# Extended quasiparticle Padé approximation for non-Fermi liquids

Klaus Morawetz1,2 1Münster University of Applied Sciences, Stegerwaldstrasse
39, 48565 Steinfurt, Germany  2International Institute of Physics- UFRN,
Campus Universitário Lagoa nova, 59078-970 Natal, Brazil

###### Abstract

The extended quasiparticle picture is adapted to non-Fermi systems by
suggesting a Padé approximation which interpolates between the known small
scattering-rate expansion and the deviation from the Fermi energy. The first
two energy-weighted sum rules are shown to be fulfilled independent of the
interpolating function for any selfenergy. For various models of one-
dimensional Fermions scattering with impurities the quality of the Padé
approximation for the spectral function is demonstrated and the reduced
density matrix or momentum distribution is reproduced not possessing a jump at
the Fermi energy. Though the two-fold expansion is necessary to realize the
spectral function and reduced density, the extended quasiparticle
approximation itself is sufficient for the description of transport properties
due to cancellation of divergent terms under integration. The T-matrix
approximation leads to the delay time as the time two particles spend in a
correlated state. This contributes to the reduced density matrix and to an
additional part in the conductivity which is presented at zero and finite
temperatures. Besides a localization at certain impurity concentrations, the
conductivity shows a maximum at small temperatures interpreted as onset of
superconducting behaviour triggered by impurities. The Tan contact reveals the
same universal behaviour as known from electron-electron scattering.

##  I Introduction

One-dimensional interacting Fermions have been investigated intensively due to
their unusual properties. Examples of interacting one-dimensional systems
range from large-scale structures like crystalline ion beams observed in high-
energy storage rings SSH01 ; SSBH02  up to one-dimensional quantum wires PMC14
experimentally found in carbon nanotubes Saito98 ; Bockrath99 ; Ishii03 ;
Shiraishi03 . Also edge states in quantum hall liquid Milliken96 ; Mandal01 ;
Chang03 , semiconducting nanowires Schafer08 ; Huang01 , cold atomic gases
Monien98 ; Recati03 ; Moritz05  and conducting molecules Nitzan03  are one-
dimensional quantum systems.

Due to these versatile applications it is important to understand the ground
state and transport properties of quantum wires. Frequently bosonization
techniques in Lu77 ; Solyom79  and out of GGM10  equilibrium are employed
which are based on the similar behaviour of long-distance correlations of
Fermi and Bose systems H81a . Exact solutions are available for Luttinger L63
; LP74 ; ES07 , Tomonaga DL74 , and Gaudin-Yang models Ram17  of contact
interaction by the Bethe ansatz EFGKK10 ; GBML13 . The ground-state properties
of one-dimensional quantum wires have been analytically and numerically
investigated LD11 ; LG16 . There the high-density expansion Loos13  of the
ground-state energy was discussed in dependence on the width GADMP22  of the
wire. The weak coupling corresponds to the high-density regime and the strong
coupling regime to low densities due to the peculiar density dependence of the
kinetic and interaction energy in one dimension GPS08 .

Green functions allow to describe interacting models beyond exactly solvable
cases and in various approximations T67 ; VMKA08 . Especially the transition
between Tomonaga-Luttinger and Fermi liquids can be investigatedSch77 ; Yo01 .
For an overview about theoretical models see V94 ; G04 ; GV08 . With the help
of the Green functions transport properties of correlated systems have been
successfully described by the extended quasiparticle picture which relies on
small scattering rates. The limit of small scattering rates was first
introduced by C66a  for highly degenerated Fermi liquids, later used in SZ79 ;
KKL84  for equilibrium nonideal plasmas. The same approximation, but under the
name of the generalised Beth-Uhlenbeck approach, has been used by SR87 ; MR94
in nuclear matter studies of the correlated density or in the kinetic equation
for nonideal gases, BKKS96 . The resulting quantum kinetic equation unifies
the achievements of transport in dense gases with the quantum transport of
dense Fermi systems SLM96 ; MLS00 ; M17b . In this kinetic equation the
quasiparticle drift of Landau’s equation is connected with a dissipation
governed by a nonlocal and non-instant scattering integral in the spirit of
Enskog corrections SLM98 . These corrections are expressed in terms of shifts
in space and time which characterize the non-locality of the scattering
process MLSK98 . In this way quantum transport is possible to recast into a
quasi-classical picture. The nonequilibrium quantum hydrodynamics resulting
from this nonlocal kinetic theory can be found in M17 ; M17b .

These successful perturbation approaches fail for one-dimensional electron
systems since the Fermi surface disappears at T = 0 indicating the breakdown
of single-particle excitation turning them into collective ones SDC19 .
Quasiparticles are not developed at the Fermi energy DBG10  as seen e.g. in
Hubbard Hamiltonian near half filling GVM93 . Nevertheless such excitations
can show up eventually at the Luttinger surface where the Green functions have
a zero at zero energy FA22 . Due to the absence of quasiparticles, expansions
are necessary beyond the quasiparticle pole approximation. Unfortunately the
expansion of small scattering rates relies on the finite behaviour of the
reduced density at the Fermi energy and fails for non-Fermi liquids due to the
divergence at the Fermi energy. Here we will suggest a twofold expansion as
possible solution and a Padé interpolation between both expansions. This will
reproduce the spectral function and the reduced density or momentum
distribution at the Fermi energy. The absence of Fermi surface is seen also in
the momentum distribution (reduced density) which shows in one-dimensional
Hubbard models Schu90 ; BVB90 , Gaudin-Yang model Ram17 , electron-impurity,
and electron-phonon systems BS19  or Tomonaga models GS67  a continuous
behaviour at the Fermi momentum controlled by the Luttinger behaviour. This
reduced density is accompanied by a universal momentum asymptotic of $1/p^{4}$
reported of both free and harmonically trapped atoms for all values of the
interaction strength OD03 . We will show that this continuous behaviour can be
reproduced if the extended quasiparticle approximation is accompanied by the
regulator of the suggested Padé approximation.

For transport properties as integrals over the reduced density we will see
that contribution of the Padé approximations related to the interpolating
function are negligible and the results of the extended quasiparticle picture
remains. In this way the validity of the extended quasiparticle picture is
extended beyond the originally derived perturbation of small scattering rates.

As illustration of this suggested Padé regularization, we consider only
scattering of electrons on impurities and assume that any electron-electron
interaction is screened off. Similar models have been investigated in second
Born approximation in BS19  with a screened Debye potential. Here we will even
consider the simpler case of contact interaction but will discuss the Born and
the T-matrix approximations M02 . The aim is not to describe the system as
exact as possible but to illustrate the expansion scheme of the spectral
function and how a two-term Padé interpolation can reproduce the spectral
function and reduced density matrix. This will repair the deficiency of the
extended quasiparticle approximation.

The exact Bethe ansatz solution shows that an impurity is dressed up by
surrounding particles and in the strongly attractive limit, it forms a dimer
with inner Fermions in the Fermi sea SZ19  similar to the highly imbalanced 1D
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state Fulde64 ; Larkin65 . In this
respect it has been reported that quasi-one-dimensional superconductivity can
be enhanced by disorder with screened Coulomb interaction PAC16 . Various
other model systems have been found for quasi-one-dimensional superconductors
Be11 ; WSLS12 ; He15 . Some synthesis and fluctuation problems are overcome by
building superconductors from inhomogeneous composites of nanofilaments PHC16
. We will show the onset of superconductivity in a one-dimensional electron
system interacting with impurities at finite momentum. It will be seen
further-on that any approximation beyond Born, like e.g. the T-matrix
summation, requires the inclusion of the correlated density due to scattering
delay. The latter one leads to a correction of transport properties, e.g. the
conductivity, due to the two-particle correlations.

The outline is as follows. In the next chapter we briefly give the many-body
scheme with the focus on the extended quasiparticle picture and discuss its
failure due to the behaviour at the Fermi energy. This leads to the suggestion
of a twofold expansion interpolated by a Padé formula. In chapter III this
twofold expansion is derived and the two first energy-weighted sum rules are
proven to be completed independent on the interpolating function. In chapter
IV the Padé expression is applied successively to higher-order approximations
starting first with a model of constant damping proceeding to a model of
impurity scattering with contact interaction in Born and T-matrix
approximation. The influence of the bound-state poles are illustrated in the
latter approximation. Two necessary amendments are found, the correction of
energy argument of the selfenergy in order to reproduce the Fermi energy and
the correlated density besides the free density. The reason of the latter is
the delay time appearing in approximations with a dynamical vertex as two-
particle correlation which is explained in chapter V. The Tan contact is given
and compared to other approaches showing a universal behaviour. In chapter VI
we discuss the effect of the delay time to the conductivity and give the
finite temperature expression. It shows a localization at certain impurity
densities and an enhancement at small temperatures as a possible onset of
superconducting behaviour. Chapter VII summarizes and the appendix gives the
Euler expansion of the $arctan$ function and the proof of cancellation of
interpolation-function related terms for transport. This concludes the
reasoning why the extended quasi-particle picture works also for correlated
one-dimensional systems.

##  II Many-body scheme

###  II.1 Extended quasiparticle approximation

The Dyson equation for the causal propagator $G(1,2)=-i\langle{\cal
T}a_{1}^{+}a_{2}\rangle$ with time ordering ${\cal T}$ reads

$\displaystyle G(1,2)=G_{0}(1,2)+G_{0}(1,3)\Sigma(3,4)G(4,2)$ (1)

in terms of the Hartree-Fock propagator

$\displaystyle\left(i\frac{\partial}{\partial
t_{1}}\\!+\\!\frac{\nabla_{1}^{2}}{2m}\right)G_{0}(1,2)\\!-\\!\Sigma^{HF}(1,3)G_{0}(3,2)=\delta(1-2)$
(2)

with the time-diagonal Hartree-Fock self energy

$\displaystyle\Sigma^{HF}(1,2)$ $\displaystyle=$
$\displaystyle\left[V(x_{1}\\!-\\!x_{2})G(2,2^{+})\\!-\\!V(x_{1}\\!-\\!x_{2})G(1,2)\right]$
(3) $\displaystyle\times\delta(t_{1}\\!-\\!t_{2}),$

and the correlation selfenergy $\Sigma(1,2)$. It is integrated about double
occurring indices, $1=(\vec{x}_{1},t_{1})$. The interaction potential is $V$
and $2^{+}$ signs an infinitesimal later time than $t_{2}$.

Since we concentrate on equilibrium, all quantities are only dependent on the
difference of coordinates. One Fourier transforms these difference coordinates
into frequency/momentum. We will denote these transformed quantities by small
letters. Nonequilibrium expressions can be found in M17b .

The real part of the selfenergy is the Hilbert transform

$\displaystyle\sigma(\omega,q)={\rm
Re}\,\sigma^{R}(\omega,q)=\int{d\bar{\omega}\over
2\pi}{\gamma(q,\bar{\omega})\over\omega-\bar{\omega}}$ (4)

of the selfenergy spectral function

$\displaystyle\gamma=\sigma^{>}+\sigma^{<}=i(\sigma^{R}-\sigma^{A})=-2{\rm
Im}\,\sigma^{R}.$ (5)

Both specifying the retarded selfenergy

$\displaystyle\sigma^{R}(\omega,k)=\sigma(\omega,k)-\frac{i}{2}\gamma(\omega,k)=\int{d\bar{\omega}\over
2\pi}{\gamma(\bar{\omega},k)\over\omega-\bar{\omega}+i\eta}.$ (6)

The retarded propagator we obtain from (1) by Langreth/Wilkins LW72  rules
leading to the same form of equation which are easily solved in equilibrium as

$\displaystyle
g^{R}(\omega,k)={1\over\omega-\epsilon_{k}^{0}-\sigma^{R}(\omega,k)}$ (7)

where we abbreviate in the following

$\displaystyle\varepsilon^{0}_{k}=\epsilon_{k}+\sigma^{\rm HF}_{k}.$ (8)

The spectral function follows

$\displaystyle a(\omega,k)$ $\displaystyle=$ $\displaystyle g^{>}\pm
g^{<}=i(g^{R}-g^{A})$ (9) $\displaystyle=$
$\displaystyle{\gamma(\omega,k)\over[\omega-\epsilon_{k}^{0}-\sigma(\omega,k)]^{2}+{\gamma(\omega,k)^{2}\over
4}}$

for Bose/Fermi particles. The poles of (9) at

$\displaystyle\varepsilon_{k}=\varepsilon^{0}_{k}+\sigma(\varepsilon_{k},k)$
(10)

describe the quasiparticle excitations by the selfconsistent energy
$\varepsilon_{k}$ and $\gamma$ represents the quasiparticle damping. For
question concerning the convergence of different many body expansions see Fa21
.

Within the extended quasiparticle picture we expand the spectral and
correlation functions with respect to the order of damping

$\displaystyle a^{\rm
EQP}\\!(\omega,k)={2\pi\delta(\omega\\!-\\!\varepsilon_{k}\\!)\over
1\\!-\\!{\partial\sigma(\omega,k)\over\partial\omega}}n_{\varepsilon_{k}}\\!+\\!\gamma(\omega,k){\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}\\!+\\!o(\gamma^{2})$
(11)

and correspondingly

$\displaystyle g^{<}(\omega,k)={2\pi\delta(\omega\\!-\\!\varepsilon_{k})\over
1\\!-\\!{\partial\sigma(\omega,k)\over\partial\omega}}n_{\varepsilon_{k}}\\!+\\!\sigma^{<}(\omega,k){\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}\\!+\\!o(\gamma^{2})$
(12)

with the derived principal value
${\wp^{\prime}\over\omega}=-\partial_{\omega}{\wp\over\omega}$ and the
equilibrium distribution function $n_{\omega}$, i.e. the Fermi- or
Bosefunction respectively. The form (12) was presented with respect to small
damping (scattering rate) expansion and with quasiparticle energies under the
name of extended quasiparticle approximation in nonequilibrium SL94 ; SL95
and used for transport in impurity systems SLMa96 ; SLMb96 . The nonlocal
kinetic theory finally is based on this expansion SLM96 ; MLS00 ; M17b .

If we integrate (12) over the energy $\omega$ we get the connection between
the reduced density matrix $\rho$ (momentum distribution) and the free
(quasiparticle) distribution $n_{\varepsilon_{k}}$ as

$\displaystyle\rho(k)$ $\displaystyle=$ $\displaystyle
n_{\varepsilon_{k}}+\int{d\omega\over
2\pi}{\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}\left[\sigma^{<}(\omega,k)-\gamma(\omega,k)n_{\varepsilon_{k}}\right]$
(13) $\displaystyle=$ $\displaystyle n_{\varepsilon_{k}}+\int{d\omega\over
2\pi}{\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}\gamma(\omega,k)\left[n_{\omega}-n_{\varepsilon_{k}}\right].$

The quasiparticle distribution $n_{\varepsilon_{k}}$ is to be taken at the
pole $\omega=\varepsilon_{k}$ of the spectral function (10) and the second
line of (13) is valid in equilibrium where we have
$\sigma^{<}(\omega,k)=n_{\omega}\gamma(\omega,k)$.

For Fermions in the ground state,
$n_{\epsilon}=\Theta(\epsilon_{F}-\epsilon)$, observing that

$\displaystyle
n_{\omega}\\!-\\!n_{\varepsilon}=\Theta(\varepsilon\\!-\\!\epsilon_{F})\Theta(\epsilon_{F}-\omega)\\!-\\!\Theta(\epsilon_{F}\\!-\\!\varepsilon)\Theta(\omega\\!-\\!\epsilon_{F})$
(14)

we can further rewrite (13) into

$\displaystyle\rho(k)=$
$\displaystyle\Theta(\epsilon_{F}-\varepsilon_{k})+\Theta(\varepsilon_{k}\\!-\\!\epsilon_{F})\int\limits_{-\infty}^{\epsilon_{F}}{d\omega\over
2\pi}{\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}\gamma(\omega,k)$
$\displaystyle-\Theta(\epsilon_{F}\\!-\\!\varepsilon_{k})\int\limits^{\infty}_{\epsilon_{F}}{d\omega\over
2\pi}{\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}\gamma(\omega,k).$ (15)

This shows that if we approach the Fermi energy from above
$\varepsilon_{k}=\epsilon_{F}+0$ we have the value of the second term in (15)

$\displaystyle z_{+}=\int\limits_{-\infty}^{0}{d\omega\over
2\pi}{\wp^{\prime}\over\omega}\gamma(\omega+\epsilon_{F},k_{F})$ (16)

and approaching from below $\varepsilon_{k}=\epsilon_{F}-0$ the step function
is reduced by a factor given by the third term in (15)

$\displaystyle z_{-}=1-\int\limits^{\infty}_{0}{d\omega\over
2\pi}{\wp^{\prime}\over\omega}\gamma(\omega+\epsilon_{F},k_{F}).$ (17)

Together one sees a jump at the Fermi energy of

$\displaystyle z_{-}-z_{+}$ $\displaystyle=$ $\displaystyle
1-\int\limits^{\infty}_{-\infty}{d\omega\over
2\pi}{\wp^{\prime}\over\omega}\gamma(\omega+\epsilon_{F},k_{F})$ (18)
$\displaystyle=$ $\displaystyle
1+\left.\partial_{\omega}\sigma(\omega,k_{F})\right|_{\omega=\epsilon_{F}}$

i.e. just the wave-function renormalization as the factor of the pole in (11).
This typical Fermi-liquid behaviour is present as long as the imaginary part
of selfenergy, the damping $\gamma$ of the quasiparticles, vanishes at the
Fermi surface

$\displaystyle\lim\limits_{\omega\to 0}\gamma(\omega+\epsilon_{F},k_{F})=0$
(19)

in order to render (16) and (17) finite as necessary condition. If this
requirement is not fulfilled then the integrals diverge and the perturbation
theory breaks down. This we explore in the following section.

###  II.2 Problems with the extended quasiparticle approximation for non-Fermi
liquids

####  II.2.1 Model of constant quasiparticle damping

The problem of the perturbation expansion and resulting extended quasiparticle
picture is best understood by a model of constant quasiparticle damping
$\gamma(\omega,k)=\gamma$ and consequently $\sigma=0$ and
$\varepsilon_{k}=\epsilon_{k}$ due to (4). The exact reduced density matrix
reads

$\displaystyle\rho(k)$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}{d\omega\over
2\pi}a(\omega,k)n_{\omega}=\int\limits_{-\infty}^{\epsilon_{F}}{d\omega\over
2\pi}{\gamma\over(\omega-\epsilon_{k})^{2}+{\gamma^{2}\over 4}}$ (20)
$\displaystyle=$ $\displaystyle\frac{1}{2}+\frac{1}{\pi}{\rm
arctan}{\epsilon_{F}-\epsilon_{k}\over\gamma/2}.$

The extended quasiparticle approximation as the expansion with respect to
$\gamma$ of (15) would be

$\displaystyle\rho(k)=\Theta(\epsilon_{F}-\varepsilon_{k})-\gamma{\wp\over\epsilon_{F}-\varepsilon_{k}}$
(21)

and shows a divergence at the Fermi energy. This puzzling failure can be seen
from the exact expression (20). The $arctan$ has a cut such that the expansion
near the Fermi energy, which means at small values of $\epsilon_{F}-\epsilon$,
is different from the expansion of small damping $\gamma$,

$\displaystyle{\rm
arctan}(x)=\left\\{\begin{array}[]{l}x+o(x^{2})\cr\frac{\pi}{2}{\rm
sig}(x)-{1\over x}+o({1\over x^{2}})\end{array}\right.$ (24)

which explains why the expansion diverges at
$x=2(\epsilon_{F}-\varepsilon_{k})/\gamma\to 0$, seen in figure 1.

A somewhat better expansion which does not diverge at the Fermi energy was
found by Euler in 1755, see appendix A,

$\displaystyle{\rm arctan}(x)=\left\\{\begin{array}[]{l}{x\over
1+x^{2}}+o(x^{2})\cr\frac{\pi}{2}{\rm sig}(x)-{x\over 1+x^{2}}+o({1\over
x^{2}})\end{array}\right.$ (27)

which does not diverge at $x=0$ but remains discontinuous as seen in figure 1.

Figure 1: The reduced density matrix (20) (thick) together with the extended
quasiparticle picture (dashed) and the Euler expansion (27) (dotted) and the
Padé interpolation (28) (thin red line) visibly not distinguishable from the
exact reduced density matrix.

This problem of the cut cannot be circumvented. A solution could be to use the
Euler-expansion (27) and to interpolate between both expansions

$\displaystyle{\rm arctan}(x\\!)\\!=\\!{x\over
1\\!+\\!x^{2}}f(x)\\!+\\!\left[\frac{\pi}{2}{\rm sig}(x)\\!-\\!{x\over
1\\!+\\!x^{2}}\right][1\\!-\\!f(x)]$ (28)

with a Padé function $f(x)$ approaching $1$ for small $x$ and $0$ for large
$x$. We choose here $f(x)=1/(1+x^{2})^{2}$ which gives an excellent agreement
with the exact result as seen in figure 1. Using $\Theta(x)=[1+{\rm
sig}(x)]/2$, the Padé interpolation (28) translates into the reduced density
matrix for (20)

$\displaystyle\rho^{\rm Pade}(k)\\!=$
$\displaystyle\left(\\!\frac{1}{2}\\!+\\!\frac{1}{\pi}{x\over
1\\!+\\!x^{2}}\\!\right)\\!f(x)$
$\displaystyle\\!+\\!\left(\\!\Theta(x)\\!-\\!\frac{1}{\pi}{x\over
1\\!+\\!x^{2}}\right)[1\\!-\\!f(x)]$ (29)

with $x=2(\epsilon_{F}-\epsilon_{k})/\gamma$.

##  III Padé approximation for the spectral function

###  III.1 Twofold expansion of spectral function

The obviously working idea of Padé approximation of the last chapter we derive
now by a systematic expansion of the spectral function. For this purpose we
observe that the Euler formula (27) is obtained if we replace formally the
standard Taylor expansion by a finite value form

$\displaystyle g(x)\approx g(0)+g^{\prime}(0)x\longrightarrow g(x)\approx
g(0)+g^{\prime}(x)x$ (30)

with $g(x)={\rm arctan}(x)$.

In the spectral function (9) we have now two expansions, one with respect to
small $\gamma$ and one with respect to deviations from the Fermi level
$\varepsilon_{k}-\epsilon_{F}$ with $\varepsilon_{k}$ the quasiparticle energy
as solution of (10). Both expansions have to be interpolated according to (28)
due to the cut. We translate now this expansion for (20) into an expansion for
the spectral function in such a way to reproduce (28). Applying the rule (30)
for both expansions, $\gamma$ and $\varepsilon_{k}-\epsilon_{F}$, we suggest

$\displaystyle a(\omega,k)\approx$ $\displaystyle a^{\rm
Pade}(\omega,k)=a_{\rm
f}(\omega,k)\,f\\!\\!\left(x\right)\\!+\\!a_{1\\!-\\!{\rm
f}}(\omega,k)\,\left[1\\!-\\!f\\!\left(x\right)\right]$ (31)

with $x=2(\epsilon_{F}\\!-\\!\varepsilon_{k})/\gamma(\epsilon_{F},k)$. The
small scattering-rate expansion of the standard extended quasiparticle
approximation (11) is denoted as

$\displaystyle a_{1\\!-\\!{\rm f}}(\omega,k)=$ $\displaystyle
2\pi\delta[\omega\\!-\\!\epsilon_{k}\\!-\\!\sigma(\omega)]$
$\displaystyle-\gamma(\omega,k)\partial_{\omega}{\omega\\!-\\!\varepsilon_{k}\over(\omega\\!-\\!\varepsilon_{k})^{2}\\!+\\!{\gamma^{2}(\varepsilon_{k},k)\over
4}}.$ (32)

Here we have extended the last term by the damping according to (30). The
additional part in (31)

$\displaystyle a_{\rm f}(\omega,k)=$
$\displaystyle{\gamma(\epsilon_{F},k)\over(\omega-\varepsilon_{F})^{2}\\!+\\!{\gamma^{2}(\epsilon_{F},k)\over
4}}$
$\displaystyle-(\epsilon_{k}\\!-\\!\epsilon_{F})\partial_{\omega}{\gamma(\epsilon_{F},k)\over(\omega-\varepsilon_{F})^{2}\\!+\\!{\gamma^{2}(\epsilon_{F},k)\over
4}}$ (33)

describes now the expansion with respect to the deviation from the Fermi
energy. The subtle difference has to be noted in the pre-factor of the second
term being the deviation of the Hartree-Fock energy
$\epsilon_{k}-\epsilon_{F}$ from the Fermi energy and not the quasiparticle
energy $\varepsilon_{k}-\epsilon_{F}$. This is not visible from the constant
model so far where both expressions are identical. That this extension is
correct we will convince ourselves now by various sum rules and the comparison
with the reduced density (29) for successively more complicated models in
chapter IV.

###  III.2 Reduced density matrix

First it is instructive to see how the reduced density matrix (13) appears
from the Padé approximation (31). One has as exact spectral expression

$\displaystyle\rho(k)\\!=\\!\int\limits_{-\infty}^{\infty}\\!\\!{d\omega\over
2\pi}a(\omega,k)n_{\omega}=\\!\int\limits_{-\infty}^{\epsilon_{F}}\\!\\!{d\omega\over
2\pi}a(\omega,k).$ (34)

The integral over the $\delta$-function as first part in (32) reads

$\displaystyle\int\limits_{-\infty}^{0}\\!\\!d\bar{\omega}\delta[\bar{\omega}\\!+\\!\epsilon_{F}\\!-\\!\epsilon_{k}\\!-\\!\sigma(\bar{\omega}\\!+\\!\epsilon_{F})]=\\!\int\limits_{-\infty}^{0}\\!\\!d\bar{\omega}{\delta(\bar{\omega}\\!-\\!\bar{\varepsilon}_{k})\over
1\\!-\\!\partial_{\bar{\omega}}\sigma}$
$\displaystyle\approx\\!\Theta(\epsilon_{F}-\varepsilon_{k})\left[1\\!+\\!\int{d\omega\over
2\pi}\gamma(\omega,k)\partial_{\omega}{\omega\\!-\\!\varepsilon_{k}\over(\omega\\!-\\!\varepsilon_{k})^{2}+{\gamma^{2}(\varepsilon_{k},k)\over
4}}\right]$ (35)

where we used (4). The approximation includes the expansion of the denominator
in accordance with first-order damping and the subsequent extension of the
principal value by the damping according to (30). This combines now with the
second part of (32) to yield

$\displaystyle\rho_{1\\!-\\!{\rm
f}}(k)=\int\limits_{-\infty}^{\epsilon_{F}}\\!\\!{d\omega\over
2\pi}a_{1\\!-\\!{\rm f}}(\omega,k)\\!=\\!\Theta(\epsilon_{F}-\varepsilon_{k})$
$\displaystyle+\int\\!\\!{d\omega\over
2\pi}\gamma(\omega,k)\left[\Theta(\epsilon_{F}\\!-\\!\varepsilon_{k})\\!-\\!\Theta(\epsilon_{F}\\!-\\!\omega)\right]\partial_{\omega}\tilde{b}(\omega,k)$
(36)

where the last line can be rewritten with the help of (14)

$\displaystyle\\!\left(\\!\\!\Theta(\epsilon_{F}\\!-\\!\varepsilon_{k})\\!\\!\int\limits_{\epsilon_{F}}^{\infty}\\!\\!{d\omega\over
2\pi}\\!-\\!\Theta(\varepsilon_{k}\\!-\\!\epsilon_{F})\\!\int\limits^{\epsilon_{F}}_{-\infty}\\!\\!{d\omega\over
2\pi}\\!\right)\\!\gamma(\omega,k)\partial_{\omega}\tilde{b}(\omega,k)$ (37)

with the abbreviation

$\displaystyle\tilde{b}(\omega,k)={\omega\\!-\\!\varepsilon_{k}\over(\omega\\!-\\!\varepsilon_{k})^{2}\\!+\\!{\gamma^{2}(\varepsilon_{k},k)\over
4}}.$ (38)

For the additional part (33) we can perform the frequency integral to get

$\displaystyle\rho_{\rm
f}(k)=\int\limits_{-\infty}^{\epsilon_{F}}\\!\\!{d\omega\over
2\pi}a_{f}(\omega,k)$ $\displaystyle=\left[\frac{1}{\pi}{\rm
arctan}{\omega-\epsilon_{F}\over\gamma(\epsilon_{F})}-{(\epsilon_{k}-\epsilon_{F})\over
2\pi}{\gamma(\varepsilon_{k})\over(\omega-\epsilon_{F})^{2}+{\gamma(\epsilon_{F})^{2}\over
4}}\right]_{-\infty}^{\omega=\epsilon_{F}}$ (39)

where the above and below integration limits have to be inserted depending on
the damping which might be zero.

As an intermediate check, adding both terms (36) and (39) yields exactly (29)
with the Padé approximation (28) observing that here in this model we have
$\gamma(\omega,k)=const$ and $\sigma=0$ and consequently
$\varepsilon_{k}=\epsilon_{k}$.

The form of Padé approximation suggested in (31) has a general validity for
more refined models, e.g. also for electron-electron scattering since it
relates to any selfenergy. As illustration we restrict to impurity models in
the next chapters. To convince ourselves about the larger validity we first
proof that the sum rules are completed independent of the used models.

###  III.3 Sum rules

We check the frequency sum rule valid for any model of the selfenergy. We can
perform the trivial frequency integrals directly analogously to (36) and (39)
but with the upper integration limit up to infinity to obtain

$\displaystyle\int{d\omega\over 2\pi}a_{1\\!-\\!{\rm
f}}(\omega,k)=\int{d\omega\over 2\pi}a_{f}(\omega,k)=1$ (40)

which means

$\displaystyle\int{d\omega\over 2\pi}a^{\rm Pade}(\omega,k)=[1-f(x)]+f(x)=1$
(41)

is completed independent of the interpolation function $f(x)$ where we
abbreviated $x=2(\epsilon_{F}\\!-\\!\varepsilon_{k})/\gamma(\epsilon_{F},k)$.

Even the first energy-weighted sum rule is also fulfilled which already the
extended quasiparticle approximation does M17b . To see this we consider the
integrals over (32) and (33) separately. Analogously to (36) we get with
abbreviation (38)

$\displaystyle\int{d\omega\over 2\pi}\omega a_{1\\!-\\!{\rm
f}}(\omega,k)=\varepsilon_{k}\\!+\\!\int\\!\\!{d\omega\over
2\pi}[\varepsilon_{k}\\!-\\!\omega]\,\gamma(\omega,k)\,\partial_{\omega}\tilde{b}(\omega,k)$
$\displaystyle\approx\varepsilon_{k}+\int\\!\\!{d\omega\over
2\pi}(\varepsilon_{k}-\omega)\gamma(\omega,k)\partial_{\omega}\\!{1\over\omega\\!-\\!\varepsilon_{k}}+o(\gamma^{3})$
$\displaystyle=\varepsilon_{k}+\int\\!\\!{d\bar{\omega}\over
2\pi}{\gamma(\omega,k)\over\bar{\omega}-\\!\varepsilon_{k}}=\varepsilon_{k}-\sigma(\varepsilon_{k}).$
(42)

The integral over (33) reads

$\displaystyle\int{d\omega\over 2\pi}\omega a_{\rm f}(\omega,k)$
$\displaystyle=\epsilon_{F}\\!-\\!\int{d\omega\over
2\pi}\omega(\epsilon_{k}\\!-\\!\epsilon_{F})\partial_{\omega}{\gamma(\epsilon_{F},k)\over(\omega\\!-\\!\epsilon_{F})^{2}+{\gamma^{2}(\epsilon_{F}.k)\over
4}}$ $\displaystyle=\epsilon_{F}\\!+\\!\int{d\omega\over
2\pi}(\epsilon_{k}\\!-\\!\epsilon_{F}){\gamma(\epsilon_{F},k)\over(\omega\\!-\\!\epsilon_{F})^{2}+{\gamma^{2}(\epsilon_{F},k)\over
4}}=\epsilon_{k}.$ (43)

Adding both parts we obtain the correct first energy-weighted sum rule

$\displaystyle\int{d\omega\over 2\pi}\omega a^{\rm
Pade}(\omega,k)=(\varepsilon_{k}-\sigma)(1-f(x))+\epsilon_{k}f(x)=\epsilon_{k}.$
(44)

It is remarkable that both sum rules are fulfilled for any interpolating
function $f[2(\epsilon_{F}-\varepsilon_{k})/\gamma(\epsilon_{F},k)]$ and any
model for the selfenergy. This suggests already that for transport, i.e.
integrated forms of the reduced density, the extended quasiparticle term
$\rho_{\rm 1-f}$ might be sufficient which we will proof indeed in appendix B
which is valid for any approximation of self energy. Therefore the Padé
approximation presented here is also valid for electron-electron scattering
though only impurity scattering examples are chose in the following for
illustration.

##  IV Examples of Application

Here we will compare the reduced density matrix from the integral over the
spectral function (34) with the analytical Padé approximations (31)-(33). This
will be illustrated on two different many-body models, the Born and T-matrix
approximation. For further models see e.g. Fa19 .

###  IV.1 Impurity scattering in Born approximation

####  IV.1.1 Integrated spectral function

We will now test the Padé interpolation (31)-(33) with an exactly integrable
model of frequency-dependent selfenergy. In Born approximation the scattering
of particles with mass $m$ on impurities of density $n_{i}$ interacting by
contact interaction $V_{q}=V_{0}$ the imaginary part of selfenergy or
quasiparticle damping (5) reads

$\displaystyle\gamma(\omega)$ $\displaystyle=$ $\displaystyle
n_{i}s\int{dq\over 2\pi\hbar}V_{q}^{2}2\pi\delta\left(\omega-{(k+q)^{2}\over
2m}\right)$ (45) $\displaystyle=$ $\displaystyle
g{\Theta(\omega)\over\sqrt{\omega}}$

with the interaction constants

$\displaystyle g$ $\displaystyle=$
$\displaystyle{sn_{i}V_{0}^{2}4m^{2}\over\hbar k_{F}^{3}}={n_{i}\over
n_{F}}{4s^{2}\over\pi}b$ $\displaystyle b$ $\displaystyle=$
$\displaystyle{\hbar^{2}\over
k_{F}^{2}a_{B}^{2}}=r_{s}^{2}{s^{4}\over\pi^{2}}.$ (46)

In the following we understand all energies, $\omega,\gamma,\sigma$ etc, in
units of Fermi energy and the momenta $k$ in units of Fermi momentum given by
the free-particle density $n_{F}$ as $k_{F}=n_{F}\hbar\pi/s$ where we denote
the spin-degeneracy by $s$. The interaction strength we express in terms of a
Bohr-radius-equivalent $V_{0}=\hbar^{2}/ma_{B}$ which allows to discuss
charged and neutral impurities on the same footing. The Bruckner parameter
$r_{s}$ is the ratio of inter-particle distance $d=1/ns$ to this Bohr radius
$r_{s}=d/a_{B}$.

In Born approximation we have only a dependence on the single parameter $g$.
In T-matrix approximation presented in the next chapter we will see the
dependence on both parameters $g$ and $b$ independently, i.e. the interaction
strength and the impurity density.

The corresponding real part of self energy reads according to (4)

$\displaystyle\sigma(\omega)=-g{\Theta(-\omega)\over 2\sqrt{-\omega}}$ (47)

as it was used in Eq. 30 of BS19  with Fermi liquid parameters. The spectral
function (9) for this Born approximation can then be written with (45) and
(47) introducing in (7)

$\displaystyle
a(\omega)=\left\\{\begin{array}[]{ll}{{g\over\sqrt{\omega}}\over(\omega-k^{2})^{2}+{g^{2}\over
4\omega}}&\omega>0\cr 2\pi\delta(\omega-k^{2}+{g\over
2\sqrt{-\omega}})=2\pi{\delta(\omega+q_{0}^{2})\over 2q_{0}+{g\over
2q_{0}^{2}}}&\omega<0\end{array}\right.$ (50)

where $q_{0}$ is the real solution of

$\displaystyle q_{0}^{2}+k^{2}-{g\over 2q_{0}}=0$ (51)

and describes a sharp bound-state pole at negative energies due to vanishing
damping. The selfenergy (47) results into the quasiparticle energy

$\displaystyle\varepsilon_{k}=\left\\{\begin{array}[]{ll}k^{2}&\omega>0\cr-
q_{0}^{2}&\omega<0\end{array}\right.$ (54)

and are the maxima of the spectral function as seen in figure 2. They follow
the quasiparticle $\omega=k^{2}$ line but with a finite damping. In contrast,
the negative bound-state poles at $\omega=-q_{0}^{2}$ are sharp.

Figure 2: The spectral function (9) in Born approximation (50). The maxima at
the quasiparticle energy (54) are indicated in the $\omega-k$ area as green
line. The sharp bound states at negative energy are artificially broadened to
make them visible.

The sum rules (41) and (44) are illustrated in figure 3. The substantial
contribution of the bound-state pole to complete the sum rules is visible.

Figure 3: The frequency sum rule (41) (above) and the energy-weighted sum rule
(44) (below) for the impurity scattering model in Born approximation (50) for
two different impurity couplings. The red area indicates the contribution of
the bound-state pole (51) at negative frequencies.

The reduced density matrix (34) can be obtained analytically though the
integral about positive frequencies is somewhat lengthy. In figure 4 we plot
the reduced density from spectral function (34) for two different coupling
constants and illustrate the influence of the bound-state pole at negative
frequencies. Its contribution is quite remarkable.

Figure 4: The reduced density matrix from the integral (34) for the impurity
scattering model in Born approximation (50) for two different impurity
couplings. The red area indicates the contribution of the bound-state pole at
negative frequencies.

####  IV.1.2 Padé approximation

The Padé approximation (36) reads now for the impurity model (47) and (45)

$\displaystyle\rho_{1\\!-\\!{\rm f}}(k)=\Theta(1-k^{2})+$
$\displaystyle-{g\over
2\pi}\\!\left(\\!\\!\Theta(1\\!-\\!k^{2})\\!\\!\int\limits_{1}^{\infty}\\!-\Theta(k^{2}\\!-\\!1)\\!\int\limits^{1}_{0}\right)\\!\\!{dq\over
q}\partial_{q}{q^{2}\\!-\\!k^{2}\over(q^{2}\\!-\\!k^{2})^{2}\\!+\\!{g^{2}\over
4q}}$ (55)

with an elementary integral. The second part (39) becomes

$\displaystyle\rho_{\rm f}(k)$ $\displaystyle={1\over 1+{g\over
4q_{0}^{3}}}+\frac{1}{\pi}{\rm arctan}{2\over g}$
$\displaystyle-{(k^{2}-1)g\over 2\pi}\left[{1\over(1-k^{2})^{2}+{g^{2}\over
4}}-{1\over(k^{2})^{2}+{g^{2}\over 4}}\right]$ (56)

where we had to break the integration over frequency into parts larger and
smaller zero whereby the latter one leads to the bound-state pole
contribution.

In figure 5 we can see the reduced density matrix from (50) together with the
Padé approximation (31). Though the contributing parts (55) and (56) are
deviating strongly out of the area of their expansion range, the Padé-weighted
form shows a good approximation for different impurity couplings. Basically
the role of the additional $\rho_{F}$ is to cure the jump of $\rho_{1-f}$ at
the Fermi energy.

Figure 5: The reduced density matrix by integration of the spectral function
(34) in Born approximation (50) together with its Padé approximation and the
contributing parts (55) and (56) for two different impurity couplings.

###  IV.2 Impurity scattering in Ladder approximation

####  IV.2.1 Integrated spectral function

Next we sum the ladder diagrams which means to solve the equation for the
retarded $T$-matrix M02

$\displaystyle T^{R}(\omega)$ $\displaystyle=$ $\displaystyle V+V\int{dk\over
2\pi\hbar}{T^{R}(\omega)\over\omega-{k^{2}\over 2m}+i0}$ (57) $\displaystyle=$
$\displaystyle{\hbar^{2}\over
ma_{B}}\left[1+\sqrt{b}\left({\Theta(-\omega)\over\sqrt{-\omega}}+i{\Theta(\omega)\over\sqrt{\omega}}\right)\right]^{-1}$

with $b$ of (46). The correlation part of the selfenergy simplifies from two-
particle scattering towards impurity scattering used here

$\displaystyle\sigma^{<}(\omega,k)$ $\displaystyle=$ $\displaystyle
s\sum\limits_{p,q}|T^{R}(\omega\\!+\\!\varepsilon_{p}^{i},k,p)|^{2}2\pi\delta(\omega\\!+\\!\varepsilon_{p}^{i}\\!-\\!\varepsilon_{k\\!+\\!q}\\!-\\!\varepsilon_{p\\!-\\!q}^{i})$
(58) $\displaystyle\times n_{k+q}n_{p-q}^{i}(1-n_{p}^{i})$ $\displaystyle=$
$\displaystyle
n_{i}s\sum\limits_{q}|T^{R}(\omega)|^{2}2\pi\delta(\omega+\varepsilon_{p}^{i}-\varepsilon_{q})n_{q}$
$\displaystyle=$
$\displaystyle{g\over\sqrt{\omega}+{b\over\sqrt{\omega}}}\Theta(\omega)\Theta(\epsilon_{F}-\omega)$

and $\sigma^{>}$ is obtained by interchanging $n\leftrightarrow 1-n$. The
resulting imaginary (5) and real (4) parts of the selfenergy read

$\displaystyle\gamma(\omega)$ $\displaystyle=$
$\displaystyle\sigma^{>}+\Sigma^{<}=i(\sigma^{R}-\Sigma^{A})={g\over\sqrt{\omega}+{b\over\sqrt{\omega}}}\Theta(\omega)$
$\displaystyle\sigma(\omega)$ $\displaystyle=$
$\displaystyle-{g(\sqrt{b}-\Theta(-\omega)\sqrt{-\omega})\over 2(b+\omega)}$
(59)

and compared to the Born approximation (45) and (47) they deviate by the
parameter $b$ given in (46). The expressions for Born and noncrossing
approximations can be found in HuS93 .

The spectral function (9) is plotted in figure 6.

Figure 6: The spectral function (9) in T-matrix approximation (59). The
maxima/minima at the quasiparticle energy (54) are indicated as thick line in
the $\omega-k$ area and their corresponding imaginary parts as dotted line.
The zero of the quasiparticle dispersion (61) is plotted as dashed line. The
sharp bound states at negative energy are artificially broadened.

One sees that the bound-state pole vanishes at a certain momentum $k_{0}$ and
resolves into an additional peak in the positive energy spectrum. This point
can be found considering the quasiparticle dispersion. For $\omega>0$ we have
two damped ($\gamma\neq 0$) solutions by

$\displaystyle\omega-\epsilon_{k}-\sigma(\omega)=0$ (60)

as

$\displaystyle(\varepsilon_{k})_{1,2}=\frac{1}{2}(k^{2}-b\pm\sqrt{(k^{2}+b)^{2}-2g\sqrt{b}}).$
(61)

For $\omega<0$ we have the bound-state pole $\varepsilon=-q_{0}^{2}$ as
solution of

$\displaystyle q_{0}^{3}+\sqrt{b}q_{0}^{2}+k^{2}q_{0}+k^{2}\sqrt{b}-{g\over
2}=0.$ (62)

It is easy to see that possible extreme points of this polynomial are always
at negative $q_{0}$. Since the polynomial increases with increasing $q_{0}$ we
have only a real positive solution as a crossing of the polynomial with the
$q_{0}$-axes if the value of the polynomial at $q_{0}=0$ is negative which
translates into

$\displaystyle k^{2}<k_{0}^{2}={g\over 2\sqrt{b}}=r_{s}{n_{i}\over
n_{F}}{2s^{3}\over\pi^{2}}$ (63)

which determines the critical $k_{0}$ above which the bound-state pole
vanishes and resolves into the positive dispersion. This value coincides with
the value where the quasiparticle energy $\varepsilon$ of (61) crosses zero as
illustrated in figure 7.

Figure 7: The solution of dispersion (61) (dashed) together with the maximum
of the spectral function (solid) and the negative bound-state pole (thick) for
three different parameter sets.

This turn-over of the bound-state pole from negative sharp values to positive
damped values is accompanied by a sharp additional excitation as seen in
figure 6. Such a localization in momentum space is typically for the onset of
superconductivity as Bose condensation of pairs. One therefore suspects that
around the momenta (63) one might have superconducting behaviour for a 1D wire
interacting solely with impurities. We will find this interpretation supported
by the temperature dependence of the conductivity in chapter VI. Since this
appears here with a finite momentum off the Fermi momentum we might have the
situation similar to the FFLO state. Of course, any electron-electron
scattering will probably smooth out this effect.

The quasiparticle dispersion (61) has only real solutions as long as

$\displaystyle k^{2}>k_{c}^{2}=\sqrt{2g\sqrt{b}}-b.$ (64)

One sees in figure 7 that as soon as the most left point of this quasiparticle
dispersion is turning from negative to positive values which is at $k^{2}=b$,
the spectral function develops two maxima which can be seen in the top panel
of figure 7. For higher momenta the quasiparticle dispersion and the points of
maxima coincide. We have no minimal value of quasiparticle dispersion if
according to (64) $2g<b^{3/2}$ which means as long as the impurity density
does not exceed

$\displaystyle n_{i}<{1\over 8a_{B}}$ (65)

which case is plotted in the bottom panel of figure 7. It is quite remarkable
that this limit does not depend on the density of electrons but only on the
strength of interaction.

As long as we have an impurity concentration higher than (65) the appearance
of quasiparticle poles only above a finite momentum (energy) describes an
energy gap. The same behaviour has been found as localized states in a
pseudogap M68 . We will see indeed in chapter VI that this leads to a minimum
in the conductivity.

Figure 8: The frequency sum rule for the impurity scattering model in T-matrix
approximation for different impurity couplings. The red area indicates the
contribution of the bound-state pole (62) at negative frequencies.

Figure 9: The reduced density matrix (34) for the impurity scattering model in
T-matrix approximation for two different impurity couplings corresponding to
figure 8. The red area indicates the contribution of the bound-state pole at
negative frequencies.

To complete the discussion we illustrate again the importance of the bound-
state pole to the sum rule (41) in figure 8. The corresponding reduced density
matrix (34) are plotted in figure 9 which again illuminates the influence of
the bound-state pole at negative frequencies. Compared to the Born
approximation in figure 3 and figure 4 the second parameter $b$ of (46)
creates different shapes of the momentum dependence.

####  IV.2.2 Padé approximation

The Padé approximation (36) reads now for the impurity model in T-matrix
approximation (59)

$\displaystyle\rho_{1\\!-\\!{\rm f}}=\Theta(1-k^{2})$
$\displaystyle-\\!\left(\\!\\!\Theta(1\\!-\\!k^{2})\\!\\!\int\limits_{1}^{\infty}\\!\\!-\Theta(k^{2}\\!-\\!1)\\!\\!\int\limits^{1}_{0}\\!\right)\\!\\!{dq\over
2\pi}\gamma(q^{2})\partial_{q}{q^{2}\\!-\\!\varepsilon_{k}\over(q^{2}\\!-\\!\varepsilon_{k})^{2}\\!+\\!{\gamma(\varepsilon_{k})^{2}\over
4}}$ (66)

with an elementary integral due to (59). The second part (39) becomes

$\displaystyle\rho_{\rm f}=$ $\displaystyle{1\over 1+{g\over
4q_{0}(\sqrt{b}+q_{0})^{2}}}+\frac{1}{\pi}{\rm arctan}{2\over\gamma}$
$\displaystyle-{(k^{2}-1)\over
2\pi}\left[{\gamma\over(1-\varepsilon_{k})^{2}+{\gamma^{2}\over
4}}-{\gamma\over\varepsilon_{k}^{2}+{\gamma^{2}\over 4}}\right]$ (67)

where $\gamma=\gamma(\epsilon_{F})$ and all energies are in terms of
$\epsilon_{F}$. We had to break the integration over frequency again into
parts larger and smaller zero where the latter leads to the bound-state pole
contribution as the first part in $\rho_{\rm f}$.

Figure 10: The reduced density matrix in Padé approximation (thin red)
compared with (34) (thick blue) and both contributions entering the Padé
approximations, $\rho_{\rm 1-f}$ of (66) (dashed) and $\rho_{\rm f}$ of (67)
(dotted). Above: without selfconsistent quasiparticle and without correlated
density, middle: with selfconsistent quasiparticle and without correlated
density, below: with selfconsistent quasiparticle and correlated density.

If we compare this Padé approximation with the reduced density matrix from the
spectral function (34) in figure 10 (above) we see a strong deviation. It is
instructive to reveal the missing parts in two steps.

##### Selfconsistent quasiparticles

First we have used the quasiparticle energy at values of the free energy
$\varepsilon_{k}=\varepsilon(k^{2})$. A selfconsistent solution of the
quasiparticle dispersion (10) would require a successive iteration which means
$\varepsilon_{k}=\varepsilon(k^{2}+\delta)$. We can determine this required
shift $\delta$ by demanding that the Fermi energy should be reproduced
$\epsilon_{F}=\varepsilon(k_{F}^{2}+\delta)$. For the quasiparticle
approximation (61) one finds explicitly $\delta=-\sigma(\epsilon_{F})$.
Including this shift leads to a smoothing of the curve parallel to the
spectral expression (34) as seen in the middle panel of figure 10.

##### Correlated density

Second, we recognize a parallel shift between the spectral result (34) and the
Padé approximation in the middle panel of figure 10. This comes from the fact
that the T-matrix approximation leads to time nonlocality in the scattering
process given by the delay time $\Delta_{t}=\partial_{\omega}\Phi$ between in-
and outgoing scattering events with the phase of the T-matrix $T^{R}=|T|{\rm
e}^{i\Phi}$. As result a correlated density $n_{c}$ appears besides the free
(quasiparticle) density adding to the total density $n=n_{F}+n_{c}$. It
describes how much particles are in the correlated state not available for the
Fermi momentum. Consequently it diminishes the Fermi momentum SLMa96

$k_{F}={\pi\hbar\over s}n_{F}={\pi\hbar\over s}{n\over 1+{n_{c}\over n}}.$
(68)

This shift due to the time delay clearly violates the Luttinger theorem
stating that the Fermi momentum is exclusively determined by the free
(quasiparticle) density $n_{F}$. Since the Luttinger theorem is valid only for
Fermi liquids, one should not be puzzled since we do have a non-Fermi liquid.
Collision delays describe the correlations beyond Fermi liquids. Other
examples of non-Fermi liquids are the correlated density for systems with
short-living bound states (resonances) or pairing.

Before presenting the explicit form we see already in the bottom panel of
figure 10 that the missing agreement between the Padé T-matrix approximation
and the spectral expression of the reduced density (34) is completed by this
time delay respectively correlated density. The comparison of the momentum
distribution with results of other methods, e.q. by variation technique with
figure 4 in SZ19 , can be done but is not sensitive enough to distinguish the
subtle differences.

##  V Time delay and correlated density

###  V.1 Time-delay and correlated density

As soon as the vertex, the T-matrix $T=|T|\exp{i\Phi}$, has a frequency-
dependent phase $\phi(\omega)$ there appears a finite duration of the
collision process. The first-order effect can be expressed (see Eq. 68 in
MLS00 ) as a time delay $\Delta_{t}=\partial_{\omega}\Phi$ correction to the
selfenergy (58)

$\displaystyle\Delta\sigma^{<}(\omega,k)=s\sum\limits_{p,q}|T^{R}(\omega\\!+\\!\varepsilon_{p}^{i},k,p)|^{2}\Delta_{t}$
$\displaystyle\times
2\pi\delta(\omega\\!+\\!\varepsilon_{p}^{i}\\!-\\!\varepsilon_{k\\!+\\!q}\\!-\\!\varepsilon_{p\\!-\\!q}^{i})(1-n_{p}^{i}){\partial\over\partial
t}n_{k+q}n_{p-q}^{i}.$ (69)

We express now

$\displaystyle|T^{R}|^{2}\Delta_{t}={\rm
Im}T^{A}\partial_{\omega}T^{R}={\hbar^{2}\over
k_{F}^{2}a_{B}^{2}}{\sqrt{b\omega}\over 2(\omega+b)^{2}}$ (70)

and use the impurity limit $1-n_{p}\approx 1$, $n_{p-q}\approx n_{p}$ to find
the additional contribution to the collision integral
$I=(1-n_{k})\sigma^{<}(\varepsilon_{k})-n_{k}\sigma^{>}(\varepsilon_{k})$ to
be

$\displaystyle\Delta I=$ $\displaystyle{\partial\over\partial
t}n_{i}s\sum\limits_{q}{\rm
Im}T^{A}\partial_{\omega}T^{R}2\pi\delta(\varepsilon_{k}\\!-\\!\varepsilon_{k\\!+\\!q})n_{k+q}.$
(71)

This means that in the balance equation for the density we obtain with (70) an
additional correlated density contribution

$\displaystyle n_{c}$ $\displaystyle=$ $\displaystyle-
n_{i}s\sum\limits_{q}{\rm
Im}T^{A}\partial_{\omega}T^{R}2\pi\delta(\varepsilon_{k}\\!-\\!\varepsilon_{k\\!+\\!q})n_{k+q}$
(72) $\displaystyle=$ $\displaystyle-n_{i}{g\over
8s}\int\limits_{-1}^{1}dq{\sqrt{b}\over(q^{2}+b)^{2}}.$

For Born approximation $b\to 0$ we do not have a phase and the collision delay
$\sim\sqrt{b}$ is zero. Therefore the integral (72) should vanish. This is the
case if we expand the integrand into a Taylor series and get

$\displaystyle\int\limits_{-1}^{1}dq{\sqrt{b}\over(q^{2}+b)^{2}}=-\frac{2}{3}\sqrt{b}+\frac{4}{5}b^{3/2}+....$
(73)

Unfortunately the integral does not smoothly converge which means the
integration and expansion cannot be interchanged. In fact one gets formally

$\displaystyle\int\limits_{-1}^{1}dq{\sqrt{b}\over(q^{2}+b)^{2}}$
$\displaystyle=$ $\displaystyle{1\over\sqrt{b}(1+b)}+\frac{1}{b}{\rm
arctan}{1\over\sqrt{b}}$ (74) $\displaystyle=$ $\displaystyle{\pi\over
2b}-\frac{2}{3}\sqrt{b}+\frac{4}{5}b^{3/2}+....$

which compared to (72) shows that the term $\pi/2b$ appears to be wrong in the
sense of perturbation expansion of the integrand which we demand as limit of
T-matrix towards Born approximation. Therefore we subtract this term which
means we first expand and then integrate and then resume again to get finally

$\displaystyle n_{c}$ $\displaystyle=$ $\displaystyle-n_{i}{g\over
8s}\left[{1\over\sqrt{b}(1+b)}-\frac{1}{b}{\rm arctan}{\sqrt{b}}\right].$ (75)

This correlated density adds to the free density for total density
$n=n_{F}+n_{c}$ and accordingly changes the Fermi momentum since the latter
one is determined with respect to the free density (68). In figure 11 the free
density increases with the square of Fermi energy while the correlated density
decreases with increasing Fermi energy dependent on the impurity density. This
different dependence will lead to localization effects in the conductivity as
we will see in chapter VI.

Figure 11: The free and correlated density vs. Fermi energy for increasing
impurity densities $a_{B}n_{i}=0.1-1.5$ (from bottom to top).

###  V.2 Tan contact and large momentum behaviour

The expansion at large momentum in the Padé approximation only the term
$\rho_{\rm 1-f}$ contributes as it is in the extended quasiparticle
approximation. From (61) and (66) we obtain

$\displaystyle\lim\limits_{k\to\infty}\rho(k)={Cn^{4}\hbar^{4}\over k^{4}}$
(76)

with the Tan contact

$\displaystyle C$ $\displaystyle=$ $\displaystyle{gk_{F}^{4}\over\pi
n_{F}^{4}\hbar^{4}}\left[\sqrt{b}\,{\rm arccot}\sqrt{b}-1\right]$ (77)
$\displaystyle=$ $\displaystyle{4n_{i}\over n_{F}}\pi^{2}b\left[\sqrt{b}\,{\rm
arccot}\sqrt{b}-1\right]$

independent of the pole correction $\delta$. The values for the e-e model of
contact interaction is compared in Ram17 . In figure 12 we plot this contact
(77) versus the coupling constant $b$ of (46). It is qualitatively very
similar to the e-e interaction considered in OD03  where it had been
conjectured its universal behaviour. This is quantitatively supported when
comparing with the result of Eq. 10 in SZ19  where an impurity immersed in
Fermions forms a polaron. Reformulating for repulsive interaction their result
reads 111The strange factor $2$ in the subtraction instead of $1$ with our
result might be a misprint.

$\displaystyle C$ $\displaystyle=$ $\displaystyle{1\over
2a_{B}n_{F}}\pi^{2}b\left[\sqrt{b}\,{\rm arccot}\sqrt{b}-2\right]$ (78)

and compared to (77) we see that we have to replace in our result the impurity
density simply by $n_{i}=1/8a_{B}$ to reproduce the result of SZ19 .
Remarkably, this is just coinciding with the limit of (65) where the spectral
function changes from one to two maxima and forming a gap.

Figure 12: The large momentum expansion of the reduced density $\rho(k)\approx
C(n\hbar)^{4}/k^{4}$ versus coupling constant (46).

##  VI Nonlocal corrections to the conductivity

For any transport property we have to consider momentum integrals over the
reduced density abbreviating
$x=2(\epsilon_{F}\\!-\\!\varepsilon_{k})/\gamma(\epsilon_{F},k)$

$\displaystyle\rho^{\rm Pade}(k)=\rho^{\rm EQP}(k)+f(x)[\rho_{\rm
f}(k)-\rho_{\rm 1-f}(k)]$ (79)

which presents the difference to the extended quasiparticle part $\rho^{\rm
EQP}=\rho_{\rm 1-f}$. In appendix B we show that the additional part is
negligible under integration for most regular cases of selfenergy. This means
though we have to use two-term Padé approximation to reproduce the spectral
function and the reduced density matrix at the Fermi energy correctly, only
the part of the extended quasiparticle part matters under integrations. In
other words the divergence of the extended quasiparticle distribution function
at the Fermi energy cancels under integration.

The impurity scattering including averaged T-matrix approximation contains the
average about impurity configurations and therefore reinforce spatial
invariance. We write the kinetic equation for homogeneous systems in an
electric field $E$

$\displaystyle\partial_{t}n_{k}\\!-\\!eE\partial_{k}n_{k}\\!=\\!$
$\displaystyle
n_{i}s\\!\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!{dq\over\hbar^{2}}|T(\varepsilon_{k\\!+\\!q})|^{2}\delta(\varepsilon_{k\\!+\\!q}\\!-\\!\varepsilon_{k})[n_{k\\!+\\!q}\\!-\\!n_{k}]$
$\displaystyle+\Delta I$ (80)

with the nonlocal corrections (71). When linearized with respect to the
electric field

$\displaystyle n(k,t)=n_{0}(|k|)+\tau(|k|)eE\partial_{k}n_{0}(|k|)$ (81)

one obtains the relaxation time

$\displaystyle\tau(|k|)$ $\displaystyle=$ $\displaystyle|k|{h^{2}\over
2mn_{i}sT^{2}(\epsilon_{k})}={\hbar\over
g\epsilon_{F}}\sqrt{\epsilon_{k}\over\epsilon_{F}}\left(1+{b\over\epsilon_{k}/\epsilon_{F}}\right)$
$\displaystyle\tau(k_{F})$ $\displaystyle=$
$\displaystyle{\hbar\over\epsilon_{F}}{\pi\over 2s^{2}}{n_{F}\over
n_{i}}{1+b\over b}.$ (82)

The energy dependence of this relaxation rate starts at zero showing a maximum
and vanishes for large energies. Coulomb interactions between the electrons
just lead to the opposite behaviour RM21 .

By integrating the kinetic equation (80) with momentum weight, the balance for
the current becomes

$\displaystyle\partial_{t}(j+j_{c})=-eEn_{F}$ (83)

and leads besides the standard current density

$\displaystyle j=-eE\int{dk\over 2\pi\hbar}{k\over m}f=\lambda_{0}E$ (84)

also to a current density due to two-particle correlations

$\displaystyle j_{c}\\!$ $\displaystyle=$
$\displaystyle\\!en_{i}s\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!{dqdk\over
2\pi\hbar^{2}}{k\\!+\\!q\over m}\,{\rm
Im}T^{R}T^{A}\,\delta(\varepsilon_{k+q}\\!-\\!\varepsilon_{k})n_{k+q}$ (85)
$\displaystyle=$ $\displaystyle\lambda_{c}E.$

The corresponding conductivities are

$\displaystyle\lambda_{0}\\!$ $\displaystyle=$
$\displaystyle\\!{n_{F}e^{2}\tau(p_{F})\over m}\\!=\\!{e^{2}\over
n_{i}\hbar}{\pi\over 2s^{4}r_{s}^{2}}(1\\!+\\!b)={e^{2}\over 2\pi\hbar
n_{i}}{1\\!+\\!b\over b}$ $\displaystyle\lambda_{c}$ $\displaystyle=$
$\displaystyle\lambda_{0}{s\over\pi}{n_{i}\over n_{F}}{b^{3/2}\over(1+b)^{2}}$
(86)

where we used (46). The results for the density (75) and total conductivity
$\lambda=\lambda_{0}+\lambda_{c}$ with (86) can be seen in figure 13. One
recognizes that both the total density and the total conductivity lead to a
development of a minimum at small electron/impurity density ratios due to the
different behaviour of the free and correlated density as seen in figure 11.

Figure 13: The total density versus Fermi energy (left) and conductivity
versus free density (right) for zero temperature (86) with
$n_{i}a_{B}=0.1-1.5$ from bottom to top (left) and $n_{i}a_{B}=1-3.5$ from top
to bottom (right).

These minima lead now to a localization effect in the conductivity when
plotted versus total number of particles as done in figure 14. The minima
appear as long as $n_{i}a_{B}\geq 1$ which all lay on the curve

$\displaystyle n_{i}^{\rm min}a_{B}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!{n_{i}a_{B}\over 2\pi}{\rm
arccot}{\sqrt{\\!\sqrt{n_{i}a_{B}}\\!-\\!1}}$
$\displaystyle-\left(\\!{\sqrt{n_{i}a_{B}}\over
2\pi}\\!-\\!{1\over\pi}\\!\right)\sqrt{\\!\sqrt{n_{i}a_{B}}\\!-\\!1}$
$\displaystyle\lambda^{\rm min}$ $\displaystyle=$
$\displaystyle{e^{2}a_{B}\over\pi\hbar\sqrt{n_{i}a_{B}}}$ (87)

easily seen from (75) and (86).

Figure 14: The conductivity versus total density for different impurity
densities and the Fermi energy as parameter denoted by the arrows. For
$n_{i}a_{B}>1$ the curves start at zero Fermi energy at
$(n_{i}a_{B}/2,1+1/n_{i}a_{B})$ and develop a minimum at the energy
$\epsilon^{\rm min}=(\sqrt{n_{i}a_{B}}-1)/2$ and $\lambda^{\rm
min}=e^{2}a_{B}/\pi\hbar\sqrt{n_{i}a_{B}}$ when the Fermi energy is
increasing. Bellow $n_{i}a_{B}<1$ no such minima appears. The localization of
the minima are plotted as dotted line according to (87).

###  VI.1 Temperature dependence

Since the expressions for the conductivity are linear in the distribution
functions we can calculate the finite temperature $T$ expressions observing
the relation Mal78

$\displaystyle
n_{k}=\int\limits_{0}^{\infty}d\bar{\mu}{\Theta(\bar{\mu}-\varepsilon_{k})\over
4T{\rm cosh}^{2}{\mu-\bar{\mu}\over 2T}}$ (88)

to be applied to the zero-temperature expressions (86)

$\displaystyle\lambda(0)\\!=\\!\lambda_{0}(0)\\!+\\!\lambda_{c}(0)\\!=\\!{e^{2}\over
2\pi
n_{i}\hbar}\\!\left(\\!1\\!+\\!{k_{F}^{2}a_{B}^{2}\over\hbar^{2}}\\!+\\!{n_{i}a_{B}\over
1\\!+\\!{k_{F}^{2}a_{B}^{2}\over\hbar^{2}}}\\!\right)$ (89)

where the last term is the correlated conductivity. The integral about the
normal conductivity can be done analytically to yield

$\displaystyle\lambda_{0}(T)={e^{2}\over 2\pi n_{i}\hbar}\left[{z\over
1+z}+2{T\over T_{0}}\ln(1+z)\right]$ (90)

with the temperature scale and fugacity

$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle{\hbar^{2}\over
ma_{B}^{2}}$ $\displaystyle z$ $\displaystyle=$ $\displaystyle{\rm
e}^{\mu\over kT}$ (91)

where the last one determines the free density

$\displaystyle n_{F}a_{B}=-\sqrt{2T\over\pi T_{0}}Li_{1/2}(z)$ (92)

by the poly-logarithm $Li_{n}(z)=\sum_{k=1}^{\infty}z^{k}/k^{n}$. The
correlated conductivity remains as integral

$\displaystyle\lambda_{c}(T)={e^{2}a_{B}\over
4\pi\hbar}\int\limits_{0}^{\infty}{dy\over\left(1+{2T\over
T_{0}}y\right)[1+\cosh(y-\ln z)]}.$ (93)

It is remarkable that this expression becomes independent on the impurity
density and is only determined by the interaction strength coded in the
equivalent Bohr radius $a_{B}$ according to $V_{0}=\hbar^{2}/ma_{B}$ and by
the density of electrons. The effect of temperature is basically to diminish
the area of localization seen as shrinking hysteresis in figure 15.

Figure 15: The conductivity versus total density and inverse temperature for
an impurity density $n_{i}a_{B}=20$.

In the next figure 16 we plot the conductivity for various temperatures versus
the chemical potential. One sees that for lower temperatures it develops a
peaked structure and a minimum. This is the behaviour observed in figure 3 of
PAC16 . Therefore we interpret this as the onset of a possible superconducting
behaviour triggered by impurities. Without the two-particle correlations
expressed by the delay time or correlated density this effect would be absent.

Figure 16: The conductivity versus chemical potential for temperatures
$T=0,3...2.5\hbar^{2}/ma_{B}^{2}$ (from top to bottom of peak) and an impurity
density $n_{i}a_{B}=20$.

##  VII Summary

A two-fold expansion of the spectral function in interacting quantum systems
is presented. One expansion with respect to small scattering rates is the
known extended quasiparticle picture. This expansion fails in non-Fermi
systems like one-dimensional Fermi gases since it diverges at the Fermi
energy. Therefore a second expansion is suggested around the Fermi energy.
Both expansions can be interpolated by a Padé expression. The advantage is
that independent on the interpolating function the first two energy-weighted
sum rules are completed as it is the case in the extended quasiparticle
picture. The validity of the approach is the same as the perturbation theory
where the extended quasiparticle picture is based on. With the help of the
presented Padé regularization the divergence of the reduced density in
perturbation theory is cured at the Fermi energy independent on the used
selfenergy approximation.

For models of electron-impurity scattering the quality of this Padé
approximation is demonstrated to reproduce the spectral function and reduced
density. In this way the divergence of the reduced density in extended
quasiparticle approximation at the Fermi energy is corrected to a finite
value. It is shown that the divergence of the extended quasiparticle
approximation at the Fermi energy cancels under integration. Consequently, in
order to describe the reduced density and spectral function a two-term Padé
interpolation is necessary while for transport processes the extended
quasiparticle approximation alone is sufficient. This extends the validity of
extended quasiparticle approximation beyond the Fermi-liquids.

For the T-matrix approximation one obtains delay times resulting into a
correlated density which describes how much particles remain in a correlated
state. Further consequences are an additional contribution to the conductivity
which results into two effects. At certain impurity densities localization
appears and a minimum of conductivity develops for small temperatures
dependent on the impurity concentration which is interpreted as a possible
onset of superconductivity. The Tan contact shows the same universal behaviour
as found in e-e scattering when we replace the impurity concentration by the
value where the gap in the spectral function vanishes.

The Padé expansion of the spectral function is derived. Here it is illustrate
on a simple exploratory example on electron-impurity scattering. In a
forthcoming paper it will be demonstrated that it works equally well for
electron-electron correlations. The expectation is that it might be fruitfully
applicable also for other non-Fermi systems.

###### Acknowledgements.

The motivating discussions with Kare N. Pathak and Vinod Ashokan are
gratefully acknowledged as well as the support from DFG-project MO 621/28-1.

##  Appendix A Derivation of Euler expansion

The expansion of $arctan$ function found by Euler in 1755 reads

$\displaystyle\arctan x=\int\limits_{0}^{1}dt{x\over
1+x^{2}t^{2}}=\int\limits_{0}^{\pi/2}du{x\sin{u}\over 1+x^{2}\cos^{2}{u}}$
$\displaystyle=\int\limits_{0}^{\pi/2}du{x\sin{u}\over 1+x^{2}}{1\over
1-{x^{2}\sin^{2}{u}\over
1+x^{2}}}=\int\limits_{0}^{\pi/2}du\sum\limits_{n=0}^{\infty}{x^{2n+1}\sin^{2n+1}{u}\over(1+x^{2})^{n+1}}$
$\displaystyle=\sum\limits_{n=0}^{\infty}\left(\int\limits_{0}^{\pi/2}du\sin^{2n+1}{u}\right){x^{2n+1}\over(1+x^{2})^{n+1}}$
$\displaystyle=\sum\limits_{n=0}^{\infty}{2^{2n}(n!)^{2}\over(2n+1)!}{x^{2n+1}\over(1+x^{2})^{n+1}}={x\over
1+x^{2}}+\frac{2}{3}{x^{3}\over(1+x^{2})^{2}}+...$ (94)

with obvious substitutions and using a geometric series.

##  Appendix B Proof that interpolating terms cancel under integrals

We want to show that the contribution in the Padé approximation beyond the
extended quasiparticle picture in (79) is negligible. This means we consider

$\displaystyle\langle\phi\rangle$ $\displaystyle=\int{dk}\phi_{k}\rho^{\rm
Pade}(k)=\langle\phi\rangle^{\rm EQP}$ $\displaystyle+\int
dk\phi_{k}f\left[{\epsilon_{F}-\varepsilon_{k}\over{\gamma(\epsilon_{F},k)\over
2}}\right]\int{d\omega\over 2\pi}n_{\omega}[a_{\rm f}(\omega,k)-a_{\rm
1-f}(\omega,k)].$ (95)

Since the interpolating function has the value $f=1$ only in a small limited
neighborhood of the Fermi energy $\varepsilon_{k}=\epsilon_{F}$ we can write
the additional term

$\displaystyle\langle\phi\rangle\\!-\\!\langle\phi\rangle^{\rm
EQP}\\!\approx\\!\phi_{k_{F}}\\!\\!\int\\!\\!\\!{d\omega\over
2\pi}n_{\omega}[a_{\rm f}(\omega,k)\\!-\\!a_{\rm
1\\!-\\!f}(\omega,k)]_{\varepsilon_{k}=\epsilon_{F}}.$ (96)

Similar to the steps leading to the frequency-weighted sum rule (44) we make a
partial integration observing $\varepsilon_{k}=\varepsilon_{k}^{0}+\sigma$
with (8) to have

$\displaystyle\int{d\omega\over 2\pi}n_{\omega}a_{\rm
1-f}(\omega,k)=n_{\varepsilon_{k}}-\sigma n_{\varepsilon_{k}}^{\prime}$
$\displaystyle+\int{d\omega\over
2\pi}{\gamma(\omega,k)(n_{\omega}^{\prime}\\!-\\!n_{\varepsilon_{k}}^{\prime})\\!+\\!\partial_{\omega}\gamma(\omega,k)(n_{\omega}\\!-\\!n_{\varepsilon_{k}})\over\omega\\!-\\!\varepsilon_{k}}\\!+\\!o(\gamma^{2})$
(97)

where the prime denotes the derivative and abbreviating
$\gamma=\gamma(\epsilon_{F},k)$

$\displaystyle\int{d\omega\over 2\pi}n_{\omega}a_{\rm
f}(\omega,k)=n_{\epsilon_{F}}+(\varepsilon_{k}^{0}-\epsilon_{F})n_{\epsilon_{F}}^{\prime}$
$\displaystyle+\\!\\!\int\\!\\!{d\omega\over
2\pi}{\gamma\over(\omega\\!-\\!\epsilon_{F})^{2}\\!+\\!{\gamma^{2}\over
4}}\left[(n_{\omega}\\!-\\!n_{\epsilon_{F}})\\!+\\!(\varepsilon_{k}^{0}\\!-\\!\epsilon_{F})(n_{\omega}^{\prime}\\!-\\!n_{\epsilon_{F}}^{\prime})\right].$
(98)

In the last expression we consider it as a first-order Taylor expansion and
write $[...]\approx
n_{\omega+\varepsilon_{k}^{0}-\epsilon_{F}}-n_{\varepsilon_{k}^{0}}$. This
expression is integrated with the Lorentzian weight in the second term of (98)
which peaks at $\omega\approx\epsilon_{F}$ where the expression vanishes.
Therefore the second term of (98) is zero up to orders of
$o(\gamma(\epsilon_{F})^{2})$.

Needed in (96) we have to subtract from (97) at $\varepsilon_{k}=\epsilon_{F}$
the term (98). Since $\varepsilon_{k}=\varepsilon_{k}^{0}+\sigma$ it remains
only

$\displaystyle\int\\!\\!{d\omega\over 2\pi}n_{\omega}[a_{\rm
f}(\omega,k)\\!-\\!a_{\rm
1\\!-\\!f}(\omega,k)]_{\varepsilon_{k}=\epsilon_{F}}$
$\displaystyle=\int{d\omega\over
2\pi}{\gamma(\omega,k)(n_{\omega}^{\prime}-n_{\epsilon_{F}}^{\prime})+\partial_{\omega}\gamma(\omega,k)(n_{\omega}-n_{\epsilon_{F}})\over\omega-\epsilon_{F}}.$
$\displaystyle=\int{d\omega\over
2\pi}{\gamma(\omega,k)\over(\omega-\epsilon_{F})^{2}}[(n_{\omega}-n_{\epsilon_{F}})-n_{\epsilon_{F}}^{\prime}(\omega-\epsilon_{F})]$
(99)

where we used a partial integration to go to the third line. Since the
integral is peaked around $\omega\sim\epsilon_{F}$ we can expand the first
term in the $[...]$ bracket up to first order which cancels exactly and we can
conclude that (96) is vanishing.

## Data Availability Statement

No Data associated in the manuscript.

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