# Electronic quantum wires in extended quasiparticle picture

Klaus Morawetz1,2, Vinod Ashokan3, Kare Narain Pathak4, Neil Drummond5,
Gianaurelio Cuniberti6 1Münster University of Applied Sciences,
Stegerwaldstrasse 39, 48565 Steinfurt, Germany  2International Institute of
Physics- UFRN, Campus Universitário Lagoa nova, 59078-970 Natal, Brazil
3Department of Physics, Dr. B. R. Ambedkar National Institute of Technology,
Jalandhar 144011, India  4Centre for Advanced Study in Physics, Panjab
University, 160014 Chandigarh, India  5Department of Physics, Lancaster
University, Lancaster LA1 4YB, United Kingdom  6Institute for Materials
Science, TU Dresden, 01062 Dresden, Germany

###### Abstract

A one-dimensional quantum wire of Fermions is considered and ground state
properties are calculated in the high density regime within the extended
quasiparticle picture and Born approximation. Expanding the two-particle Green
functions determines the selfenergy and the polarization as well as the
response function on the same footing. While the on-shell selfenergies are
strictly zero due to Pauli-blocking of elastic scattering, the off-shell
behaviour shows a rich structure of a gap in the damping of excitation which
is closed when the momentum approaches the Fermi one. The consistent spectral
function is presented completing the first two energy-weighted sum rules. The
excitation spectrum shows a splitting due to holons and antiholons as non-
Fermi liquid behaviour. A renormalization procedure is proposed by subtracting
an energy constant to render the Fock exchange energy finite. The effective
mass derived from meanfield shows a dip as onset of Peierls instability. The
correlation energy is calculated with the help of the extended quasiparticle
picture which accounts for off-shell effects. The corresponding response
function leads to the same correlation energy as the selfenergy in agreement
with perturbation theory. The reduced density matrix or momentum distribution
is calculated with the help of a Padé regularization repairing deficiencies of
the perturbation theory. A seemingly finite step at the Fermi energy
indicating Fermi-liquid behaviour is repaired in this way.

##  I Introduction and motivation

The one-dimensional correlated electron gas is especially interesting since
the quasiparticle picture breaks down and non-Fermi liquid behaviour appears.
Such non-Fermi liquid behaviour has been observed in various physical systems
ranging from large-scale structures like crystalline ion beams SSH01 ; SSBH02
, quantum wires PMC14 , carbon nanotubes Saito98 ; Bockrath99 ; Ishii03 ;
Shiraishi03 , edge states in quantum hall liquid Milliken96 ; Mandal01 ;
Chang03 , semiconducting nanowires Schafer08 ; Huang01 , cold atomic gases
Monien98 ; Recati03 ; Moritz05  up to conducting molecules Nitzan03 . Mostly
it is claimed that perturbation theory breaks down due to divergences in the
expansion and the absence of quasiparticles since single-particle excitations
turn into collective ones SDC19 ; DBG10 ; 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. This is achieved if one expands with respect to small damping
(scattering rate) resulting into the extended quasiparticle approximation SL94
; SL95  and used for transport in impurity systems SLMa96 ; SLMb96  or
nonlocal kinetic theory SLM96 ; MLS00 ; M17b .

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 . This extended quasiparticle picture is plagued by the
same divergence at the Fermi energy as it is typical for non-Fermi liquids.
Recently this deficiency of quasiparticle picture has been cured by a Padé
approximation M23  which shows that the extended quasiparticle picture works
and perturbation theory can be applied. This renewed interest in perturbation
theory is motivated by the fact that in one-dimensional systems the strongly
correlated case coincides with the small-density limit due to the special
density dependencies of kinetic and correlation energy GPS08 . Therefore it is
worth to investigate how far the extended quasiparticle picture which takes
into account first-order damping and off-shell behaviour is able to capture
the physics.

The aim of the article is twofold. First in a pedagogical sense, different
many-body schemes of Dyson equation and selfenergy on one side and the
variational technique with response and correlation functions on the other
side are unified and it is shown how they are yielding to identical results.
The second aim is to demonstrate how with a proper Padé regularization a
finite momentum distribution can be achieved.

There exists of course a fast literature on many-body theories among them only
Green function techniques are mentioned here T67 ; V94 ; G04 ; GV08 ; VMKA08 ;
M17b . Specially for one-dimensional systems exact solutions are known for
Luttinger L63 ; LP74 ; ES07 , Tomonaga DL74 , and Gaudin-Yang models Ram17 ;
Pan22  by the Bethe ansatz EFGKK10 ; GBML13 . Frequently bosonization
techniques are used GGM10 ; Lu77 ; Solyom79 ; H99  due to the similar
behaviour of long-distance correlations of Fermi and Bose systems H81a ;
Emery79 . If one wants to consider more realistic systems like the width
dependence GADMP22  of quantum wires one does not have any exact solutions and
perturbation methods have been used to investigate analytically and
numerically the ground-state properties LD11 ; LG16 ; Loos13 . In MVBP18  it
was shown that the Random Phase Approximation (RPA) becomes exact in the high-
density limit for one-dimensional systems. An overview about one-dimensional
models can be found in V94 ; G04 ; GV08

The paper consists of two main parts. First, in the second chapter two
approaches to correlations are presented, i.e. the structure factor and the
Dyson equation with selfenergy both rooted in the two-particle Green function.
It will be shown that the extended quasiparticle picture reproduces the
correlation results of structure factor and correlation energy. It is
explicitly demonstrated on the Born approximation level. The coupling
parameter integration as a special form of variational method is demonstrated
to yield the same correlation energy. Second, in chapter III we present the
model of finite width Fermion wires and discuss systematically the meanfield
and Born approximation. From the meanfield the effective mass is calculated in
chapter III.3. A dip occurs at twice the Fermi momentum indicating the onset
of Peierls instability. The selfenergy in Born approximation is then
calculated in chapter III.4 revealing an energy gap. The resulting spectral
function is presented which requires a re-adaption of the correct pole when
considered selfconsistently. The extended quasiparticle approximation
describes this correct poles and completes the first two energy-weighted
sumrules. In chapter III.6 we collect the results of structure factor and pair
correlation function together with the correlation energy. The reduced density
matrix is explicitly in chapter III.8 calculated which shows a divergence at
the Fermi energy due to perturbation theory. With the help of a proper Padé
regulator this divergence can be subtracted and the momentum distribution
takes a finite value at the Fermi momentum. The contact potentials as well as
finite-size potentials show seemingly a finite jump at the Fermi energy like a
Fermi liquid which is corrected by a proper Padé regulator. Chapter IV
summarizes the finding. In the appendices the corresponding integration
schemes are presented for the selfenergy and the momentum distribution.

##  II Many-body scheme

###  II.1 Correlation energy and pair correlation

We consider Hamiltonians of the form

$\displaystyle\hat{H}=\hat{H}_{0}+\hat{V}=\sum\limits_{1}\epsilon_{1}\hat{\Psi}_{1}^{+}\hat{\Psi}_{1}+\frac{1}{2}\sum\limits_{12}V_{12}\hat{\Psi}_{1}^{+}\hat{\Psi}_{2}^{+}\hat{\Psi}_{2}\hat{\Psi}_{1}$
(1)

with creation operators $\Psi^{+}$ and the free single-particle band
dispersion $\epsilon_{1}={k_{1}^{2}\over 2m}$, e.g. for free particles.
Numbers are cumulative variables, $1\equiv x_{1},t_{1},...$. The probability
to find a particle at $1$ and another at $2$ is expressed by the one-time
pair-correlation function

$\displaystyle\langle\hat{\Psi}_{1}^{+}\hat{\Psi}_{2}^{+}\hat{\Psi}_{2}\hat{\Psi}_{1}\rangle_{t_{1}=t_{2}}=\langle\hat{n}\left(x_{1}\right)\hat{n}\left(x_{2}\right)\rangle-n(x_{1})\delta(x_{1}-x_{2})$
$\displaystyle=g(x_{1},x_{2})n\left(x_{1}\right)n\left(x_{2}\right).$ (2)

This can be used to express the mean correlation energy in space
representation by SL13

$\displaystyle E_{int}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
dx_{1}dx_{2}V(x_{1},x_{2})\langle\hat{\Psi}_{x_{1}}^{+}\hat{\Psi}_{x_{2}}^{+}\hat{\Psi}_{x_{2}}\hat{\Psi}_{x_{1}}\rangle$
(3) $\displaystyle=$ $\displaystyle\frac{1}{2}\int
dx_{1}dx_{2}V(x_{1},x_{2})n(x_{1})n(x_{2})g(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dx_{1}dx_{2}V(x_{1},x_{2})n(x_{1})n(x_{2})[g(x_{1},x_{2})-1]$
$\displaystyle+\frac{1}{2}\int dx_{1}dx_{2}V(x_{1},x_{2})n(x_{1})n(x_{2})$

where the definition (2) is used in the second line and the Hartree energy is
split off in the third line. Changing to difference and centre-of-mass
coordinates $r=x_{1}-x_{2}$, $R=(x_{1}+x_{2})/2$ and introducing the liquid
structure function

$\displaystyle
n(R)S(r,R)=(g(r,R)\\!-\\!1)n\left(\\!R\\!+\\!\frac{r}{2}\right)\\!n\\!\left(\\!R\\!-\\!\frac{r}{2}\\!\right)\\!+\\!n(R)\delta(\\!r\\!)$
(4)

the correlation energy without Hartree term $E_{H}$ which is the correlation
energy $E_{c}$ with Fock (exchange) term, becomes

$\displaystyle E_{int}\\!-\\!E_{H}=E_{c}\\!+\\!E_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\\!\int\\!\\!drdR\,V(r)n(R)[S(r,R)\\!-\\!1]$ (5)
$\displaystyle=$
$\displaystyle\frac{N}{2}\int{dq\over(2\pi)^{d}}V_{-q}(S_{q}-1)$

where the last line is valid if $S(r,R)\approx S(r)$. Integrating (4) yields
the density fluctuation correlator

$\displaystyle\int dr{\rm e}^{-irq}\int dR\,n(R)S(r,R)$ $\displaystyle=$
$\displaystyle\langle\hat{n}_{q}\hat{n}_{-q}\rangle-n_{q}n_{-q}$ (6)
$\displaystyle=$ $\displaystyle\langle\delta n_{q}\delta n_{-q}\rangle$

where the left side can be integrated again for $S(r,R)\approx S(r)$ and
finally

$\displaystyle
S_{q}=\frac{1}{N}\langle\delta\hat{n}_{q}\delta\hat{n}_{-q}\rangle$ (7)

with the total number of particles $N$. It means that the structure function
is the density fluctuation correlator. Therefore it is advisable to
investigate the density fluctuations.

###  II.2 Two-particle Green’s function and selfenergy

For this aim we consider the causal Green’s function

$\displaystyle
G(1,2)=\frac{1}{i}\Theta(t_{1}\\!-\\!t_{2})G^{>}(1,2)\\!\mp\\!\frac{1}{i}\Theta(t_{2}\\!-\\!t_{1})G^{<}(1,2)$
(8)

for Fermi/Bose systems which time-orders the two double-time correlation
functions $G^{<}(1,2)=\langle\hat{\psi}^{\dagger}_{2}\hat{\psi}_{1}\rangle$
and $G^{>}(1,2)=\langle\hat{\psi}_{1}\hat{\psi}^{\dagger}_{2}\rangle$ with
averaging about the unknown statistical operator. Applying the Heisenberg
equation of motion one obtains the Martin-Schwinger hierarchy MS59 ; KB62
where the one-particle Green’s function couples to the two-particle,

$\displaystyle G_{2}(1,3;2,4)={1\over
i^{2}}\langle\hat{T}\hat{\psi}_{1}\hat{\psi}_{3}\hat{\psi}_{4}^{\dagger}\hat{\psi}_{2}^{\dagger}\rangle,$
(9)

the two-particle to the three-particle and so on. A formal closure is reached
by introducing the selfenergy

$\displaystyle\mp
i\\!\int\\!d3\,V(1,3)G_{2}(13,23^{+})=\int\limits_{C}\\!d3\,\Sigma(1,{3})G({3},2),$
(10)

as illustrated in figure 1. About double occurring indices we understand
integration in the following.

Figure 1: Formal closure of the Martin-Schwinger hierarchy for Fermi/Bose
systems introducing the selfenergy (10). About variable $3$ one integrates and
$3^{+}$ means infinitesimal time later that at $3$. We introduce diagrammatic
rules that a broken line means $iV$ and an arrow line means a causal Green’s
function $G$.

The integration path is determined by the demand that in the infinite past the
two-particles are uncorrelated which leads to M17b

$\displaystyle\int\limits_{C}d3\Sigma(1,{3})G(3,2)$
$\displaystyle=\int\limits_{t_{0}}^{+\infty}d{3}\biggl{\\{}\Sigma(1,3)G(3,2)\mp\Sigma^{<}(1,3)G^{>}(3,2)\biggr{\\}}.$
(11)

This allows to set up conveniently the Langreth-Wilkins rules LW72  to recover
the correlation parts from causal functions. If one has $C(1,2)=\int
d3A(1,3)B(3,2)$ these rules provide

$C^{\gtrless}=A^{\gtrless}B^{A}+A^{R}B^{\gtrless}$ (12)

where the retarded/advanced functions are

$\displaystyle C^{R}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle-i\theta(t-t^{\prime})\left[C^{>}(t,t^{\prime})\pm
C^{<}(t,t^{\prime})\right],$ $\displaystyle C^{A}(t,t^{\prime})$
$\displaystyle=$ $\displaystyle\ \
i\theta(t^{\prime}-t)\left[C^{>}(t,t^{\prime})\pm C^{<}(t,t^{\prime})\right].$
(13)

Now we can systematically expand the two-particle Green’s function in terms of
interaction as given in figure 2. Introducing this into (10), the first term
gives the Hartree, the second the Fock, and the third and fourth term the
first Born approximations of the selfenergy.

Figure 2: Expansion of the two-particle Green function up to linear order in
V.

The closure (10) leads to the Dyson equation for the full propagator

$\displaystyle G(1,2)=G_{0}(1,2)+G_{0}(1,3)\bar{\Sigma}(3,4)G(4,2)$ (14)

or graphically

where the thin line is $G_{0}$ and we absorb the Hartree selfenergy together
with the external potential $U$ into the induced potential

$\displaystyle{\bar{U}}_{11^{\prime}}=U_{11^{\prime}}\mp
iV_{12}G_{22^{+}}\delta_{11^{\prime}},$ (15)

such that the free propagator reads

$\displaystyle\left(i\frac{\partial}{\partial
t_{1}}+\frac{\nabla_{1}^{2}}{2m}-\bar{U}(1)\right)G_{0}(1,2)=\delta(1-2).$
(16)

One Fourier transforms the difference coordinates into frequency/momentum.
Since we concentrate on equilibrium all quantities are only dependent on the
difference of coordinates. Nonequilibrium expressions can be found in M17b .

###  II.3 Correlation energy and extended quasiparticle picture

From the Heisenberg equation of motion the Hamiltonian (1) leads to

$\displaystyle(i\partial_{t_{1}}\\!-\\!i\partial_{t_{2}})\Psi_{2}^{+}\Psi_{1}=$
$\displaystyle(\epsilon_{1}\\!+\\!\epsilon_{2})\Psi_{2}^{+}\Psi_{1}$
$\displaystyle+\sum\limits_{3}(V_{13}+V_{12})\Psi_{2}^{+}\Psi_{3}^{+}\Psi_{3}\Psi_{1}$
(17)

and therefore after averaging

$\displaystyle\left.\frac{1}{2}(i\partial_{t_{1}}-i\partial_{t_{2}})G^{<}_{12}\right|_{1=2}=\langle
K\rangle+2\langle V\rangle$ (18)

with the kinetic energy $\langle
K\rangle=\operatorname{Tr}[\hat{\rho}\hat{H}_{0}]$ and the potential energy
$\langle V\rangle=\operatorname{Tr}[\hat{\rho}\hat{V}]$. If one knows the
correlation function, the correlation energy density can be expressed
therefore as

$\displaystyle E_{c}+E_{\rm HF}=\sum\limits_{k}\int{d\omega\over
2\pi}\frac{1}{2}\left(\omega-{k^{2}\over 2m}\right)G^{<}(k,\omega)$ (19)

where the total energy would be the plus sign instead of minus sign in the
bracket.

Within in the extended quasiparticle picture we expand the correlation
functions with respect to the order of selfenergy

$\displaystyle G^{<}(k,\omega)={2\pi\delta(\omega\\!-\\!\varepsilon_{k})\over
1-{\partial\Sigma(\omega)\over\partial\omega}}n_{k}\\!+\\!\Sigma^{<}(\omega){\wp^{\prime}\over\omega\\!-\\!\varepsilon_{k}}+o(\Sigma^{2})$
(20)

where the real part of the spectral function is the Hilbert transform

$\displaystyle\Sigma={\rm Re}\Sigma^{R}=\int{d\bar{\omega}\over
2\pi}{\Gamma(\bar{\omega})\over\omega-\bar{\omega}}$ (21)

of the selfenergy spectral function

$\displaystyle\Gamma=\Sigma^{>}+\Sigma^{<}=-2{\rm Im}\Sigma^{R}.$ (22)

Both specifying the retarded selfenergy

$\displaystyle\Sigma^{R}(q,\omega)=\Sigma(q,\omega)-\frac{i}{2}\Gamma(q,\omega)=\int{d\bar{\omega}\over
2\pi}{\Gamma(\bar{\omega})\over\omega-\bar{\omega}+i\eta}.$ (23)

We abbreviate

$\displaystyle\varepsilon^{0}_{k}=\epsilon_{k}+\Sigma^{\rm
HF},\quad\varepsilon_{k}=\varepsilon^{0}_{k}+\Sigma(k,\varepsilon_{k})$ (24)

according to the poles of (20).

If we integrate (20) over the energy $\omega$ we get the connection between
reduced density matrix $\rho$ and the distribution $n_{k}$ as

$\displaystyle\rho_{k}=n_{k}+\int{d\omega\over 2\pi}{\Sigma^{<}(\omega)(1\mp
n_{k})-\Sigma^{>}(\bar{\omega})n_{k}\over(\omega-\epsilon_{k})^{2}}.$ (25)

The quasiparticle (Bose/Fermi) distribution $n_{k}$ is to be take at the pole
$\varepsilon_{k}$. The form (20) was derived with respect to small scattering
rate expansion and with quasiparticle energies under the name of extended
quasiparticle approximation in nonequilibrium SL94 ; SL95  and the history of
this expansion was given in the introduction above. Details can also be found
in M17b .

###  II.4 Born approximation

Since we want to consider the terms up to $V^{2}$ in the selfenergy, the terms
exceeding (20) would start with fourth-order interaction. Using the expansion
of figure 2 to calculate the selfenergy in figure 1 one sees that the Hartree
and Fock terms are time-diagonal and therefore do not possess any frequency
dependence which leads to no second part of (20). The Born terms lead to the
selfenergies

$\displaystyle\Sigma^{<}(k,\omega)$ $\displaystyle=$
$\displaystyle\sum\limits_{qp}2\pi\delta(\omega\\!+\\!\epsilon_{p}\\!-\\!\epsilon_{p\\!-\\!q}\\!-\\!\epsilon_{k\\!+\\!q})n_{p\\!-\\!q}n_{k\\!+\\!q}(1\\!-\\!n_{p})$
(26) $\displaystyle\times V_{q}\left[g_{s}V_{q}-V_{p-k-q}\right]$

where the spin degeneracy $g_{s}$ does only apply to the direct and not to the
exchange terms. The $\Sigma^{>}$ selfenergy is obtained by interchanging
$n\leftrightarrow 1-n$. Using this Born approximation in (25) we get

$\displaystyle\langle\epsilon\rho\rangle=\sum\limits_{k}\epsilon_{k}\rho_{k}=\sum\limits_{k}\epsilon_{k}n_{k}+\sum\limits_{kpq}V_{q}\left[g_{s}V_{q}-V_{p-k+q}\right]\epsilon_{k}$
$\displaystyle\times{n_{k+q}n_{p-q}(1\\!\mp\\!n_{k})(1\mp
n_{p})\\!-\\!n_{k}n_{p}(1\\!\mp\\!n_{k+q})(1\\!\mp\\!n_{p-q})\over(\epsilon_{k+q}+\epsilon_{p-q}-\epsilon_{p}-\epsilon_{k})^{2}}$
(27)

and

$\displaystyle\langle\rho\Sigma\rangle=\langle n\Sigma\rangle+o(V^{4}).$ (28)

Using symmetry to replace
$\epsilon_{k}\to-\frac{1}{4}(\epsilon_{k+q}+\epsilon_{p-q}-\epsilon_{p}-\epsilon_{k})$
in (27) and subtracting (28) we obtain

$\displaystyle\langle\varepsilon^{0}\rho\rangle=\langle\varepsilon^{0}n\rangle-\frac{1}{4}\sum\limits_{kpq}V_{q}\left[g_{s}V_{q}-V_{p-k+q}\right]\epsilon_{k}$
$\displaystyle\times{n_{k+q}n_{p-q}(1\mp n_{k})(1\mp n_{p})-n_{k}n_{p}(1\mp
n_{k+q})(1\mp
n_{p-q})\over\epsilon_{k+q}+\epsilon_{p-q}-\epsilon_{p}-\epsilon_{k}}.$ (29)

We identify the difference of kinetic energy calculated with the reduced
density matrix and the quasiparticle distribution function

$\displaystyle K_{\rho}$ $\displaystyle=K_{n}-\frac{1}{2}E_{c}$ (30)

with the first non-vanishing correlation energy (19)

$\displaystyle{E_{c}}=$
$\displaystyle\int{dkdpdq\over(2\pi\hbar)^{3}}V_{q}[g_{s}V_{q}-V_{p-k-q}]$
$\displaystyle\times{(1-n_{p-q})(1-n_{k+q})n_{p}n_{k}\over\epsilon_{p}+\epsilon_{k}-\epsilon_{k+q}-\epsilon_{p-q}}.$
(31)

The difference of reduced density matrix and quasiparticle distribution
function accounts for half of correlation energy KM01  and we can write
alternatively for the total energy

$\displaystyle E=K_{n}+\frac{1}{2}E_{c}=K_{\rho}+E_{c}.$ (32)

We want to note here already that the on-shell selfenergy
$\Sigma^{\gtrless}(k,\epsilon_{k})$ will vanish since two particles can
scatter only by exchanging their momenta. In (20) the off-shell selfenergy is
required which has been presented in a different scheme and discussed in MA23
.

Integrating (20) over frequency we get the reduced density matrix or Wigner
distribution $\rho_{k}=n_{k}+\rho_{k}^{>}-\rho_{k}^{<}$ in terms of the Fermi-
Dirac distribution $n_{k}$. We can explicitly using (26) in (25) to obtain

$\displaystyle\rho_{k}^{>}$ $\displaystyle=$
$\displaystyle(1-n_{k})\int{dpdq\over(2\pi\hbar)^{2}}{V_{q}(g_{s}V_{q}-V_{p-k-q})\over(\epsilon_{p}+\epsilon_{k}-\epsilon_{k+q}-\epsilon_{p-q})^{2}}$
(33) $\displaystyle\times n_{p-q}n_{k+q}(1-n_{p})$

and $\rho_{k}^{<}$ by replacing $n\leftrightarrow 1-n$.

###  II.5 Density fluctuation and response function

Besides the selfenergy we can extract also the density fluctuation
$\delta\hat{n}(11^{\prime})=\Psi^{+}(1^{\prime})\Psi(1)-\langle\Psi^{+}(1^{\prime})\Psi(1)\rangle$
from the two-particle Green’s functions. One observes that the fluctuation
correlations are linked to the two-particle Green’s function subtracted by the
Hartree term

$-i\chi(121^{\prime}2^{\prime})=G_{2}(121^{\prime}2^{\prime})-G^{c}(11^{\prime})G^{c}(22^{\prime})$
(34)

and

$\displaystyle\langle\delta\hat{n}(11)\delta\hat{n}(22)\rangle$
$\displaystyle=\langle\Psi_{1}^{+}\Psi_{1}\Psi_{2}^{+}\Psi_{2}\rangle-G^{<}(11)G^{<}(22)$
$\displaystyle=i^{2}\Theta(t_{1}-t_{2})G_{121^{+}2^{+}}+G^{c}(11^{+})G^{c}(22^{+})$
$\displaystyle=\chi^{>}(12)=\chi^{<}(21)$ (35)

which is illustrated in figure 3.

and

Figure 3: The causal functions representing equation (35).

In other words, instead of closing (10) in an s-channel way, the density
fluctuations consider the two-particle Green function in a t-channel, i.e. a
different way to reduce the four times to two times. We can use now the causal
two-time correlation function

$\chi(12)=\frac{1}{i}\Theta(t_{1}-t_{2})\chi^{>}(12)+\frac{1}{i}\Theta(t_{2}-t_{1})\chi^{<}(12)=\chi_{121^{+}2^{+}}$
(36)

where we obey the Bose character of fluctuations by the plus sign in
accordance with (8).

On the other hand one can express the two-particle Green’s function by a
variation of the one-particle Green’s functions $iG_{12}=\langle
T_{c}\hat{\Psi}_{1}\hat{\Psi}_{2}^{+}\rangle$ with respect to the external
potential KB62 ; kker86

$\displaystyle
G_{121^{\prime}2^{\prime}}=G_{11^{\prime}}G_{22^{\prime}}\mp{\delta
G_{11^{\prime}}\over\delta U_{2^{\prime}2}}$ (37)

corresponding to Fermi/Bose systems. It shows that the response function is
just the density fluctuation (36) when we consider the times
$t_{2}=t_{1}^{+}=t_{1}+0$

$\displaystyle\chi_{12}^{<}$ $\displaystyle=$ $\displaystyle{\delta
n_{1}\over\delta U_{2^{+}2}}={\delta G^{<}_{11^{+}}\over\delta U_{2^{+}2}}=\mp
i{\delta G_{11^{+}}\over\delta U_{2^{+}2}}$ (38) $\displaystyle=$
$\displaystyle i(G_{121^{+}2^{+}}-G_{11^{+}}G_{22^{+}})=i\chi_{12}.$

Now we make the link to the structure function of chapter II.1. From the
definitions (2) and (4) we see the relation of the pair correlation function
and structure function to the fluctuation function (35) are

$\displaystyle i\chi(x_{1}t_{1},x_{2}t_{1+})$ $\displaystyle=$
$\displaystyle\chi^{<}(12)=\langle\Psi_{2}^{+}\Psi_{2}\Psi_{1}^{+}\Psi_{1}\rangle-n(1)n(2)$
(39) $\displaystyle=$
$\displaystyle\langle\Psi_{1}^{+}\Psi_{2}^{+}\Psi_{2}\Psi_{1}\rangle+\delta_{12}n(1)-n(1)n(2)$
$\displaystyle=$ $\displaystyle n(1)n(2)[g(12)-1]+\delta_{12}n(1)$
$\displaystyle=$ $\displaystyle n(1)S(x_{1},x_{2},t,t)$

which provides the static pair-correlation (2) and structure function (4) in
terms of the time-diagonal of $\chi^{<}$. Neglecting gradients in the density
$n(R\pm r/2)\approx n(R)$ we obtain the known Fourier transform

$\displaystyle S_{q}$ $\displaystyle=$ $\displaystyle 1+n\int dr{\rm
e}^{-iqr}[g(r)-1]={1\over n}\chi^{>}(q,t,t).$ (40)

We can extend this to the dynamical structure factor by

$\displaystyle nS(q,\omega)$ $\displaystyle=$ $\displaystyle\int
d(t-t^{\prime}){\rm e}^{i\omega(t-t^{\prime})}\chi^{>}(q,t,t^{\prime})$ (41)
$\displaystyle=$ $\displaystyle\chi^{>}(q,\omega)=-2{\rm
Im}\chi(q,\omega)[1+n_{B}(\omega)]$

with the Bose distribution $n_{B}$ accounting for the Bose character of the
fluctuations. Since the imaginary part of the response function is odd in
frequency we can write

$\displaystyle S_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\int\limits_{-\infty}^{\infty}{d\omega\over
2\pi}[1+n_{B}(\omega)][-2{\rm Im}\chi(q,\omega)]$ (42) $\displaystyle=$
$\displaystyle\frac{1}{n}\int\limits_{0}^{\infty}{d\omega\over 2\pi}{\rm
coth}\left({\beta\omega\over 2}\right)[-2{\rm Im}\chi(q,\omega)].$

If we consider the ground state at $\beta=1/k_{B}T\to\infty$ the $coth$
approaches unity.

###  II.6 Coupling parameter integration

So far we have two possibilities to calculate the correlation energy. We need
the first-order expansion in the two-particle Green’s function, Fig. 2, in
order to achieve the second order in the selfenergy in figure 1 and the
correlation energy (19). The same approximation of the two-particle Green’s
function is used now to calculate also the density response in figure 3 as
t-channel closing which provides the density fluctuations, the structure
factor (42) and the correlation energy (5). Due to the same rooting of
approximations to the two-particle Green’s function the correlation energies
(5) and (19) will coincide.

The total ground-state energy can be obtained by coupling-constant
integration. Therefore we add a constant $\lambda$ in front of the potential.
Since the ground-state wave function is normalized independently of this
parameter $\langle\Psi_{0}(\lambda)|\Psi_{0}(\lambda)\rangle=1$, the
derivative of the ground-state energy reads

$\displaystyle{dE(\lambda)\over d\lambda}$
$\displaystyle={d\over\lambda}\langle\Psi_{0}(\lambda)|\hat{H}(\lambda)|\Psi_{0}(\lambda)\rangle=\langle\Psi_{0}(\lambda)|{d\over\lambda}\hat{H}(\lambda)|\Psi_{0}(\lambda)\rangle$
$\displaystyle={E_{int}(\lambda)\over\lambda}$ (43)

which provides the ground-state energy beyond the free one $E_{0}$ as

$\displaystyle
E=E_{0}+\int\limits_{0}^{1}{d\lambda\over\lambda}E_{int}(\lambda)$ (44)

which we will call charging formula. Using (42) in (5) we get the ground-state
energy without Hartree term per particle as

$\displaystyle{E-E_{H}\over
N}=-\frac{1}{2}\int\\!{dq\over(2\pi)^{d}}V_{q}\left[\frac{1}{n}\int\limits_{0}^{1}\\!{d\lambda}\int\limits_{0}^{\infty}{d\omega\over\pi}{\rm
Im}\chi(q,\omega,\lambda)\\!+\\!1\right].$ (45)

Both forms (45) as well as (19) allow a systematic expansion with the help of
the two-particle Green’s function. Some pitfalls of the coupling-constant
integration are discussed in O76 .

###  II.7 RPA-like integral equations

Sometimes it is useful to express the response function (38) with respect to
the external potential $U$ in terms of the polarization function which is the
response with respect to the induced potential (15)

$\displaystyle\Pi_{12^{\prime}1^{\prime}2}=\mp{\delta
G_{11^{\prime}}\over\delta{\bar{U}}_{22^{\prime}}}.$ (46)

Using $\delta G=-G\delta G^{-1}G$ we can write the polarisation function with
the help of the Dyson equation (14)-(16)

$\displaystyle\Pi_{12^{\prime}1^{\prime}2}=\mp
G_{12}G_{2^{\prime}1^{\prime}}\mp
G_{13}{\delta{\bar{\Sigma}}_{34}\over\delta{\bar{U}}_{22^{\prime}}}G_{41^{\prime}}.$
(47)

On the other hand we can express the response function (38) in terms of the
fluctuation $L$ with the help of (37) as

$\displaystyle L_{121^{\prime}2^{\prime}}$ $\displaystyle=$ $\displaystyle
G_{121^{\prime}2^{\prime}}-G_{11^{\prime}}G_{22^{\prime}}$ (48)
$\displaystyle=$ $\displaystyle\mp G_{12^{\prime}}G_{21^{\prime}}\mp
G_{13}{\delta\Sigma_{34}\over\delta U_{2^{\prime}2}}G_{41^{\prime}}$
$\displaystyle=$ $\displaystyle\mp
G_{12^{\prime}}G_{21^{\prime}}+G_{13}{\delta\Sigma_{34}\over\delta
G_{56}}L_{5262^{\prime}}G_{41^{\prime}}.$

Comparing (48) and (47) and using the chain rule to express variations with
respect to $U$ by variations with respect to ${\bar{U}}$ we find a relation
between $L$ and $\Pi$ expressed in figure 4.

Figure 4:  The integral equation connecting the response $\chi=iL$ with the
polarization $\Pi$.

Closing the upper and lower edges in the t-channel manner we obtain the RPA-
like integral equation for the causal functions

$\displaystyle L(12)=-i\chi(12)=\Pi(12)+\Pi(13)V(34)L(42)$ (49)

which in equilibrium is solved and reads for the retarded functions

$\displaystyle L^{R}(q,\omega)=\chi^{R}(q,\omega)={\Pi^{R}(q,\omega)\over
1-V(q)\Pi^{R}(q,\omega)}.$ (50)

There is some care to be observed since the polarization itself has a kernel
to be determined selfconsistently with the selfenergy. In fact closing (47) in
the t-channel manner we get for the polarization just Fig. 5 which shows that
any approximation beyond the lowest RPA are calculated as variations of the
selfenergy.

Figure 5:  Polarization (47) in terms of the variation of the selfenergy with
respect to the induced potential (15).

Let us illustrate the procedure with the first-order expansion of the
selfenergy which is the Fock term

$\displaystyle\Sigma^{F}(34)=\mp V(3-4)G(34)$ (51)

and the Hartree term is absorbed in (15). We obtain

$\displaystyle{\delta\Sigma^{F}(3,4)\over\delta\bar{U}(2,2^{\prime})}=\mp
V(3,4){\delta G(3,4)\over\delta\bar{U}(2,2^{\prime})}$ $\displaystyle=\pm
V(3,4)G(3,5){\delta[\bar{G}_{0}^{-1}(5,6)+\Sigma(5,6)]\over\delta\bar{U}(2,2^{\prime})}G(6,4)$
$\displaystyle=\mp
V(3,4)G(3,5)\delta_{2,6}\delta_{5,2^{\prime}}G(6,4)+o(V^{2})$
$\displaystyle=\mp V(3,4)G(3,2^{\prime})G(2,4)$ (52)

and introduced in (47) one gets the expansion in figure 6. Besides the non-
interacting polarization function

$\displaystyle\Pi_{0}(q,\omega)=g_{s}\sum_{k}\frac{n_{k}-n_{k+q}}{\omega+\Omega_{k,q}}$
(53)

there appear the selfenergy and vertex correction given by BMSP12

$\displaystyle\Pi_{se}(q,\omega)=g_{s}\sum_{k,p}\frac{v(k-p)(n_{k}-n_{k+q})(n_{p}-n_{p+q})}{(\omega+\Omega_{k,q})^{2}}$
(54)

and

$\displaystyle\Pi_{ex}(q,\omega)=\mp
g_{s}\\!\sum_{k,p}\frac{v(k\\!-\\!p)(n_{k}\\!-\\!n_{k+q})(n_{p}\\!-\\!n_{p+q})}{(\omega+\Omega_{k,q})(\omega+\Omega_{p,q})},$
(55)

respectively. Here $\Omega_{k,q}=\omega_{k}-\omega_{k+q}$,
$\Omega_{p,q}=\omega_{p}-\omega_{p+q}$ and $n_{k}$ represents the Fermi-Dirac
or Bose distribution function for Fermions or Bosons respectively.

Figure 6:  The polarization diagrams (47) when introducing (52), the second
line is the expansion with the help of the Dyson equation (14) up to first-
order interaction $\mp\Pi_{0}\mp\Pi_{se}+\Pi_{ex}$.

This completes the many body scheme where we have shown how the variational
expressions resulting into pair correlation or structure function as well as
response functions appear on the same footing as the single-particle self
energy from Dyson equation. Both lead to the same expression for the
correlated energy and are rooted to the two-particle Green function.

##  III Wire of Fermions

###  III.1 Model

We apply now the many-body scheme to the model of a one-dimensional wire of
charged Fermions. Due to the strong divergence of the Coulomb interaction we
model it by a soften Coulomb potential of a cylindrical wire
$V(r)=e^{2}/4\pi\epsilon_{0}\sqrt{r^{2}+\bar{b}^{2}}$ and consider the limit
$b\to 0$. Its Fourier transform reads

$\displaystyle V(q)$ $\displaystyle={e^{2}\over 4\pi\epsilon_{0}}v(q)$
$\displaystyle v(q)$
$\displaystyle=2K_{0}(\bar{b}q)=-2\left[\ln\left(\frac{q}{2}\right)+\gamma\right]-2\ln\bar{b}+o\left(\bar{b}^{2}\right)$
(56)

where $\bar{b}$ is related to the transverse width parameter of the wire and
$K_{0}$ is the modified Bessel function of $2^{nd}$ kind.

Within the jellium model of electron density $\rho(x)$ one considers an
oppositely charged background density $\rho_{b}(x)$. The background potential
$V_{b}(x)=-\int dx^{\prime}V(x-x^{\prime})\rho_{b}(x^{\prime})$ gives the
interaction energy of electrons with the background

$\displaystyle E_{e-b}=-\int dx\rho(x)V_{b}(x).$ (57)

This energy is compensated by the selfenergy of the background itself

$\displaystyle\frac{1}{2}\int
dxdx^{\prime}\rho_{b}(x)\rho_{b}(x^{\prime})V(x-x^{\prime})$ (58)

together with the Hartree selfenergy of the electrons

$\displaystyle\frac{1}{2}\int
dxdx^{\prime}\rho(x)\rho(x^{\prime})V(x-x^{\prime})$ (59)

if charge neutrality $\rho_{b}(x)=\rho(x)$ is assumed. Therefore the Hartree
term does not count and we can use directly the formulas (19) and (45)
starting from the Fock term.

We consider first spin-polarized densities $n_{\uparrow\downarrow}=n(1\pm
p)/2$ with arbitrary polarization $p=(n_{\uparrow}-n_{\downarrow})/n$.
Therefore the Fermi momentum is $k_{\uparrow\downarrow}=\pi\hbar
n/g_{s}=k_{F}/g_{s}$ with $g_{s}=2/(1\pm p)$. For the paramagnetic case we
have $p=0$ and $g_{s}=2$ which means $k_{\uparrow}=k_{\downarrow}=k_{F}/2$.
Correspondingly for the ferromagnetic case $g_{s}=1$ and $k_{\uparrow}=k_{F}.$
The $r_{s}$ parameter as the number of particles in the Wigner size radius
$2a_{B}$ is $r_{s}=1/2na_{B}$.

###  III.2 Fock term or exchange term

First we investigate the lowest order Fock term

$\displaystyle\Sigma_{F}(k)=\mp\int\limits_{-\infty}^{\infty}{dq\over
2\pi\hbar}V_{q}n_{k-q}$ (60)

which would be the upper sign for for the spin-polarized electrons, $g_{s}=1$.
The spectral function in the Fock-propagator
$G^{<}(k,\omega)=a(k,\omega)n_{k}$ becomes
$a(k,\omega)=2\pi\delta(\omega-{k^{2}\over 2m}-\Sigma_{\rm HF})$ and (19)
leads to the Fock correlation energy density

$\displaystyle{E_{F}\over\Omega}=\frac{1}{2}\int\limits_{-\infty}^{\infty}{dk\over
2\pi\hbar}n_{k}\Sigma_{F}(k).$ (61)

It is instructive to see how this formula appears from the charging formula
(44) or (45). For any temperature we have (41) and from (50) ${\rm Im}L={\rm
Im}\chi={\rm Im}\Pi_{0}+o(V)$. With the help of
$n_{B}(-\omega)=-1-n_{B}(\omega)$ we write

$\displaystyle{E_{F}\over
N}=\frac{1}{2}\int\\!{dq\over(2\pi)^{d}}V_{q}\left[\frac{1}{n}\int\limits_{-\infty}^{\infty}{d\omega\over\pi}n_{b}(-\omega){\rm
Im}\Pi_{0}(q,\omega)\\!-\\!1\right].$ (62)

Using

$\displaystyle{\rm
Im}\Pi_{0}(q,\omega)=g_{s}\pi\\!\int\\!\\!{dk\over(2\pi)^{d}}[n(\epsilon_{k})\\!-\\!n(\epsilon_{k\\!-\\!q})]\delta(\epsilon_{k}\\!-\\!\epsilon_{k\\!-\\!q}\\!-\\!\omega)$
(63)

one calculates with the help of $n(a)[\pm 1-n(b)]=n(a)n(-b)=\pm
n_{B}(a-b)[n(b)-n(a)]$

$\displaystyle\int\limits_{-\infty}^{\infty}{d\omega\over\pi}n_{b}(-\omega){\rm
Im}\Pi_{0}(q,\omega)$
$\displaystyle=g_{s}\int{dk\over(2\pi)^{d}}[n(\epsilon_{k})-n(\epsilon_{k-q})]n_{b}(\epsilon_{k-q}-\epsilon_{k})$
$\displaystyle=\pm
g_{s}\int{dk\over(2\pi)^{d}}n(\epsilon_{k-q})n(-\epsilon_{k})$
$\displaystyle=g_{s}\int{dk\over(2\pi)^{d}}n(\epsilon_{k-q})[1\mp
n(\epsilon_{k})]$ $\displaystyle=n\mp
g_{s}\int{dk\over(2\pi)^{d}}n(\epsilon_{k-q})n(\epsilon_{k})$ (64)

and introducing into (62) one gets exactly the Fock energy (61)

$\displaystyle{E_{F}\over N}=\mp{g_{s}\over
2n}\int{dkdq\over(2\pi)^{2d}}V_{q}n(\epsilon_{k-q})n(\epsilon_{k}).$ (65)

Figure 7:  The Fock correlation energy for a width of $b=1a_{B}/\hbar$ versus
polarization

We can give this exchange energy analytically at zero temperature with
$n_{k}=\Theta(k^{2}_{\uparrow\downarrow}-k^{2})$. In the following we scale
the momenta in units of $k_{\uparrow\downarrow}=k_{f}(1\pm p)/2$ with spin-
polarization $p=(n_{\uparrow}-n_{\downarrow})/(n_{\uparrow}+n_{\downarrow})$.
The Fock correlation energy per particle and in units of ${\rm Ryd}$ can be
integrated with the potential (56) by interchanging integration orders

$\displaystyle{E_{F}\over N_{\uparrow}{\rm Ryd}}={-1\over
16r_{s\uparrow}}\\!\\!\int\limits_{-1}^{1}\\!\\!dk\\!\\!\\!\int\limits_{k-1}^{k+1}\\!\\!\\!dqK_{0}(bq)={-1\over
8r_{s\uparrow}}\\!\\!\int\limits_{0}^{2}\\!\\!\\!dq(2\\!-\\!q)K_{0}(bq)$
$\displaystyle=-{1\over 8r_{s\uparrow}}\\!\\!\left[\\!\frac{2b(\pi
b\boldsymbol{L}_{0}(2b)\\!+\\!1)K_{1}(2b)\\!-\\!1}{b^{2}}\\!+\\!2\pi\boldsymbol{L}_{-1}(2b)K_{0}(2b)\\!\right]$
(66)

with ${\rm Ryd}=e^{2}/4\pi\epsilon_{0}a_{B}$ and
$b=\bar{b}k_{\uparrow\downarrow}=\bar{b}k_{F}(1\pm p)/2$ and the StruveL
function $L_{n}(x)$. One sees that in this scaling the meanfield appears in
orders $1/r_{s}$ or $r_{s\uparrow}=r_{s}/(1+p)$ respectively. If one wants to
present the energies in terms of Fermi energy one has the relation

$\displaystyle{\rm Ryd}={e^{2}\over
4\pi\epsilon_{0}a_{B}}=\epsilon_{F}{8\over\pi^{2}}r_{s}^{2}$ (67)

and the mean field would start with $r_{s}$.

In figure 7 the scaled Fock selfenergy per particle is plotted as a function
of polarization. We see that it is increasing with increasing polarization.

Figure 8:  The ferro- and paramagnetic Fock correlation energy for a width of
$b=0.1a_{B}/\hbar$

Now we can investigate whether there is a symmetry-broken ground state by
comparing the para- (p=0) with the ferromagnetic (p=1) ground state as
illustrated in figure 8. We see that for any specific width the ferromagnetic
ground state is higher than the paramagnetic one. This is in agreement with
the Lieb-Schultz-Mattis theorem LSM61  up to spin-up Bruckner parameter of
$r_{s\uparrow}\sim 0.7$ which shows the limit of meanfield approach. If we
scale the $b$ parameter we see that this is true for any width $b$ as
illustrated in figure 9. We conclude that in one-dimensional systems there is
no symmetry-broken Hartree-Fock state as found in 2D and 3D systems. For an
overview see LG16 .

Figure 9:  The difference of ferro- and paramagnetic Fock correlation energy
for any width.

###  III.3 Renormalization of potential and effective mass

The result for the Fock selfenergy is

$\displaystyle{\Sigma_{F}\over n{\rm Ryd}}=$ $\displaystyle-{1\pm p\over
4nr_{s}}\int\limits_{k-1}^{k+1}dqK_{0}(bq)$ $\displaystyle=$
$\displaystyle-\frac{\pi}{4r_{s}g_{s}}\biggl{[}(1-k)\boldsymbol{L}_{-1}(b(1-k))K_{0}(b\left|1-k\right|)$
$\displaystyle+\left|1-k\right|\boldsymbol{L}_{0}(b(1-k))K_{1}(b\left|1-k\right|)$
$\displaystyle+(k+1)\boldsymbol{L}_{-1}(b(k+1))K_{0}(b(k+1))$
$\displaystyle+(k+1)\boldsymbol{L}_{0}(b(k+1))K_{1}(b(k+1))\biggr{]}$ (68)

which is plotted in figure 10.

Figure 10:  The un-renormalized Fock selfenergy (68).

When we are approaching the Coulomb limit for vanishing width $b\to 0$ the
Fock term diverges to $-\infty$. This is cured by a summation of higher-order
diagrams which are the RPA ones to produce an appropriate screened potential.
Here we suggest the following procedure. In principal we can fix the energy
scale as we want. Therefore adding a constant does not alter the physics.
Within the jellium model we simply assume a background bias. Therefore we are
allowed to subtract from any potential a constant, $v(q)\rightarrow
v(q)+v_{0}$. Conveniently we chose $v_{0}=2\ln b$ which renders the Fock
selfenergy finite. We will see in the next chapter that this constant $v_{0}$
drops out exactly in second-order selfenergy. In figure 11 we plot the
renormalized Fock selfenergy which shows how it converges to a finite value
for $b\to 0$, the result being

$\displaystyle{\Sigma_{F}\over n{\rm
Ryd}}=\frac{\pi}{r_{s}g_{s}}\biggl{[}\gamma\\!-\\!\ln{e^{2}\over
2}\\!+\\!{k\\!+\\!1\over 2}\ln(k\\!+\\!1)\\!+\\!{1\\!-\\!k\over
2}\ln|1\\!-\\!k|\biggr{]}.$ (69)

Figure 11:  The renormalized Fock selfenergy (68).

The effective mass

$\displaystyle{k\over m_{\rm eff}}={\partial\varepsilon^{0}\over k}={k\over
m}+{\partial\Sigma_{F}\over\partial k}$ (70)

is independent of such added constant. In figure 12 we see that the Coulomb
limit is reached with a finite value

$\displaystyle{m\over m_{\rm eff}}=1+r_{s}{2(1\pm p)\over
x\pi^{2}}\ln{1+x\over|1-x|}$ (71)

with $x=k/k_{\uparrow\downarrow}$. At the (polarized) Fermi momentum we see
that the effective mass is zero indicating the break down of the Fermi liquid
picture.

Figure 12:  The effective mass for different width in the ferromagnetic case
and $r_{s}=1$.

This Coulomb effective mass we plot in figure 13 for different $r_{s}$
parameter.

Figure 13:  The Coulomb effective mass ($b=0.01$) for different $r_{s}$ in the
ferromagnetic case.

We see that with increasing density the effective mass is more suppressed. The
dip in the effective mass is dependent on the polarization as seen in figure
14. In fact for the paramagnetic case we see that the dip occurs at twice the
Fermi momentum and indicates the onset of Peierls instability KE87 ; KoS96 .

Figure 14:  The Coulomb effective mass for $r_{s}=0.1$ and various
polarizations ranging from paramagnetic (p=0) to the ferromagnetic (p=1) case.

###  III.4 Selfenergy in first Born approximation

Now we calculate the selfenergy in Born approximation (26) which represents
the next order in $r_{S}$ beyond the meanfield. The $\delta$-function we use
to perform the $q$-integration which gives two poles $q=(p-k\pm\eta)/2$ with
$\eta=\sqrt{p^{2}-k^{2}-2kp+2\omega}$ with the residue $1/2\eta$. This
restricts the integration to render the root real. The sum of this two poles
yields finally in dimensionless units

$\displaystyle{\Sigma^{<}\over{\rm
Ryd}}={g_{s}\over\pi}\int{dp\over\eta}\left[v\left({p-k+\eta\over
2}\right)-v\left({p-k-\eta\over 2}\right)\right]^{2}$
$\displaystyle\Theta[4-(k+p+\eta)^{2}]\Theta[4-(k+p-\eta)^{2}]\Theta[p^{2}-1].$
(72)

The expression for $\Sigma^{>}$ is given by inter-changing the sign in the
$\Theta$ functions. It is remarkable that any constant shift of the potential
$v(q)+v_{0}$ drops out. Therefore we can work with the renormalized potential
as introduced in the meanfield section. The last integration can be done
numerically. The integration range for $p$ is in fact quite involved and given
in appendix (A). In MA23  an alternative analytical way is presented to
express the selfenergy in terms of one integral about any used potential.

The first observation is that both selfenergies vanish on-shell
$\Sigma^{\gtrless}(k,\omega=k^{2}/2m)=0$. This is a specific feature of 1D
systems. One can understand this as suppression of any elastic scattering
event by Pauli-blocking allowing only exchange of momenta.

Figure 15:  The spectral function of selfenergy (22) for $b=1$ and various
momenta. The left curves are $\Sigma^{<}$ and the right ones $\Sigma^{>}$.

Figure 16:  The spectral function of selfenergy (22) for $k=0.6$ (above) and
$k=1.6$ (below) and various $b$.

We discuss the spectral function (22) of the selfenergy in figure 15. One sees
that below the Fermi momentum a gap in the dissipation spectrum appears which
is closed when the Fermi momentum is approached. For momenta above the Fermi
momentum, the dispersion splits which can be interpreted as holons and
antiholons EFGKK10 , i.e. the excitation out of Fermi see above $k_{f}$ and
$-k_{f}$ respectively. This results into the two excitations above and below
$\omega=k_{f}^{2}$. The $\Sigma^{<}$ as selfenergy due to hole damping
represents the left curves and develop a sharp peak when approaching the Fermi
momentum. It never overcomes the on-shell value. Exceeding the Fermi momentum
$\Sigma^{<}$ shrinks and form a large background. The opposite behaviour one
sees for the particle contribution $\Sigma^{>}$ which are the curves on the
right side respectively. The sharp peak developed above the Fermi momentum is
moving to higher values and broadens for higher momenta. Please note that at
the on-shell value $\Sigma^{>}$ is also exactly zero. If we approach the
Coulomb limit for $b\to 0$ we see in figure 16 that the selfenergy is
increasing and converging visibly at $b=0.05$.

###  III.5 One-particle spectral function

####  III.5.1 Selfconsistent spectral function

Next we calculate the real part of the selfenergy (21) which allows to discuss
the spectral function of the electrons from the Dyson equation

$\displaystyle a(k,\omega)=-2{\rm Im}\left[\omega-{k^{2}\over
2m}-\Sigma_{F}(k)-\Sigma^{R}(k,\omega)\right]^{-1}.$ (73)

In figure 17 we present this spectral function (73) for various momenta. We
see that the pole increases according to the expected dispersion
$k^{2}/2m+\Sigma_{F}+\Sigma$ until the Fermi energy. Above, the spectral
function shows quite a fragmented behaviour indicating that we have missed the
correct pole.

Figure 17:  The (unrenormalized) non-selfconsistent electron spectral function
(73) for various momenta.

Moreover, there are two sum rules known for the spectral function, for
derivation see M17b . The first one, the norm conservation, is

$\displaystyle\int{d\omega\over 2\pi}a(k,\omega)=1$ (74)

and the second one, the energy-weighted sum rule, reads

$\displaystyle\int{d\omega\over 2\pi}\omega a(k,\omega)={k^{2}\over
2m}+\Sigma_{F}(k).$ (75)

Checking, one finds that below the Fermi momentum both sum rules are completed
only within $5-10\%$ but with higher than Fermi momentum both are badly off.
The reason by deeper inspection is that the energy argument of the selfenergy
is not the energy $\omega$. In principle one has to meet the energy at the
pole of the spectral function there. This creates a selfconsistency loop which
has to be performed by iteration. As consequence this leads away from the
$\omega$ argument of $\Sigma(\omega)$ to a position $\omega+\Delta_{k}$. A
very good short-cut is to consider this shift at the Fermi momentum but
corrected by $\Delta_{k}\approx-\Sigma_{F}(k_{F})-\Sigma(k_{F},\epsilon_{F})$.
In fact this selfconsistent spectral function completes both sum rules better
that $0.01\%$ and are given in figure 18. The difference to figure 17 is
visible.

Figure 18:  The selfconsistent electron spectral function (73) with
$\Sigma(k,\omega+\Delta_{k})$ for various momenta of figure 17.

We see that below the Fermi momentum a sharp side peak develops which is
vanishing at the Fermi momentum. Above this sharp side peak is suppressed
again. Below zero a bound state pole is visible which vanishes for momenta
around $2k_{f}$ indicating that nesting is destroying the appearance of bound
states.

####  III.5.2 Extended quasiparticle spectral function

According to the extended quasiparticle picture for the correlation function
(20) we can also write the spectral function as

$\displaystyle a_{\rm
EQP}(k,\omega)=G^{>}+G^{<}={2\pi\delta(\omega-\epsilon_{k})\over
1-{\partial\Sigma(\omega)\over\partial\omega}}+{\Gamma(k,\omega)\over(\omega-\epsilon_{k})^{2}}$
(76)

with the dispersion
$\epsilon_{k}=k^{2}/2m-\Sigma_{F}(k)-\Sigma(k,\epsilon_{k})$. This spectral
function is the consistent expansion in second order potential according to
(20). The residue renormalizes the weight of the pole and the sum rules (74)
and (75) are completed M17b .

In figure 19 we compare the selfconsistent spectral function with the extended
quasiparticle one. One sees how the selfconsistent one approximates the
correct pole which is indicated by an arrow and approaches the side band for
higher momenta.

Figure 19:  The selfconsistent electron spectral function (73) with the non-
selfconsistent as well as the extended quasiparticle one (76 for momenta
$k=0.7k_{F}$ above and $k=1.6k_{F}$ below. The arrow indicates the pole of the
$\delta$-function.

####  III.5.3 Quasiparticle energy and density of states

The spectral function describes the one-particle excitations of the electrons.
The quasiparticle excitation of the electrons are given by the main pole of
the spectral function and according to the above discussion can be
approximated by

$\displaystyle\epsilon_{k}={k^{2}\over 2m}+\Sigma_{F}(k)+\Sigma(k,{k^{2}\over
2m}+\Delta_{k}).$ (77)

In figure 20 we plot various contributions to the dispersion. If we compare
the case of $b=1$ with the Coulomb limit $b=0.1$ we see that the first-order
selfenergy becomes remarkable and compensates partially the strong meanfield
contribution. Of course this is dependent on $r_{S}$. For illustrative purpose
we plot also the case of $r_{s}=2$ seeing how the influence of the meanfield
is further reduced.

Figure 20:  The quasiparticle energy dispersion (77) together with mean field
and selfenergy contribution for $b=1,r_{s}=1$ (above), $b=0.1,r_{s}=1$
(middle) and $b=1,r_{s}=2$ (below).

Form the spectral function we can also calculate the density of states

$\displaystyle{\rm DOS}(\omega)=\int{dk\over 2\pi\hbar}a(k,\omega)$ (78)

which is plotted in figure 21. One sees how the meanfield density of states is
approached at higher frequencies. At lower frequencies we get a reduction from
the meanfield value showing no divergence. The shift of the bottom is nearly
identical to the mean field value. The dip is the reminiscence of the gap in
the excitation seen in the spectral functions in figures 17 and 18.

Figure 21:  The density of states (78) with the spectral functions (73) of
figures 17 and 18 (thick line) together with the mean field one
$\sqrt{m/2(w-\Sigma_{F}(k_{F}))}$ (thin line) for $b=1,r_{s}=1$. The meanfield
value of frequency is indicated by a vertical grid line.

###  III.6 Structure factor and pair correlation function

Now we are going to evaluate the response function (50) structure function
(42) and the pair correlation (40). Since we consider the first-order high-
density expansion equivalent to the second-order expansion in the potential,
we can expand Eq. (50) as

$\displaystyle\chi(q,\omega)$ $\displaystyle=$
$\displaystyle\Pi_{0}(q,\omega)+\lambda\;V(q)\Pi_{0}^{2}(q,\omega)$ (79)
$\displaystyle+\lambda\;\Pi_{se}(q,\omega)+\lambda\;\Pi_{ex}(q,\omega)$

where we indicate the order of interaction by $\lambda$. The first-order
static structure factor (42) can be written according to (79)

$\displaystyle S_{1}(x)=S_{V\Pi_{0}^{2}}(x)+S_{se}(x)+S_{ex}(x)$ (80)

where we will use $x=q/2k_{F}$ in the following. The analytical evaluation of
$S_{V\Pi_{0}^{2}}$ and $S_{ex}$ for an infinitely-thin cylindrical wire can be
found in VBMP17 . The contribution of the selfenergy to the structure factor
$S_{se}(q,\omega)$ turns out to be zero due to the $\omega$ integration. The
sum of both corrections $S_{V\Pi_{0}^{2}}$ and $S_{ex}$ is given by VBMP19

$\displaystyle S_{1}(x)$
$\displaystyle=\frac{g_{s}^{2}r_{s}}{\pi^{2}x}\left\\{\begin{array}[]{ll}\zeta(x)&,x<1\cr\zeta(x)-2x\ln
x\ln e^{2}x&,x>1\end{array}\right.$ (83)

with

$\displaystyle\zeta(x)$ $\displaystyle=$
$\displaystyle(x+1)\ln(x+1)\ln\left({x^{2}e^{2}\over x+1}\right)$ (84)
$\displaystyle+|x-1|\ln|x-1|\ln\left({x^{2}e^{2}\over|x-1|}\right).$

###  III.7 Correlation energy

Next we discuss the expression for the correlation energy per particle in
second-order perturbation theory LL79 , i.e. second Born approximation with
exchange (31). For contact potentials we have to subtract an infinite value
$\sim n_{p_{1}}n_{p_{2}}$ in order to reach convergence which is a
renormalization of contact potential. For finite-range potentials we have an
intrinsic cut-off due to the range of interaction and such problem does not
occur as we see a posteriori.

We scale all momenta again by the Fermi momentum
$k_{\uparrow\downarrow}=\pi\hbar n/g_{s}$ as $p_{1}=k/k_{\uparrow\downarrow}$,
$p_{2}=p/k_{\uparrow\downarrow}$, and $x=q/2k_{f}$. The occupation factors
restrict the integration range that from $1>p_{1}^{2}$, $1>p_{2}^{2}$,
$(p_{1}+2x)^{2}>1$ and $(p_{2}+2x)^{2}>1$ follows the two cases $0<x<1$ with
$1-2x<p_{1}<1,-1<p_{2}<2x-1$ and $1<x$ with $-1,p_{1},p_{2}<1$. Presenting the
energy in terms of ${\rm Ryd}=e^{2}/4\pi\epsilon_{0}a_{B}$ as
$\epsilon_{c}=E_{c}/n/{\rm Ryd}$ we obtain

$\displaystyle{\epsilon_{c}}=$ $\displaystyle-{1\over
4\pi^{2}}\left[\int\limits_{0}^{1}{dx\over
x}\,\Lambda_{x}^{<}+\int\limits_{1}^{\infty}{dx\over
x}\,\Lambda_{x}^{>}\right]$ (85)

with $b=\bar{b}2k_{f}$. The $p_{1}$ and $p_{2}$ integrations can be carried
out analytically and yield for $x>1$

$\displaystyle\Lambda_{x}^{>}=\\!\int\limits_{-1}^{1}\\!\\!dp_{1}\\!\int\limits_{-1}^{1}\\!\\!dp_{2}{v(2k_{f}x)[v(2k_{f}x)\\!-\\!v(k_{f}p_{1}\\!-\\!k_{f}p_{2}\\!+\\!2k_{f}x)]\over
2x(p_{1}\\!-\\!p_{2}\\!+\\!2x)}$
$\displaystyle=\frac{K_{0}(2bx)}{b}\left\\{G_{2,4}^{3,1}\left({bx}-b,\frac{1}{2}|\begin{array}[]{c}1,\frac{3}{2}\\\
\frac{1}{2},\frac{1}{2},\frac{1}{2},0\\\ \end{array}\right)\right.$ (88)
$\displaystyle\left.-2G_{2,4}^{3,1}\\!\left(\\!{bx},\frac{1}{2}|\begin{array}[]{c}1,\frac{3}{2}\\\
\frac{1}{2},\frac{1}{2},\frac{1}{2},0\\\
\end{array}\\!\\!\right)\\!+\\!G_{2,4}^{3,1}\left({bx}\\!+\\!b,\frac{1}{2}|\begin{array}[]{c}1,\frac{3}{2}\\\
\frac{1}{2},\frac{1}{2},\frac{1}{2},0\\\ \end{array}\\!\\!\right)\right.$ (93)
$\displaystyle+4bK_{0}(2bx)\left[2x\ln\left({x^{2}\\!-\\!1\over
x^{2}}\right)\\!+\\!\ln\left({x\\!+\\!1\over
x\\!-\\!1}\right)^{2}\right]\biggr{\\}}$ (94)

and for $0<x<1$

$\displaystyle\lambda_{x}^{<}=\\!\\!\\!\int\limits_{1\\!-\\!2x}^{1}\\!\\!\\!\\!dp_{1}\\!\\!\\!\int\limits_{-1}^{2x\\!-\\!1}\\!\\!\\!\\!dp_{2}{v(2k_{f}x)[v(2k_{f}x)\\!-\\!v(k_{f}p_{1}\\!-\\!k_{f}p_{2}\\!+\\!2k_{f}x)]\over
2x(p_{1}\\!-\\!p_{2}\\!+\\!2x)}$
$\displaystyle=\frac{K_{0}(2bx)}{b}\left\\{G_{2,4}^{3,1}\left(\\!b\\!+\\!{bx}),\frac{1}{2}|\begin{array}[]{c}1,\frac{3}{2}\\\
\frac{1}{2},\frac{1}{2},\frac{1}{2},0\\\ \end{array}\\!\right)\right.$ (97)
$\displaystyle\left.+G_{2,4}^{3,1}\left(\\!b\\!-\\!{bx}),\frac{1}{2}|\begin{array}[]{c}1,\frac{3}{2}\\\
\frac{1}{2},\frac{1}{2},\frac{1}{2},0\\\
\end{array}\right)-2G_{2,4}^{3,1}\left(2x,\frac{1}{2}|\begin{array}[]{c}1,\frac{3}{2}\\\
\frac{1}{2},\frac{1}{2},\frac{1}{2},0\\\ \end{array}\\!\right)\right.$ (102)
$\displaystyle+4bK_{0}(2bx)\left(2\ln\left(1-x^{2}\right)+2x\ln\left(\frac{x+1}{1-x}\right)\right)\biggr{\\}}$
(103)

with the the Meijer G function. The last x-integral can be done numerically
and the result is seen in figure 22. It shows that the ground state
correlation energy is decreasing continuously with increasing width. This
means that the one-dimensional systems is unstable compared to the two-
dimensional system which is the large-b limit presented as well in figure 22.
We see how the exact expression (85) interpolates between both limits.

Figure 22: The correlation energy per particle (85) together with the first-
order analytical expansion (104) as well as the large-b expansion.

####  III.7.1 Small-b expansion

We have two ways to calculate the correlation energy via the charging formula
(5) and via the selfenergy (19). The Fock or exchange term has been given
already by (66) and was shown to yield equivalent results with the calculation
by the selfenergy (64). Lets check this with the small-b expansion.

First we give the result via the charging formula. The correlation energy per
particle in Eq. (45) in the small-$b$ limit for a cylindrical wire is given by
VBMP19

$\displaystyle\epsilon_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{4r_{s}}\bigg{\\{}\Lambda_{(x<1)}+\Lambda_{(x>1)}\bigg{\\}}$
(104)

where we use the small $b$ expansion of $v(x)$. The result for $x<1$ is

$\displaystyle\Lambda_{(x<1)}=\int^{1}_{0}v(x)[S_{1}(x)]_{x<1}\;dx$
$\displaystyle=\frac{r_{s}g_{s}^{2}}{12\pi^{2}}\bigg{\\{}42\zeta(3)\ln\left(\frac{bk_{F}}{8}\right)+48(\ln(2)-2)\ln(2)\ln(bk_{F})$
$\displaystyle+48\left(-2\text{Li}_{4}\left(\frac{1}{2}\right)+\ln^{2}(2)+\gamma\left(\ln^{2}(2)-\ln(4)\right)+\ln(4)\right)$
$\displaystyle+42(\gamma-1)\zeta(3)+\pi^{4}-4\log^{3}(2)(12+\ln(2))+4\pi^{2}\ln^{2}(2)\bigg{\\}},$
(105)

and for $x>1$ it is

$\displaystyle\Lambda_{(x>1)}=\int^{\infty}_{1}v(x)[S_{1}^{Cy.}(x)]_{x>1}\;dx$
$\displaystyle=-\frac{2r_{s}g_{s}^{2}}{\pi^{2}}\bigg{\\{}\frac{7}{4}\zeta(3)\left(\ln\left(\frac{bk_{F}}{8}\right)+\gamma-1\right)-4\text{Li}_{4}\left(\frac{1}{2}\right)$
$\displaystyle+\frac{17\pi^{4}}{360}+(\ln(2)-2)\ln(4)\ln(bk_{F})-\frac{\ln^{4}(2)}{6}-2\ln^{3}(2)$
$\displaystyle+\frac{1}{6}\pi^{2}\ln^{2}(2)+2\gamma\ln^{2}(2)+2\ln^{2}(2)+\ln(16)-4\gamma\ln(2)\bigg{\\}},$
(106)

where $\zeta(s)$ is the Riemann zeta function and $\text{Li}_{n}(z)$ is the
polylogarithm function a84 . Adding Eq.(III.7.1) and (III.7.1), major
cancellations occur and one obtains the correlation energy as

$\displaystyle\epsilon_{c}(r_{s})=-\frac{\pi^{2}}{360}$ (107)

which is the result of the conventional perturbation theoryLoos13 ; LG16  and
in excellent agreement with variational quantum Monte Carlo simulation
Vinod18c .

As comparison we now calculate the small-b expansion via the second Born
approximation (85). We use first the lowest-order of (56)

$\displaystyle v(2k_{f}x)[v(2k_{f}x)-v(k_{f}p_{1}-k_{f}p_{2}+2k_{f}x)]$
$\displaystyle=4(\gamma+\ln{b/2}+\ln{2x})[\ln
2x-\ln(p_{1}-p_{2}+2x)]+o(b^{2}).$ (108)

The $p_{1}$ and $p_{2}$ integrals read

$\displaystyle\Lambda_{x}^{>}=4(\gamma+\ln{b/2}+\ln 2x)\left[\xi(x)-2x\ln x\ln
e^{2}x\right].$ (109)

For $0<x<1$ one gets

$\displaystyle\Lambda_{x}^{<}=4(\gamma+\ln{b/2}+\ln(2x))\xi(x).$ (110)

Comparing with (III.6) we see exactly the same expressions $\xi(x)$. This
means that in the static perturbation theory the structure factor is silently
contained but not possible to identify directly here.

Integrating further we obtain

$\displaystyle\int\limits_{1}^{\infty}{dx\over
x}\Lambda_{x}^{>}=(\gamma+\ln{b/2})[8\ln^{2}(2)-16\ln 2+7\zeta(3)]$
$\displaystyle-16\text{Li}_{4}\left(\frac{1}{2}\right)-7\zeta(3)[1+\ln
2]+\frac{17\pi^{4}}{90}+\frac{2}{3}\pi^{2}\ln^{2}2$ $\displaystyle+16\ln
2-\frac{2}{3}(\ln 2-6)^{2}\ln^{2}2$ (111)

and

$\displaystyle\int\limits_{0}^{1}{dx\over
x}\Lambda_{x}^{<}=-(\gamma+\ln{b/2})[8\ln^{2}(2)+16\ln 2+7\zeta(3)]$
$\displaystyle 16\text{Li}_{4}\left(\frac{1}{2}\right)+7\zeta(3)(1+\ln
2)-{\pi^{4}\over 6}-{2\pi^{2}\over 3}\ln^{2}2$ $\displaystyle\\!-\\!16\ln
2\\!+\\!{2\over 3}(\ln 2\\!-\\!6)^{2}\ln^{2}2\biggr{]}.$ (112)

Adding (111) and (112) we get

$\displaystyle\epsilon_{c}=-{\pi^{2}\over 360}$ (113)

which is exactly (107). So both ways, the charging energy formula and the
selfenergy gives the same results.

With the same means we can calculate the next term of (56). We obtain the next
order in $b$

$\displaystyle\int\limits_{1}^{\infty}{dx\over
x}\Lambda_{x}^{>}=(\gamma+\ln{b/2})^{2}{16\over 3}(1-\ln 2)$
$\displaystyle+(\gamma+\ln{b/2}){2\over 9}[3\pi^{2}-68-4\ln 2(9\ln 2-29)]$
$\displaystyle+{1\over 54}[452+(36\pi^{2}-1448)\ln 2-48\ln^{2}2(3\ln 2-19)$
$\displaystyle-30\pi^{2}+315\zeta(3)]$ (114)

and

$\displaystyle\int\limits_{0}^{1}{dx\over
x}\Lambda_{x}^{<}=-(\gamma+\ln{b/2})^{2}{16\over 3}(1-\ln 2)$
$\displaystyle+(\gamma+\ln{b/2}){2\over 9}[-\pi^{2}+74+4\ln 2(9\ln 2-29)]$
$\displaystyle-{1\over 54}[656+(12\pi^{2}-1510)\ln 2-48\ln^{2}2(3\ln 2-19)$
$\displaystyle+10\pi^{2}+27\zeta(3)]$ (115)

which results into

$\displaystyle\epsilon_{c}=-\frac{\pi^{2}}{360}$
$\displaystyle\\!-\frac{b^{2}\left(6\left(3\\!+\\!\pi^{2}\right)\ln(b)\\!+\\!72\zeta(3)\\!+\\!6\gamma\left(3\\!+\\!\pi^{2}\right)\\!-\\!5\pi^{2}\\!-\\!51\right)}{108\pi^{2}}.$

###  III.8 Reduced density matrix

Finally we calculate the reduced density matrix (25) or explicitly (33). The
integration ranges due to the occupation factors are worked out in appendix B.
For small $b$ parameter this can be evaluated analytically. One finds that at
$k=1k_{F}$ the reduced density matrix has a singularity in that it diverges of
both sides with opposite sign. The result for the numerical integration over
the selfenergies (25) are given in figure 23 for various $b$ parameter. One
sees that in the Coulomb limit the divergence is seen as a small wiggle around
$0.5$.

Figure 23:  The reduced density matrix (25) for various width parameter $b$
together with the Fermi function and artificially $4r_{s}^{2}/\pi^{4}=1$.

This divergence has been discussed in M23  and a Padé regularization was
suggested. It consists of the extended quasiparticle approximation used so far
and an additional expansion around the Fermi energy both interpolated by a
function rapidly vanishing outside the Fermi energy. We can therefore assume
such regularization and it would subtract the divergence on both sides which
has here the form $c_{1}\ln(k-1)+c_{2}\ln(k-1)^{2}+c_{3}\ln(1-k)^{3}$. One
obtains the interesting limiting law for the harmonic potential (56)

$\displaystyle\lim\limits_{k\to 1\pm
0}\rho_{k}=\left\\{\begin{array}[]{l}{1\over 12}[3+\ln(2)^{3}]\approx
0.277752\cr 1+{1\over 12}[-3+\ln(2)^{3}]\approx 1-0.222248\end{array}\right.$
(119)

which exactly approaches the jump

$\displaystyle\Delta\rho=\frac{1}{2}$ (120)

at the Fermi momentum. Since we know that the momentum distribution is finite
at the Fermi momentum, the interpolation between expansion at the Fermi energy
and the extended quasiparticle approximation easily accounts for this finite
jump subtracting not only the divergent terms on both sides but also the jump
M23 . The corresponding analytical expression for the momentum distribution is
somewhat lengthy but trivially obtained by the formulas in appendix B.

It is instructive to consider the limit of contact potentials $v(q)=1$. Then
one obtains analytically

$\displaystyle\rho_{k}=1+{r_{s}^{2}(g_{s}-1)\over\pi^{4}}$
$\displaystyle\times\\!\left\\{\\!\begin{array}[]{cc}{\ln(1\\!+\\!k)\over(1\\!-\\!k)^{2}}\\!+\\!{\ln(1\\!-\\!k)\over(1\\!+\\!k)^{2}}\\!-\\!2(1\\!+\\!2\ln
2){1\\!+\\!k^{2}\over(k^{2}\\!-\\!1)^{2}}&0<k<1\cr&\cr{1+2\ln
2-2\ln(k-1)\over(1+k)^{2}}&1<k<3\cr&\cr{4\over
k^{2}-1)^{2}}&k>3\end{array}\right.$ (126)

and one sees that near the Fermi momentum $k=1\pm\eta$ we have from both sides

$\displaystyle\rho_{k}\approx\left\\{\begin{array}[]{cc}1+{1-2\ln 2\over
4}+\frac{1}{2}\ln\eta&k=1-\eta\cr{1+2\ln 2\over
4}-\frac{1}{2}\ln\eta&k=1+\eta\end{array}\right.+o(\eta).$ (129)

Again due to Padé expansion we subtract a regularizing term
$\rho_{k}=1+(\rho^{>}_{k}-\rho^{r}_{k})-(\rho^{<}_{k}-\rho^{r}_{k})$ to get
rid of divergences and see that the jump at the Fermi momentum approaches a
smaller value than (120) of

$\displaystyle\Delta\rho=1-\ln 2\approx 0.3068.$ (130)

Again this can be included additionally into the Padé term rendering the value
of the momentum distribution unique at the Fermi energy.

##  IV Summary

We have presented two approaches to the correlation energy, one by the
structure factor with the pair correlation function and one by the Dyson
equation with the selfenergy. Both are rooted in approximating the two-
particle Green function appropriately. Different resulting forms are compared
and it is shown how they coincide if the same level of approximations are
used. The equivalence is obtained within the extended quasiparticle picture
for the single-particle propagators and self energies.

For a one-dimensional quantum wire of Fermions the approximations are
illustrated and the self energies are explicitly discussed. A gap appears
which results into a splitting of excitation lines in the spectral function of
holons and antiholons. Also bound states are visible destroyed by higher
momentum around nesting. The meanfield leads to an effective mass which shows
the onset of Peierls transition at twice the Fermi energy. The density of
states in Born approximation and meanfield are compared and the correlation
effects are identified. The width dependence of the correlation energy is
calculated and compared with the analytical results of small and large width
expansions. The momentum distribution shows a divergence in approaching the
left and right side of the Fermi energy. The occurring divergences and jumps
at the Fermi energy are subtracted due to a regularization scheme of Padé
which interpolates between the extended quasiparticle approximation and an
expansion at the Fermi energy.

###### Acknowledgements.

This paper is to the memory of Paul Ziesche (1933-2022) with whom the authors
had many enlightening discussions. K.M. acknowledges support from DFG-project
MO 621/28-1. K.N.P. acknowledges the grant of honorary senior scientist
position by National Academy of Sciences of India (NASI) Prayagraj.

##  Appendix A Integration range for the selfenergy (72)

Performing the restrictions of the $\Theta$ functions the following
integration range for $\Sigma^{<}$ appears. It is only non-zero for
$W=k^{2}-\omega>0$ and for $0\leq k<1$

$\displaystyle p\\!>\\!1,W\\!<\\!2(k\\!-\\!1)^{2}:{\rm
Max}(1,k\\!+\\!\sqrt{2W})\\!<\\!p\\!<\\!1\\!+\\!{W\over 2(1\\!-\\!k)}$
$\displaystyle
p\\!<\\!\\!-\\!1,W\\!<\\!2(1\\!+\\!k)^{2}:\\!-\\!1\\!-\\!{W\over
2(1\\!+\\!k)}\\!<\\!p\\!<\\!{\rm Min}(\\!-\\!1,k\\!-\\!\sqrt{2W})$ (131)

and for $k\\!>\\!1$

$\displaystyle p\\!<\\!-\\!1,2k^{2}\\!<\\!W\\!<\\!2(1\\!+\\!k)^{2}:$
$\displaystyle\qquad\\!-\\!1\\!-\\!{W\over 2(1\\!+\\!k)}\\!<\\!p\\!<\\!{\rm
Min}(\\!-\\!1,k\\!-\\!\sqrt{2W})$ $\displaystyle
p\\!<\\!-\\!k\\!<\\!-\\!1:\\!-\\!1\\!-\\!{W\over
2(1\\!+\\!k)}\\!<\\!p\\!<\\!-\\!1$
$\displaystyle\\!-\\!k\\!<\\!p\\!<\\!-\\!1,2(k\\!-\\!1)^{2}\\!<\\!W\\!<\\!2(k^{2}\\!-\\!1),k\\!>\\!\frac{5}{3}:$
$\displaystyle\qquad{\rm Max}(\\!-\\!k,1\\!-\\!{W\over
2(k\\!-\\!1)}\\!<\\!p\\!<\\!{\rm Min}(\\!-\\!1,k\\!-\\!\sqrt{2W})$
$\displaystyle\qquad\qquad k\leq\frac{5}{3}:{\rm Max}(\\!-\\!k,1\\!-\\!{W\over
2(k\\!-\\!1)})\\!<\\!p\\!<\\!-\\!1$ $\displaystyle
2(k^{2}\\!-\\!1)\\!<\\!W\\!<\\!2k^{2}:\\!-\\!k\\!<\\!p\\!<\\!{\rm
Min}(\\!-\\!1,k\\!-\\!\sqrt{2W}).$ (132)

The integration range for $\Sigma^{>}$ is somewhat simpler. for $0\leq k<1$

$\displaystyle{\rm Max}(-1,1\\!+\\!{W\over 2(1\\!-\\!k)})\\!<\\!p\\!<\\!{\rm
Min}(1,\\!-\\!1\\!-\\!{W\over 2(1\\!+\\!k)})$ (133)

and for $k\\!>\\!1$ and $W>0$

$\displaystyle k\geq 3\,{\rm or}\,(k<3\,{\rm and}\,2(k-1)^{2}>W):$
$\displaystyle\qquad{\rm Max}(-1,1\\!+\\!{W\over
2(1\\!-\\!k)})\\!<\\!p\\!<\\!{\rm Min}(1,k\\!-\\!\sqrt{2W})$ $\displaystyle
1<k<3:-1<\\!p\\!<\\!{\rm Min}(1,\\!-\\!1\\!-\\!{W\over 2(1\\!-\\!k)}).$ (134)

##  Appendix B Integration range for the reduced density matrix

We scale all momenta by the Fermi momentum to obtain

$\displaystyle\rho_{k}$ $\displaystyle=n_{k}+{4r_{s}^{2}\over\pi^{4}}\iint
dpdq{V_{q}(g_{s}V_{q}-V_{p-k-q})\over[2q(k-p+q)]^{2}}$ (135)
$\displaystyle\times\left\\{[k>1][p^{2}>1][1>(k+q)^{2}][1>(p-q)^{2}]\right.$
$\displaystyle\left.-[1>k][1>p^{2}][(k+q)^{2}>1][(p-q)^{2}>1]\right\\}$

where we can restrict to positive $k$ since $\rho_{-k}=\rho_{k}$ which one
sees by interchanging sign of $p,q$. The first part appears for momenta larger
than Fermi momentum, $k>1$ and the second part for $0<k<1$. Discussing the
integration range for $0<k<1$ we have $-1<p<1$ and $q<p-1$ or $q>p+1$ as well
as $q<-1-k$ or $q>1-k$. This provides two cases

$\displaystyle(a):\,-1\\!<\\!-k\\!<\\!p\\!<\\!1:\,-\infty\\!<\\!q<-1\\!-\\!k\,{\rm
or}\,p\\!+\\!1\\!<\\!q\\!<\\!\infty$
$\displaystyle(b):\,-k>p>-1:\,-\infty<q<p-1\,{\rm or}\,1-k<q<\infty.$ (136)

Together this provides the integration range

$\displaystyle\int\limits_{-1}^{-k}\\!\\!dp\left[\int\limits_{-\infty}^{p-1}\\!\\!dq+\int\limits^{\infty}_{1-k}\\!\\!dq\right]+\int\limits^{1}_{-k}\\!\\!dp\left[\int\limits_{-\infty}^{-k-1}\\!\\!dq+\int\limits^{\infty}_{p+1}\\!\\!dq\right]$
$\displaystyle=\int\limits_{2}^{\infty}dq\int\limits_{-1}^{1}\\!\\!dp+\int\limits_{1-k}^{2}dq\int\limits_{-1}^{q-1}\\!\\!dp+(k\leftrightarrow-k).$
(137)

For $k>1$ we have $p>1$ or $p<-1$ and two conditions for $q$

$\displaystyle p-1<q<p+1,\qquad-1-k<q<1-k.$ (138)

For $p>1$ we have $1-k<0<p-1$ and there is no common overlap for the range of
$q$. Since both ranges (138) have the length of $2$ we have to cases of finite
overlap

$\displaystyle(1):$ $\displaystyle p-1<q<1-k$ $\displaystyle{\rm
if}\,-1-k<p-1<1-k<p+1<0$ $\displaystyle(2):$ $\displaystyle-k-1<q<p+1$ (139)
$\displaystyle{\rm if}\,p-1<-k-1<p+1<1-k<0.$

Case (1) translates into $-k<p<Min(-1,2-k)$ which divides into two cases

$\displaystyle(a):\,1\\!<\\!k\\!<\\!3,\,-k\\!<\\!p\\!<\\!-1;\,(b):\,k\\!>\\!3,\,-k\\!<\\!p\\!<\\!2\\!-\\!k.$
(140)

Case (2) yields $-2-k<p<-k$. Combining case (1) and (2) we find for $1\leq
k<3$

$\displaystyle\int\limits_{-2-k}^{-k}\\!\\!\\!\\!dp\int\limits_{-1-k}^{p+1}\\!\\!\\!\\!dq+\int\limits_{-k}^{-1}\\!\\!dp\int\limits_{p-1}^{1-k}\\!\\!dq=\int\limits^{-2}_{k-1}\\!\\!dq\int\limits_{q-1}^{q+1}\\!\\!dp+\int\limits_{-1-k}^{1-k}\\!\\!\\!\\!dq\int\limits_{q-1}^{-1}\\!\\!\\!\\!dp$
(141)

and for $k>3$

$\displaystyle\int\limits_{-2-k}^{-k}\\!\\!\\!dp\int\limits_{-1-k}^{p+1}\\!\\!\\!\\!dq+\int\limits_{-k}^{2-k}\\!\\!\\!dp\int\limits_{p-1}^{1-k}\\!\\!dq=\int\limits_{-1-k}^{1-k}\\!\\!\\!dq\int\limits_{q-1}^{q+1}\\!\\!dp.$
(142)

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