following $v$. We say that $v$ is a non-resonant vertex if in
${\widetilde{\psi}}(P_{v})$ there are at least two different coordinates, and
a resonant vertex when all coordinates are equal. We define
$S_{v}=S^{L}_{v}+S^{H}_{v}$ where $S^{L}_{v}$ is the number of non resonant
subtrees (including trivial ones) and $S^{H}_{v}$ the number of resonant ones
(inluding trivial ones). We also call $H$ the set of $v\in V_{\chi}$ which are
resonant and $L$ the $v\in V_{\chi}$ which are non resonant. Consider a non
resonant vertex $v$ so that there are at least two fields in $P_{v}$ with
different spatial coordinates $\vec{x}$, say
$\vec{x}_{w_{1}}\not=\vec{x}_{w_{2}}$. The fields ${\widetilde{\psi}}^{(\leq
h_{v})}(P_{v})$ have scale $\leq\gamma^{h_{v^{\prime}}}$, $v^{\prime}\in
V_{\chi}$ the first vertex belonging to $V_{\chi}$ after $v$ so that

$||(\vec{\omega}\vec{x}_{w_{1}})-\rho_{1}\alpha||_{\hbox{\msytw T}}\leq
cv_{0}^{-1}\gamma^{h_{v^{\prime}}-1}\quad\quad||(\vec{\omega}\vec{x}_{w_{2}})-\rho_{2}\alpha||_{\hbox{\msytw
T}}\leq cv_{0}^{-1}\gamma^{h_{v^{\prime}}-1}$ (72)

so that

$2cv_{0}^{-1}\gamma^{h_{v^{\prime}}}\geq||(\vec{\omega}\vec{x}_{w_{1}})-\rho_{1}\alpha||_{\hbox{\msytw
T}}+||(\vec{\omega}\vec{x}_{w_{2}})-\rho_{2}\alpha||_{\hbox{\msytw
T}}\geq||\vec{\omega}(\vec{x}_{w_{1}}-\vec{x}_{w_{2}})-(\rho_{1}-\rho_{2})\alpha||_{\hbox{\msytw
T}}$ (73)

and by (65)

$2cv_{0}^{-1}\gamma^{h_{v^{\prime}}}\geq||\vec{\omega}(\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}})+(\rho_{1}-\rho_{2})\alpha||_{\hbox{\msytw
T}}\geq{C_{0}\over|\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}}|^{\tau}}$ (74)

where the Diophantine conditions have been used. Therefore

$\sum_{w\in c_{w_{1},w_{2}}}|\vec{\delta}_{w}^{i_{w}}|\geq|\sum_{w\in
c_{w_{1},w_{2}}}\vec{\delta}_{w}^{i_{w}}|\geq C\gamma^{-h_{v^{\prime}}/\tau}$
(75)

and, if $N_{v}$ is the number of end-points following $v$ in $\tau$

$\sum_{w\in c_{w_{1},w_{2}}}|\vec{\delta}_{w}^{i_{w}}|\leq N_{v}$ (76)

as $|\vec{\delta}_{w}^{i_{w}}|=0,1$ so that

$N_{v}\geq C\gamma^{-h_{v^{\prime}}/\tau}$ (77)

Note that to each endpoint is associated a small factor $\varepsilon_{0}$ and
the fact that $N_{v}$ is large by (77) produces a gain for the $v$ with the
fields with different $\vec{x}$. Of course there can be several $\bar{T}_{v}$
with different $v$ passing through the same end-points. Therefore, given a
constant $c<1$, we can multiply the contribution to each tree $\tau$ with
$n$-endpoints by $c^{-n}c^{n}$ (the factor $c^{-n}$ is of course armless); we
can then write

$c=\prod_{h=-\infty}^{0}c^{2^{h-1}}$ (78)

and associate to each $v$ a factor $c^{N_{v}2^{h-1}}$. If there are two fields
in $P_{v}$ (that is external to the cluster $v$) with different $\vec{x}$ we
get in the bounds, by assuming $\gamma^{1\over\tau}/2\equiv\gamma^{\eta}>1$
than, for any $N$

$c^{A\gamma^{-h\over\tau}2^{h}}=e^{-|\log c|A\gamma^{-\eta
h}}\leq\gamma^{N\eta h}{N\over[|\log|c||A]^{N}e^{N}}$ (79)

as $e^{-\alpha x}x^{N}\leq[{N\over\alpha}]^{N}e^{-N}$, and we can choose
$N=3/\eta$; therefore given a couple of fields external to a vertex $v$ with
different $\vec{x}$, we can associate a factor $\gamma^{2h_{v^{\prime}}}$ in
the bounds.

On the other hand if there is a $D$ field we get in the bound an extra
$\gamma^{h_{v^{\prime}}-h_{v}}$ from the expression

$\bar{g}^{(h_{v^{\prime}})}(\vec{\omega}\vec{x},x_{0,1}-z_{0})-\bar{g}^{(h_{v^{\prime}})}(\vec{\omega}\vec{x},x_{0,2}-z_{0})=(x_{0,1}-x_{0,2})\int_{0}^{1}dt\partial\bar{g}^{(h_{v^{\prime}})}(\vec{\omega}\vec{x},\widehat{x}_{0,1,2}(t)-z_{0})$
(80)

where $\widehat{x}_{0,1,2}(t)=x_{0,1}+t(x_{0,2}-x_{0,1})$. In conclusion

  1. 1.

To each non-resonant $v$ we associate a factor (79) so that we get in the
bound an extra factor $\prod_{v\in V_{\chi}}\gamma^{2h_{v}S_{v}^{L}}$

  2. 2.

There is a factor $\prod^{*}_{v}\gamma^{h_{v^{\prime}}}$ where $v$ are the
endpoints $\nu,\alpha,\xi$ (it comes from the definition of $\nu$ and the
presence $(x_{0}-y_{0})$ or $(\vec{\omega}\vec{x}-\rho\alpha$).

  3. 3.

In the resonant $v$ with $l\geq 2$ fields there is a factor $\prod_{v\in
H}\gamma^{2(h_{v^{\prime}}-h_{v})}$. For $l=2$ this it is due to the ${\cal
R}$ definition, for $l\geq 4$ by anticommutativity.

  4. 4.

In the terms with $|P_{v}|\geq 8$ we can consider the fields
$\psi^{\varepsilon}_{x}$ whose number is maximal; we can group them in couples
connected by path in $\bar{T}$ non overlapping, and or have different
$\vec{x}$, hence there is a path in $\bar{T}$ connecting them giving an extra
$\gamma^{2h_{v^{\prime}}}$, or they have the same $\vec{x}$ so that there is
an extra $\gamma^{2(h_{v^{\prime}}-h_{v})}$. This produces an extra
$\gamma^{-\alpha|P_{v}|}$, see §F in [36].

We bound first the effective potential ($J=0$). If $\tau\in{\cal T}_{h,n}$,
the set of trees with $n$ end-points and defining

$||K_{\tau,{\bf P},T}^{(h+1)}||={1\over\beta L^{d}}\sum_{\vec{x}}\int
dx_{0,v_{0}}|K_{\tau,{\bf P},T}^{(h+1)}|$ (81)

we get

$||K_{\tau,{\bf P},T}^{(h+1)}||\leq C^{n}\varepsilon_{0}^{n}\prod_{v}{1\over
S_{v}!}\gamma^{-h_{v}(S_{v}-1)}\prod_{v\in
V_{\chi}}\gamma^{2h_{v}S_{v}^{L}}\prod^{*}_{v}\gamma^{h_{v^{\prime}}}\prod_{v\in
H}\gamma^{2(h_{v^{\prime}}-h_{v})}\prod_{v\in
V_{\chi}}\gamma^{-\alpha|P_{v}|}$ (82)

If the first vertex $v_{0}\in V_{\chi}$ is non resonant we get

$\prod_{v\in
V_{\chi}}\gamma^{-h_{v}S_{v}}\prod_{v}\gamma^{h_{v}S^{L}_{v}}\prod^{*}_{v}\gamma^{h_{v^{\prime}}}\prod_{v\in
H,v\not=v_{0}}\gamma^{h_{v^{\prime}}}=1\quad\quad\prod_{v\in
V_{\chi}}\gamma^{h_{v}}\prod_{v\in
H,v\not=v_{0}}\gamma^{-h_{v}}\leq\gamma^{h_{v_{0}}}$ (83)

We use that $S_{v}=S_{v}^{L}+S_{v}^{H}$,
$\prod_{v}\gamma^{h_{v}S^{L}_{v}}=\prod_{v\in
L}\gamma^{h_{v^{\prime}}}\prod^{**}_{v}\gamma^{h_{v}}$, with $\prod^{**}_{v}$
is over the first vertex $v\in V_{\chi}$ after the $\varepsilon,\lambda$
endpoints, and that $\prod_{v\in L}\gamma^{h_{v^{\prime}}}\leq\prod_{v\in
L}\gamma^{h_{v^{\prime}}-h_{v}}$

$\displaystyle||K_{\tau,{\bf P},T}^{(h+1)}||\leq
C^{n}\varepsilon_{0}^{n}\gamma^{h_{v_{0}}}\prod_{v}{1\over S_{v}!}\prod_{v\in
V_{\chi}}\gamma^{(h_{v^{\prime}}-h_{v})}\prod^{**}_{v}\gamma^{h_{v}}\prod_{v\in
V_{\chi}}\gamma^{-\alpha|P_{v}|}$ (84)

where $\prod^{**}_{v}$ is over the vertices $v\in V_{\chi}$ following from the
end-points associated to $\varepsilon,\lambda$. Note that $\sum_{\bf
P}[\prod_{v\in V_{\chi}}\gamma^{-{1\over 8}|P_{v}|}]\leq C^{n}$; moreover
$\sum_{\bf T}[\prod_{v}{1\over S_{v}!}]\leq C^{n}$. The sum over the trees
$\tau$ is done performing the sum of unlabeled trees and the sum over scales.
The unlabeled trees can be bounded by $4^{n}$ by Caley formula, and the sum
over the scales reduces to the sum over $h_{v}$, with $v\in V_{\chi}$, as
given a tree with such scales assigned, the others are of course determined.

Let us consider now the case in which the first vertex $v_{0}$ is resonant; we
can distinguish two cases. If we are considering the contribution to the beta
function then there is no ${\cal R}$ applied in $v_{0}$ so that the same bound
as above is found with $h_{v_{0}}=h+1$. Instead if ${\cal R}$ is applied we
get instead of (83), as there is an extra
$\gamma^{h_{v^{\prime}_{0}}-h_{v_{0}}}$

$\prod_{v\in
V_{\chi}}\gamma^{-h_{v}S_{v}}\prod_{v}\gamma^{h_{v}S^{L}_{v}}\prod^{*}_{v}\gamma^{h_{v^{\prime}}}\prod_{v\in
H}\gamma^{h_{v^{\prime}}}=\gamma^{h_{v^{\prime}_{0}}}\quad\quad\prod_{v\in
V_{\chi}}\gamma^{h_{v}}\prod_{v\in H}\gamma^{-h_{v}}\leq 1$ (85)

and the same bound is found, as $h_{v^{\prime}_{0}}=h+1$. In conclusion we get

$\sum_{\tau\in{\cal T}_{h,n}}\sum_{{\bf P},T}||K_{\tau,{\bf
P},T}^{(h+1)}||\leq C^{n}\varepsilon_{0}^{n}\gamma^{h}$ (86)

The running coupling constant $\alpha_{h},\xi_{h}$ verify

$\alpha_{h-1}=\alpha_{h}+O(\varepsilon_{0}^{2}\gamma^{h\over
2})\quad\quad\xi_{h-1}=\xi_{h}+O(\varepsilon_{0}^{2}\gamma^{h\over 2})$ (87)

where the factor $\gamma^{h\over 2}$ is due to the fact that the trees have at
least an $\varepsilon,\lambda$ endpoint, from the factor
$\prod^{**}_{v}\gamma^{h_{v}}$ in (84) (short memory property). The flow of
$z_{h},\alpha_{h}$ is therefore summable; in addition one can choose $\nu$ so
that $\nu_{h}$ is bounded, by proceeding as in Lemma 2.7 of citeM3.

##  8 Decay of correlations

We consider now the current correlations, which can be written as

$H_{\mu,\nu}({\bf x},{\bf y})=\sum_{h,n}\sum_{\tau\in{\cal
T}_{h,n+2}}\sum_{{\bf P},T}G_{\tau,{\bf P},T}({\bf x},{\bf y})$ (88)

where ${\cal T}_{h,n+2}$ is the set of trees with $n+2$ end-points, two of
them associated to the $J$ end-points. In the trees $\tau$ we can identify a
vertex $v_{x}$ for the end-point corresponding to $J_{\bf x}$, and $v_{y}$ for
the end-point corresponding to $J_{\bf y}$ with $h_{v_{x}}=h_{v_{y}}=+2$; we
call $\widehat{v}$, with scale $\widehat{h}$, the first vertex $v\in V_{\chi}$
such that $v_{x},v_{y}$ follows $\widehat{v}$, and $v_{0}$ the first vertex
$\in V_{\chi}$, with scale $h$. There are several constraints.

  1. 1.

By (65) and using that $\vec{x}-\vec{y}=\sum_{w\in
C_{v_{x},v_{y}}}\vec{\delta}_{w}^{i_{w}}$ we get $n\geq\sum_{w\in
C_{v_{x},v_{y}}}|\vec{\delta}_{w}^{i_{w}}|\geq|\vec{x}-\vec{y}|$

  2. 2.

$h\geq\bar{h}(n)$ with, if $|\vec{z}|=1+\min(|\vec{x}|,|\vec{y}|)$

$\gamma^{-\bar{h}}\leq\sup_{\vec{q}=\sum_{i=1}^{n}\vec{e}_{i}}{1\over||{\vec{\omega}(\vec{x}+\vec{q})-\rho\alpha}||}\leq
C(|\vec{z}|+n)^{\tau}$ (89)

With respect to the bound for the $J=0$ case there are the following
differences. If $T_{\widehat{v}}$ is the tree connecting the 2 $J$ endpoints,
we have an extra $\gamma^{\widehat{h}}$ due to the fact that we do not
integrate over the coordinates of the $J$ fields, and we can extract from the
the propagators in $\prod_{l\in\bar{T}_{\widehat{v}}}g^{(h_{l})}$,
$h_{l}\geq\widehat{h}$ a decay factor

${1\over 1+(\gamma^{\widehat{h}}|x_{0}-y_{0}|)^{N}}$ (90)

Moreover there is no ${\cal R}$ in the resonant terms with one or two external
$J$ lines. We can multiply and divide by
$\gamma^{-4\bar{h}}\gamma^{4\bar{h}}$: we can select two paths in $\tau$
$v_{0}<v_{1}<..v_{x}$ and $v_{0}<v^{\prime}_{1}<..v_{y}$, writing

$\gamma^{2\bar{h}}=\gamma^{2(\bar{h}-h_{v_{1}})}...\gamma^{2h_{v^{\prime}_{x}}}\quad\quad\gamma^{2\bar{h}}=\gamma^{2(\bar{h}-h_{v^{\prime}_{1}})}...\gamma^{2h_{v^{\prime}_{y}}}$
(91)

where $v^{\prime}_{x}$, $v^{\prime}_{y}$ are the first vertex $\in V_{\chi}$
after $v_{x}$, $v_{y}$. We get therefore the following bound

$|G_{\tau,{\bf P},T}({\bf x},{\bf
y})|\leq\gamma^{-4\bar{h}}{C^{n}|\varepsilon|^{n}\gamma^{\widehat{h}}\over(\gamma^{\widehat{h}}|x_{0}-y_{0}|)^{N}}\prod_{v}{1\over
S_{v}!}\gamma^{-h_{v}(S_{v}-1)}\prod_{v\in
V_{\chi}}\gamma^{2h_{v}S_{v}^{L}}\prod^{*}_{v}\gamma^{h_{v}}\prod_{v\in
H}\gamma^{2(h_{v^{\prime}}-h_{v})}\prod_{v\in
V_{\chi}}\gamma^{-\alpha|P_{v}|}$ (92)

where $H$ now includes also resonant terms with one or two $J$ fields.
Proceeding as in §7 and for $|x_{0}-y_{0}|>1$, if ${\cal T}_{n}$ are the trees
with $n$ end-points

$\sum_{\tau\in{\cal T}_{h,n}}\sum_{{\bf P},T}|G_{\tau,{\bf P},T}({\bf x},{\bf
y})|\leq\gamma^{-3\bar{h}}{C^{n}|\varepsilon|^{n}\over
1+(\gamma^{\bar{h}}|x_{0}-y_{0}|)^{N}}\leq
C^{n}|\varepsilon|^{n}{|\vec{z}|^{3\tau}\over(|\vec{z}|^{-3\tau}|x_{0}-y_{0}|)^{N}}(1+{n\over|\vec{z}|})^{(N+3)\tau}$
(93)

The sum over $h\geq\bar{h}$ can be bounded by an an extra $\gamma^{-\bar{h}}$.
As $|\vec{z}|\geq 1$ and $n/|\vec{z}|\leq n$; we can sum over $n$ obtaining,
remembering the constraint $n\geq|\vec{x}-\vec{y}|$

$|H_{\mu,\nu}({\bf x},{\bf y})|\leq
C{|\vec{z}|^{4\tau}\over(|\vec{z}|^{-3\tau}|x_{0}-y_{0}|)^{N}}|\varepsilon|^{|\vec{x}-\vec{y}|/4}$
(94)

The analysis of the 2-point function is done in a similar way; there are 2
endpoints associated with the externl fields, so with respect to the bound for
the effective potential there is an extra factor $\gamma^{-2\bar{h}}$ and an
extra $\gamma^{\bar{h}}$ from the lack of integration; the sum over the scales
produces an extra $|\bar{h}|$. Acknowledgements. This work has been supported
by MIUR, PRIN 2017 project MaQuMA, PRIN201719VMAST01.

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