# Know the enemy: 2D Fermi liquids

Sankar Das Sarma  Yunxiang Liao  Condensed Matter Theory Center and Joint
Quantum Institute,
Department of Physics, University of Maryland, College Park, MD 20742, USA

###### Abstract

We describe an analytical theory investigating the regime of validity of the
Fermi liquid theory in interacting, via the long-range Coulomb coupling, two-
dimensional Fermi systems comparing it with the corresponding 3D systems. We
find that the 2D Fermi liquid theory and 2D quasiparticles are robust up to
high energies and temperatures of the order of Fermi energy above the Fermi
surface, very similar to the corresponding three-dimensional situation. We
calculate the phase diagram in the frequency-temperature space separating the
collisionless ballistic regime and the collision-dominated hydrodynamic regime
for 2D and 3D interacting electron systems. We also provide the temperature
corrections up to third order for the renormalized effective mass, and comment
on the validity of 2D Wiedemann-Franz law and 2D Kadawoki-Woods relation.

###### keywords:

Fermi liquid , electron self-energy , random phase approximation

††journal: Annals of Physics

###### Contents

  1. 1 Introduction
  2. 2 Theory
    1. 2.1 General formulas for electron self-energy
      1. 2.1.1 Keldysh approach to interacting electrons
      2. 2.1.2 RPA dynamically screened interaction
      3. 2.1.3 RPA self-energy
    2. 2.2 2D electron self-energy
      1. 2.2.1 Momentum integration
      2. 2.2.2 Frequency integration
    3. 2.3 3D electron self-energy
    4. 2.4 Discussion of the analytical results
  3. 3 Results
    1. 3.1 Applicability of the FL theory
    2. 3.2 Effective mass
    3. 3.3 Hydrodynamic and ballistic regimes
  4. 4 Wiedemann-Franz (WF) and Kadowaki-Woods (KW) relations for 2D interacting systems
    1. 4.1 WF Law
    2. 4.2 KW law
  5. 5 Conclusion
  6. A Derivation of self-energy formulas using Matsubara technique
  7. B Integrals involving hyperbolic functions

##  1 Introduction

In a famous talk at the 1989 Kathmandu Summer School, Phil Anderson questioned
the validity of the Fermi liquid theory and the applicability of the
quasiparticle concept to high-temperature cuprate superconductors specifically
and to 2D interacting electron systems generally [1]. One of his lectures,
which was also quoted in his published lecture note, has the memorable slogan
“Know the enemy”, alluding specifically to the Fermi liquid theory as the
enemy. Anderson was the first person vehemently and tirelessly pushing the
idea that the 2D physics of cuprate superconductors is beyond the Fermi
liquid-BCS paradigm, and represents new physics, which is now commonly
referred to as non-Fermi liquids, a terminology virtually unknown in 1989.

The basic challenges Anderson posed were simple: (1) Is it possible that
interactions destroy the 2D Fermi surface just as they do in one dimension
leading to a Luttinger liquid? (2) Is it possible that cuprates represent a
new emergent form of superconductivity which simply cannot be explained and
understood using the highly successful Fermi liquid-BCS formalism of electron
pairing around the Fermi surface leading to a superconducting instability of
Copper pairs condensing into a BCS ground state?

Amazingly, there is still no answer to the second question even after 30 years
and many thousands of theoretical papers as there is no consensus on the
accepted theory of cuprate superconductivity. In fact, even the precise
mechanism for cuprate superconductivity is still actively debated in the
theoretical community, and Anderson himself worked on developing an
appropriate theory for the cuprates for the rest of his life. It is, however,
a great testimonial to Anderson’s early insight that most theorists working on
cuprate superconductivity accept that its explanation most likely lies outside
the standard Fermi liquid-BCS paradigm. Our current work is on the continuum
2D interacting Coulombic electron system whereas most work on cuprates uses
lattice models with short-range interactions.

But the answer to Anderson’s first question is definitively known. Two
dimensional interacting Fermi systems are Fermi liquids similar to 3D
interacting Fermi systems, and not non-Fermi liquids like interacting one
dimensional Luttinger liquids. In fact, Anderson’s trenchant questioning of
the nature of interacting 2D electron systems (“Know the enemy”) led to many
theoretical developments establishing conclusively, and in fact, even
rigorously, that interacting 2D electron systems are indeed normal Fermi
liquids with well-defined renormalized Fermi surfaces very much like 3D normal
metals and normal He-3 [2, 3, 4]. The basic idea, which is now established
with reasonable mathematical rigor, is rather simple and essentially the same
as it is for 3D interacting Fermi systems. The imaginary part of the
interacting 2D self-energy at energy $E$ goes as $(E-E_{\mathsf{F}})^{2}$ up
to some logarithmic corrections, with $E_{\mathsf{F}}$ being the Fermi energy,
and this holds to all orders in perturbation theory. Thus, the single-particle
spectral function is a delta function at the Fermi surface, guaranteeing the
existence of an interacting Fermi surface and low-energy quasiparticles with
one to one correspondence to the noninteracting Fermi gas. This behavior is
similar to 3D systems within logarithmic accuracy, and thus 2D systems are
similar to 3D systems. This is very different from interacting 1D systems,
which within the same perturbation theory gives an imaginary self-energy going
as $(E-E_{F})(\ln(E-E_{F}))^{1/2}$ [5]. Although a perturbation theory is
neither meaningful nor valid for a 1D system, it is clear that the Fermi
surface cannot exist in 1D already based on this simple perturbative argument.
On the other hand, the perturbative analysis applies in 2D and shows that
Fermi surface exists in the 2D interacting systems just as in 3D.

In the current work, we extend Anderson’s first question, trying to understand
‘the enemy’ better. The question we ask is the extent to which the Fermi
liquid theory applies in 2D electron liquids in the presence of long-range
Coulomb interactions. In particular, how far in energy from the Fermi surface
and how high in temperature can we go and still find well-defined
quasiparticles in 2D interacting systems? What is the regime of applicability
of the concept of 2D Fermi liquids? We answer these questions analytically,
both in 2D and 3D comparing the two situations, using a many-body perturbation
theory which is exact for the Coulomb-interacting system in the high-density
limit.

We also calculate the temperature dependent effective mass renormalization,
comment on the 2D Wiedemann-Franz law and Kadowaki-Woods relation, and
estimate the hydrodynamic regime in Fermi systems interacting through long-
range Coulomb interactions.

Our work is completely analytical involving expansions in inverse density,
energy, and temperature on an equal footing. The theory itself uses the well-
established leading-order dynamical screening or random phase approximation
(RPA) for the self-energy, which is exact in the high-density limit.

##  2 Theory

In this section, we derive the self-energy for an electron system with long-
range Coulomb interactions in both 2D and 3D [6, 7, 8, 9]. In particular, we
provide the analytical expressions for both the real and imaginary parts of
the on-shell self-energy, for arbitrary energy-to-temperature ratio
$\varepsilon/T$. Here, $\varepsilon(=E-E_{\mathsf{F}})$ is the quasiparticle
energy measured from the Fermi surface and $T$ is the temperature. The result
is valid to the leading order in $r_{s}$ and to several orders in
$\varepsilon/E_{\mathsf{F}}$ and $T/T_{\mathsf{F}}$. Here, $r_{s}$ is the
standard dimensionless Coulomb coupling parameter (the ratio of the
interparticle separation to the effective Bohr radius), and $T_{\mathsf{F}}$
is the noninteracting Fermi temperature. As usual, $r_{s}$ is the many body
perturbation parameter (or the density-dependent effective fine structure
constant) for the theory which is strictly valid only for $r_{s}\ll 1$, but in
practice works well empirically for metallic electron densities $(r_{s}\sim
3-6)$ [10]. In Sec. 2.1, we start with reviewing the derivation of general
self-energy formulas which are expressed as integrals involving RPA
dynamically screened interaction and are used to extract the explicit self-
energy expressions. Sec. 2.2 is devoted to the detailed calculation of the 2D
electron self-energy, which is presented before in Ref. [9] and reviewed here
for completeness. In Sec. 2.3, we provide the analytical expressions for the
3D electron self-energy when energy $\varepsilon$ and temperature $T$ are
arbitrary with respect to each other, which, to the best of our knowledge, are
new for long-range Coulomb interactions. See Refs. [11, 12, 13] for the
results of self-energy for the case of short-range interactions in both 2D and
3D. We also present the subleading terms in $\varepsilon/E_{\mathsf{F}}$
($T/T_{\mathsf{F}}$) for the 3D self-energy in the $\varepsilon\gg T$
($T\gg\varepsilon$) limit.

###  2.1 General formulas for electron self-energy

####  2.1.1 Keldysh approach to interacting electrons

In the framework of Keldysh formalism (see Ref. [14] for a review), we now
rederive the general formulas for self-energy which are then used to obtain
the analytic expressions presented in Secs. 2.2 and 2.3. An alternative
derivation employing Matsubara technique (see for example Refs. [6, 7]) will
be reviewed in Appendix A.

We start with the partition function $Z$ of a $d-$dimensional electron system
with Coulomb interactions on a Keldysh contour which runs from $t=-\infty$ to
$t=\infty$ and back to $t=-\infty$. The system is assumed to be in the thermal
equilibrium and noninteracting at the distant past $t=-\infty$, after which
the interactions are then adiabatically switched on. In terms of coherent
state functional integral, the partition function can be expressed as

$\displaystyle\begin{aligned}
Z=&\int\mathcal{D}\left(\bar{\psi},\psi\right)\,\exp\left(iS_{0}+iS_{\mathsf{int}}\right),\end{aligned}$
(2.1)

where $S_{0}$ and $S_{\mathsf{int}}$, which stand for the free and interacting
parts of the action, respectively, are given by

$\displaystyle\begin{aligned}
S_{0}=\,&\int_{-\infty}^{\infty}dt\int_{-\infty}^{\infty}dt^{\prime}\int
d^{d}\bm{\mathrm{r}}\int
d^{d}\bm{\mathrm{r}}^{\prime}\bar{\psi}(\bm{\mathrm{r}},t)\;\hat{G}_{0}^{-1}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})\;\psi(\bm{\mathrm{r^{\prime}}},t^{\prime}),\\\
S_{\mathsf{int}}=\,&-{\frac{1}{2}}\,\sum_{a=\pm}\zeta_{a}\int_{-\infty}^{\infty}dt\int
d^{d}\bm{\mathrm{r}}\int
d^{d}\bm{\mathrm{r}}^{\prime}\,\bar{\psi}_{\sigma}^{a}(\bm{\mathrm{r}},t)\bar{\psi}_{\sigma^{\prime}}^{a}(\bm{\mathrm{r}}^{\prime},t)V(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime})\psi_{\sigma^{\prime}}^{a}(\bm{\mathrm{r}}^{\prime},t)\psi_{\sigma}^{a}(\bm{\mathrm{r}},t).\end{aligned}$
(2.2)

Here $\psi^{+}$($\bar{\psi}^{+}$) and $\psi^{-}$($\bar{\psi}^{-}$) represent
Grassmann fields residue on the forward and backward paths of the Keldysh
contour, respectively. They are also labeled by a spin index
$\sigma=\uparrow,\downarrow$. The overall sign factor $\zeta_{a}$ takes the
value of $1$ ($-1$) for $a=+$ ($-$). $V(\bm{\mathrm{r}})=e^{2}/r$ is the bare
Coulomb interaction potential and its Fourier transform is
$V(\bm{\mathrm{q}})=2\pi e^{2}/q$ ($V(\bm{\mathrm{q}})=4\pi e^{2}/q^{2}$) in
2D (3D). In the present paper, we work in the units of
$\hbar=k_{\mathsf{B}}=1$, and use $E_{\mathsf{F}}$ and $T_{F}$
interchangeably. $G_{0}$ is the noninteracting contour ordered electron
Green’s function, and has the following structure in Keldysh space:

$\displaystyle\begin{aligned}
\hat{G}_{0}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})=-i\braket{\psi(\bm{\mathrm{r}},t)\bar{\psi}(\bm{\mathrm{r}}^{\prime},t^{\prime})}_{0}=\begin{bmatrix}G_{0}^{(\mathrm{T})}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})&G_{0}^{(<)}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
G_{0}^{(>)}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})&G_{0}^{(\bar{\mathrm{T}})}(\bm{\mathrm{r}}-\bm{\mathrm{r}}^{\prime},t-t^{\prime})\end{bmatrix},\end{aligned}$
(2.3)

with $G_{0}^{(\mathrm{T})}$, $G_{0}^{(\bar{\mathrm{T}})}$, $G_{0}^{(<)}$ and
$G_{0}^{(>)}$ being the time-ordered, anti-time-ordered, lesser, and greater
noninteracting electron Green’s function, respectively. The angular bracket
with subscript $0$ indicates functional integration over Grassmann fields
$\psi$ and $\bar{\psi}$ with weight $e^{iS_{0}}$ (Eq. 2.2).

With the help of a real bosonic field $\phi$, we can perform a Hubbard-
Stratonovich transformation to decouple the interactions

$\displaystyle\begin{aligned}
&e^{iS_{\mathsf{int}}}=\int\mathcal{D}\phi\exp\left[\frac{i}{2}\sum_{a=\pm}\zeta_{a}\int\limits_{\bm{\mathrm{q}},\omega}\phi_{a}(\bm{\mathrm{q}},\omega)V^{-1}(\bm{\mathrm{q}})\phi_{a}(-\bm{\mathrm{q}},-\omega)-i\sum_{a=\pm}\zeta_{a}\int\limits_{\varepsilon,\bm{\mathrm{k}}}\int\limits_{\omega,\bm{\mathrm{q}}}\phi^{a}(\bm{\mathrm{q}},\omega)\bar{\psi}^{a}_{\sigma}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\psi^{a}_{\sigma}(\bm{\mathrm{k}},\varepsilon)\right].\end{aligned}$
(2.4)

Here we have used the short-hand notations $\int_{\bm{\mathrm{k}}}\equiv\int
d^{d}\bm{\mathrm{k}}/(2\pi)^{d}$ and
$\int_{\omega}\equiv\int_{-\infty}^{\infty}d\omega/(2\pi)$. Introducing the
classical and quantum components of the bosonic field $\phi$ :

$\displaystyle\phi_{\mathsf{cl}}=\left(\phi_{+}+\phi_{-}\right)/\sqrt{2},\qquad\phi_{\mathsf{q}}=\left(\phi_{+}-\phi_{-}\right)/\sqrt{2},$
(2.5)

we rewrite Eq. 2.4 as

$\displaystyle\begin{aligned}
e^{iS_{\mathsf{int}}}=\,\int\mathcal{D}\phi\,\exp&\left[i\int\limits_{\bm{\mathrm{q}},\omega}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)V^{-1}(\bm{\mathrm{q}})\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega)-\frac{i}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\varepsilon,\bm{\mathrm{q}},\omega}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)\,\bar{\psi}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{\tau}^{3}\psi(\bm{\mathrm{k}},\varepsilon)\right.\\\
&\left.-\frac{i}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\varepsilon,\bm{\mathrm{q}},\omega}\phi_{\mathsf{q}}(\bm{\mathrm{q}},\omega)\,\bar{\psi}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\,\psi(\bm{\mathrm{k}},\varepsilon)\right],\end{aligned}$
(2.6)

where $\hat{\tau}^{i}$ denotes the Pauli matrix in the Keldysh space.

It is usually convenient to apply the Keldysh rotation to the fermionic fields
$\psi$ and $\bar{\psi}$:

$\displaystyle\begin{aligned}
\psi(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\frac{\hat{\tau}^{3}+\hat{\tau}^{1}}{\sqrt{2}}\psi(\bm{\mathrm{k}},\varepsilon),\qquad\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\,\frac{\hat{1}-i\hat{\tau}^{2}}{\sqrt{2}},\end{aligned}$
(2.7)

after which the bare electron Green’s function $\hat{G}_{0}$ assumes the
following form in the Keldysh space

$\displaystyle\begin{aligned}
\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)=\,\begin{bmatrix}G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)&G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)\\\
0&G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\end{bmatrix}.\end{aligned}$ (2.8)

In the present paper, we use superscripts $``R"$, $``A"$ and $``K"$ to denote
the retarded, advanced and Keldysh components respectively. The three
components of the bare electron Green’s function
$\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)$ are connected by the fluctuation-
dissipation theorem (FDT):

$\displaystyle
G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)=\left[G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)-G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\right]\tanh\left(\frac{\varepsilon}{2T}\right),$
(2.9)

and are given by, respectively,

$\displaystyle\begin{aligned}
&G_{0}^{(R/A)}(\bm{\mathrm{k}},\varepsilon)=\left(\varepsilon-\xi_{\bm{\mathrm{k}}}\pm
i\eta\right)^{-1}\\!\\!\\!\\!\\!\\!\\!,\\\
&G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)=-2\pi
i\delta\left(\varepsilon-\xi_{\bm{\mathrm{k}}}\right)\tanh\left(\frac{\varepsilon}{2T}\right).\end{aligned}$
(2.10)

Here $\eta$ is a positive infinitesimal, and $\xi_{\bm{\mathrm{k}}}\equiv
k^{2}/2m-\mu$, with $\mu$ being the chemical potential and $m$ the bare mass.
We note that $G_{0}^{(R/A)}(\bm{\mathrm{k}},\varepsilon)$ contains information
about the spectrum only, while $G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)$
knows also about the distribution function.

We then apply another transformation to the fermionic fields $\psi$ and
$\bar{\psi}$

$\displaystyle\begin{aligned}
\psi(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\hat{M}_{F}(\varepsilon)\,\psi(\bm{\mathrm{k}},\varepsilon),\qquad\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\rightarrow\,\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\,\hat{M}_{F}(\varepsilon),\end{aligned}$
(2.11)

where $\hat{M}_{F}(\varepsilon)$ is distribution function dependent and is
defined as, in Keldysh space,

$\displaystyle\begin{aligned}
\hat{M}_{F}(\varepsilon)=\begin{bmatrix}1&\tanh\left(\varepsilon/2T\right)\\\
0&-1\end{bmatrix}.\end{aligned}$ (2.12)

The advantage of this transformation is that the noninteracting electron
Green’s function becomes diagonal and acquires the form

$\displaystyle\begin{aligned}
\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)=\,\begin{bmatrix}G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)&0\\\
0&G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\end{bmatrix}.\end{aligned}$ (2.13)

After this transformation, the partition function can now be expressed as

$\displaystyle\begin{aligned}
&Z=\,\int\mathcal{D}\left(\bar{\psi},\psi\right)\mathcal{D}\phi\,\exp\left(iS_{\psi}+iS_{\phi}+iS_{c}\right),\\\
&S_{\phi}=\,\int\limits_{\bm{\mathrm{q}},\omega}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)V^{-1}(q)\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega),\\\
&S_{\psi}=\,\int\limits_{\bm{\mathrm{k}},\varepsilon}\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\left(\varepsilon-\xi_{\bm{\mathrm{k}}}+i\eta\hat{\tau}^{3}\right)\psi(\bm{\mathrm{k}},\varepsilon),\\\
&S_{c}=\,-\frac{1}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\bm{\mathrm{k}}^{\prime},\varepsilon,\varepsilon^{\prime}}\phi_{\mathsf{cl}}(\bm{\mathrm{k}}-\bm{\mathrm{k}}^{\prime},\varepsilon-\varepsilon^{\prime})\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\hat{M}_{F}(\varepsilon)\hat{M}_{F}(\varepsilon^{\prime})\psi(\bm{\mathrm{k}}^{\prime},\varepsilon^{\prime})\,\\\
&\qquad-\frac{1}{\sqrt{2}}\int\limits_{\bm{\mathrm{k}},\bm{\mathrm{k}}^{\prime},\varepsilon,\varepsilon^{\prime}}\phi_{\mathsf{q}}(\bm{\mathrm{k}}-\bm{\mathrm{k}}^{\prime},\varepsilon-\varepsilon^{\prime})\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\hat{M}_{F}(\varepsilon)\hat{\tau}^{1}\hat{M}_{F}(\varepsilon^{\prime})\psi(\bm{\mathrm{k}}^{\prime},\varepsilon^{\prime}).\end{aligned}$
(2.14)

####  2.1.2 RPA dynamically screened interaction

To proceed, we first integrate out the fermionic fields $\psi$ and
$\bar{\psi}$ to get an effective theory of the bosonic field $\phi$:

$\displaystyle\begin{aligned}
&Z=\int\mathcal{D}\phi\,\exp\left(iS_{\phi}+\ln\braket{\exp(iS_{c})}_{\psi}\right).\end{aligned}$
(2.15)

Here the angular bracket with a subscript $\psi$ is used to indicate the
functional averaging over the fermionic fields $\psi$ and $\bar{\psi}$ with
the weight $\exp\left(iS_{\psi}\right)$, so $\braket{\exp(iS_{c})}_{\psi}$
represents

$\displaystyle\begin{aligned}
&\braket{\exp(iS_{c})}_{\psi}=\,\int\mathcal{D}\left(\bar{\psi},\psi\right)\exp\left(iS_{\psi}+iS_{c}\right).\end{aligned}$
(2.16)

Within the framework of the RPA approximation, we approximate
$\braket{\exp(iS_{c})}_{\psi}$ by the leading order term in the cumulant
expansion:

$\displaystyle\begin{aligned}
&\ln\left\langle\exp(iS_{c})\right\rangle_{\psi}\approx\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}.\end{aligned}$
(2.17)

The effective action $\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}$
can be expressed in terms of the polarization operator $\hat{\Pi}$ which can
be considered as the self-energy of the bosonic field
$\phi=(\phi_{\mathsf{cl}},\phi_{\mathsf{q}})^{T}$:

$\displaystyle\begin{aligned}
\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}=-\frac{i}{2}&\int_{\bm{\mathrm{q}},\omega}\phi^{\mathsf{T}}(-\bm{\mathrm{q}},-\omega)\hat{\Pi}(\bm{\mathrm{q}},\omega)\phi(\bm{\mathrm{q}},\omega),\\\
\Pi^{ab}(\bm{\mathrm{q}},\omega)=\,-i\int\limits_{\bm{\mathrm{k}},\varepsilon}&\operatorname{Tr}\left\\{\left[\frac{1+\zeta_{a}}{2}+\frac{1-\zeta_{a}}{2}\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\right.\\\
&\qquad\left.\times\left[\frac{1+\zeta_{b}}{2}+\frac{1-\zeta_{b}}{2}\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon)\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)\hat{M}_{F}(\varepsilon)\right\\}.\end{aligned}$
(2.18)

Utilizing the causality relation
$\int_{\bm{\mathrm{k}},\varepsilon}G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)=0,$
and inserting the explicit expression for $\hat{M}_{F}(\varepsilon)$ (Eq.
2.12) into Eq. 2.18, one can verify that the polarization operator $\hat{\Pi}$
acquires the following triangular structure in the Keldysh space:

$\displaystyle\begin{aligned}
&\hat{\Pi}(\bm{\mathrm{q}},\omega)=\,\begin{bmatrix}0&\Pi^{(A)}(\bm{\mathrm{q}},\omega)\\\
\Pi^{(R)}(\bm{\mathrm{q}},\omega)&\Pi^{(K)}(\bm{\mathrm{q}},\omega)\end{bmatrix},\end{aligned}$
(2.19)

and its components are connected by the FDT for bosons

$\displaystyle\begin{aligned}
&\Pi^{(K)}(\bm{\mathrm{q}},\omega)=\,\left[\Pi^{(R)}(\bm{\mathrm{q}},\omega)-\Pi^{(A)}(\bm{\mathrm{q}},\omega)\right]\coth\left(\frac{\omega}{2T}\right).\end{aligned}$
(2.20)

One can also show that the retarded (advanced) component
$\Pi^{(R/A)}(\bm{\mathrm{q}},\omega)$ is given by

$\displaystyle\begin{aligned}
\Pi^{(R)}(\bm{\mathrm{q}},\omega)=\,\left(\Pi^{(A)}(\bm{\mathrm{q}},\omega)\right)^{*}=\,&-i\int\limits_{\bm{\mathrm{k}},\varepsilon}\left[G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)G_{0}^{(K)}(\bm{\mathrm{k}},\varepsilon)+G_{0}^{(K)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\right].\end{aligned}$
(2.21)

Employing the explicit form of the bare electron Green’s function (Eq. 2.10),
Eq. 2.21 can be reduced to the following integral representing the non-
interacting polarization function or the irreducible polarizability (“bare
bubble”):

$\displaystyle\begin{aligned}
\Pi^{(R)}(\bm{\mathrm{q}},\omega)=\,&\int\limits_{\bm{\mathrm{k}}}\frac{\tanh(\xi_{\bm{\mathrm{k}}+\bm{\mathrm{q}}}/2T)-\tanh(\xi_{\bm{\mathrm{k}}}/2T)}{\omega+\xi_{\bm{\mathrm{k}}}-\xi_{\bm{\mathrm{k}}+\bm{\mathrm{q}}}+i\eta},\end{aligned}$
(2.22)

whose result at zero temperature has been found by Stern [15] and Lindhard
[16] in dimensions $d=2$ and $d=3$, respectively. In terms of the
dimensionless parameters $u\equiv\omega/v_{\mathsf{F}}q$ and $x\equiv
q/2k_{\mathsf{F}}$, the zero temperature polarization operator takes the
following form in 2D

$\displaystyle\begin{aligned}
\operatorname{Re}\Pi_{0}^{(R)}(u>0,x)=&-\nu\left\\{1-\frac{\operatorname{sgn}{(x+u)}}{2x}\sqrt{\left(x+u\right)^{2}-1}\,\Theta\left(x+u-1\right)\right.\\\
&\left.\qquad\quad-\frac{\operatorname{sgn}{(x-u)}}{2x}\sqrt{\left(x-u\right)^{2}-1}\,\Theta\left(\lvert
x-u\rvert-1\right)\right\\},\\\
\operatorname{Im}\Pi_{0}^{(R)}(u>0,x)=&-\nu\left\\{-\frac{1}{2x}\sqrt{1-\left(x+u\right)^{2}}\Theta\left(1-(x+u)\right)+\frac{1}{2x}\sqrt{1-\left(x-u\right)^{2}}\Theta\left(1-\lvert
x-u\rvert\right)\right\\},\end{aligned}$ (2.23)

and in 3D it is given by

$\displaystyle\begin{aligned}
&\operatorname{Re}\Pi_{0}^{(R)}(u>0,x)=-\nu\left\\{\frac{1}{2}+\frac{1}{8x}\left[1-\left(x-u\right)^{2}\right]\ln\bigg{\lvert}\frac{x-u+1}{x-u-1}\bigg{\rvert}+\frac{1}{8x}\left[1-\left(x+u\right)^{2}\right]\ln\bigg{\lvert}\frac{x+u+1}{x+u-1}\bigg{\rvert}\right\\},\\\
&\operatorname{Im}\Pi_{0}^{(R)}(u>0,x)=-\pi\nu\left\\{\frac{u}{2}\Theta\left(\lvert
1-x\rvert-u\right)+\frac{1}{8x}\left[1-\left(x-u\right)^{2}\right]\Theta(1+x-u)\Theta\left(u-\lvert
1-x\rvert\right)\right\\}.\end{aligned}$ (2.24)

Here $\nu$ represents the density of states at Fermi level and is given by
$\nu=m/\pi$ and $\nu=mk_{\mathsf{F}}/\pi^{2}$ in 2D (Eq. 2.23) and 3D (Eq.
LABEL:eq:pi-3), respectively. We note that Eqs. 2.23 and LABEL:eq:pi-3 give
the forms of the polarization operator with frequency $\omega\geq 0$. The
expressions for $\omega<0$ can be deduced from these equations with the help
of
$\operatorname{Im}\Pi^{(R)}(\bm{\mathrm{q}},\omega)=-\operatorname{Im}\Pi^{(R)}(\bm{\mathrm{q}},-\omega)$
and
$\operatorname{Re}\Pi^{(R)}(\bm{\mathrm{q}},\omega)=\operatorname{Re}\Pi^{(R)}(\bm{\mathrm{q}},-\omega)$.

Once the polarization operator is known, it is straightforward to deduce
$\hat{D}(\bm{\mathrm{q}},\omega)$, the interaction dressed Green’s function
for the bosonic field $\phi$, also known as the RPA dynamically screened
interaction potential, from the Dyson equation

$\displaystyle\begin{aligned}
\hat{D}(\bm{\mathrm{q}},\omega)\equiv\,&-i\left\langle\begin{bmatrix}\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)\\\
\phi_{\mathsf{q}}(\bm{\mathrm{q}},\omega)\end{bmatrix}\begin{bmatrix}\phi_{\mathsf{cl}}(-\bm{\mathrm{q}},-\omega)&\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega)\end{bmatrix}\right\rangle=\left[\hat{D}_{0}(\bm{\mathrm{q}},\omega)-\hat{\Pi}(\bm{\mathrm{q}},\omega)\right]^{-1}\\!\\!\\!\\!\\!\\!\\!.\end{aligned}$
(2.25)

Here $\hat{D}_{0}$ is the bare bosonic Green’s function arising from the free
action $S_{\phi}$ (Eq. 2.14), and takes the form

$\displaystyle\begin{aligned}
\hat{D}_{0}(\bm{\mathrm{q}},\omega)=\begin{bmatrix}0&V(\bm{\mathrm{q}})\\\
V(\bm{\mathrm{q}})&0\end{bmatrix}.\end{aligned}$ (2.26)

From Eq. 2.25, one can see that $\hat{D}$ acquires the same structure in
Keldysh space as $\Pi^{-1}$:

$\displaystyle\begin{aligned}
\hat{D}(\bm{\mathrm{q}},\omega)=\,&\begin{bmatrix}D^{(K)}(\bm{\mathrm{q}},\omega)&D^{(R)}(\bm{\mathrm{q}},\omega)\\\
D^{(A)}(\bm{\mathrm{q}},\omega)&0\end{bmatrix},\end{aligned}$ (2.27)

with components given by

$\displaystyle\begin{aligned}
D^{(R)}(\bm{\mathrm{q}},\omega)=\,&\left[D^{(A)}(\bm{\mathrm{q}},\omega)\right]^{*}=\,\left[V^{-1}(q)-\Pi^{(R)}(\bm{\mathrm{q}},\omega)\right]^{-1}\\!\\!\\!\\!\\!\\!\\!,\end{aligned}$
(2.28a) $\displaystyle\begin{aligned}
D^{(K)}(\bm{\mathrm{q}},\omega)=\,&\left[D^{(R)}(\bm{\mathrm{q}},\omega)-D^{(A)}(\bm{\mathrm{q}},\omega)\right]\coth\left(\frac{\omega}{2T}\right).\end{aligned}$
(2.28b)

A diagrammatic representation of the Dyson equation (Eq. 2.28a) for the
bosonic field $\phi$ is depicted in Fig. 1(a) where the bare (dressed) bosonic
propagator $D_{0}$ ($D$), i.e., the bare (dynamically screened) Coulomb
interaction, is represented by the red wavy line with a open (solid) dot in
the middle. The black bare bubble corresponds to the polarization operator,
with each line representing a bare electron Green’s function $G_{0}$ in Eq.
2.21.

Figure 1: A diagrammatic representation of (a) the Dyson equation for the
dynamically screened RPA interaction (Eq. 2.28a) and (b) the RPA electron
self-energy. The dynamically screened (bare) interaction can be considered as
the dressed (bare) propagator $D$ ($D_{0}$) of the bosonic field $\phi$
introduced to decouple the interactions, and is represented diagrammatically
by the red wavy line with a solid (open) dot in the middle. The polarization
operator $\Pi$ can be considered as the self-energy for bosonic field $\phi$
and is shown by the black bubble where each line represents a bare electron
Green’s function $G_{0}$. To the leading order in the dynamically screened
interaction, the electron self-energy is given by the diagram in panel (b).

####  2.1.3 RPA self-energy

We now return to Eq. 2.14, the partition function before the fermionic fields
$\psi$ and $\bar{\psi}$ are integrated out. To the leading order in RPA
interaction, the electron self-energy can be obtained from $(iS_{c})^{2}$ the
square of the action which couples the fermionic and bosonic fields (Eq.
2.14):

$\displaystyle\begin{aligned}
-i\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)=&-\frac{1}{2}\int\limits_{\bm{\mathrm{q}},\omega}\left\langle\hat{M}_{F}(\varepsilon)\left[\phi_{\mathsf{cl}}(-\bm{\mathrm{q}},-\omega)+\phi_{\mathsf{q}}(-\bm{\mathrm{q}},-\omega)\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon+\omega)\psi_{\sigma}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right.\\\
&\left.\times\bar{\psi}_{\sigma}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\left[\phi_{\mathsf{cl}}(\bm{\mathrm{q}},\omega)+\phi_{\mathsf{q}}(\bm{\mathrm{q}},\omega)\hat{\tau}^{1}\right]\hat{M}_{F}(\varepsilon)\right\rangle.\end{aligned}$
(2.29)

The angular bracket here represents averaging with respect to the weight:

$\displaystyle\begin{aligned}
\exp\left(iS_{\psi}+iS_{\phi}+\left\langle\frac{1}{2}(iS_{c})^{2}\right\rangle_{\psi}\right)=\exp\left(i\int\limits_{\bm{\mathrm{k}},\varepsilon}\bar{\psi}(\bm{\mathrm{k}},\varepsilon)\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)\psi(\bm{\mathrm{k}},\varepsilon)+\frac{i}{2}\int\limits_{\bm{\mathrm{q}},\omega}\phi(\bm{\mathrm{q}},\omega)\hat{D}(\bm{\mathrm{q}},\omega)\phi(-\bm{\mathrm{q}},-\omega)\right).\end{aligned}$
(2.30)

Employing Wick’s theorem, we then equate
$-i\braket{\phi(\bm{\mathrm{q}},\omega)\phi(-\bm{\mathrm{q}},-\omega)}$ to the
RPA dynamically screened interaction $\hat{D}(\bm{\mathrm{q}},\omega)$ and
$-i\braket{\psi(\bm{\mathrm{k}},\varepsilon)\bar{\psi}(-\bm{\mathrm{k}},-\varepsilon)}$
to the bare electron Green’s function
$\hat{G}_{0}(\bm{\mathrm{k}},\varepsilon)$ in Eq. 2.29, and obtain

$\displaystyle\begin{aligned}
\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)=&\frac{i}{2}\int_{\bm{\mathrm{q}},\omega}D^{(K)}(-\bm{\mathrm{q}},-\omega)\hat{M}_{F}(\varepsilon)\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\hat{M}_{F}(\varepsilon)\\\
+\,&\frac{i}{2}\int_{\bm{\mathrm{q}},\omega}D^{(R)}(-\bm{\mathrm{q}},-\omega)\hat{M}_{F}(\varepsilon)\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\hat{\tau}^{1}\hat{M}_{F}(\varepsilon)\\\
+\,&\frac{i}{2}\int_{\bm{\mathrm{q}},\omega}D^{(A)}(-\bm{\mathrm{q}},-\omega)\hat{M}_{F}(\varepsilon)\hat{\tau}^{1}\hat{M}_{F}(\varepsilon+\omega)\hat{G}_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\hat{M}_{F}(\varepsilon+\omega)\hat{M}_{F}(\varepsilon).\end{aligned}$
(2.31)

$\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)$ is shown diagrammatically by Fig.
1(b) where the black solid line and the red wavy line with a closed dot
represent, respectively, the bare electron Green’s function $G_{0}$ and the
RPA dynamically screened interaction $D$. Inserting Eqs. 2.12 and 2.13 into
the equation above and using the causality relation
$\int\limits_{\bm{\mathrm{q}},\omega}D^{(R)}(\bm{\mathrm{q}},\omega)G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)=0,$
one can prove that $\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)$ is diagonal in
the Keldysh space

$\displaystyle\begin{aligned}
&\hat{\Sigma}(\bm{\mathrm{k}},\varepsilon)=\,\begin{bmatrix}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)&0\\\
0&\Sigma^{(A)}(\bm{\mathrm{k}},\varepsilon)\end{bmatrix},\end{aligned}$ (2.32)

and its retarded (advanced) component is given by

$\displaystyle\begin{aligned}
&\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,\left[\Sigma^{(A)}(\bm{\mathrm{k}},\varepsilon)\right]^{*}=\frac{i}{2}\int\limits_{\bm{\mathrm{q}},\omega}\left\\{D^{(K)}(\bm{\mathrm{q}},\omega)G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)+D^{(A)}(\bm{\mathrm{q}},\omega)G_{0}^{(K)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right\\}.\end{aligned}$
(2.33)

With the help of Kramers-Krönig relation,

$\displaystyle
f^{(R)}(\bm{\mathrm{k}},\varepsilon)=\int_{-\infty}^{\infty}\dfrac{d\varepsilon^{\prime}}{\pi}\dfrac{\operatorname{Im}f^{(R)}(\bm{\mathrm{k}},\varepsilon^{\prime})}{\varepsilon^{\prime}-\varepsilon-i\eta},$
(2.34)

one can rewrite Eq. 2.33 as

$\displaystyle\begin{aligned}
&\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,-2\int\limits_{\bm{\mathrm{q}},\omega,\omega^{\prime}}\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\operatorname{Im}G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega^{\prime})\frac{1}{\omega^{\prime}-\omega-i\eta}\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\varepsilon+\omega^{\prime}}{2T}\right)\right].\end{aligned}$
(2.35)

In two dimensions, after inserting the explicit expression for the bare
electron Green’s function (Eq. 2.10) and integrating over the angular
direction of the momentum $\bm{\mathrm{q}}$, Eq. 2.35 reduces to

$\displaystyle\begin{aligned}
\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{m}{\pi
k}\int\limits_{\omega,\omega^{\prime}}\int_{0}^{\infty}dq\Theta\left(1-\left|\frac{m\omega^{\prime}}{kq}\right|\right)\dfrac{\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left(\frac{m\omega^{\prime}}{kq}\right)^{2}}}\\\
&\times\dfrac{1}{\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}-\omega-i\eta}\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\varepsilon+\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}}{2T}\right)\right],\end{aligned}$
(2.36)

where for simplicity we have defined

$\displaystyle\begin{aligned}
\Delta\varepsilon\equiv\varepsilon-\xi_{\bm{\mathrm{k}}}.\end{aligned}$ (2.37)

From Eq. 2.36, one then find that the imaginary part of electron self-energy
is given by the following integral

$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,\frac{m}{4\pi^{2}k}\int_{-\infty}^{\infty}d\omega\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\omega+\varepsilon}{2T}\right)\right]\int_{q_{-}(\omega)}^{q_{+}(\omega)}dq\dfrac{\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left[\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right]^{2}}},\end{aligned}$
(2.38)

where

$\displaystyle q_{\pm}(\omega)=\left|\pm
k+\sqrt{k^{2}+2m\left(\omega+\Delta\varepsilon\right)}\right|.$ (2.39)

For any $q$ within the regime $q_{-}(\omega)\leq q\leq q_{+}(\omega)$,
$\left|\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right|\leq
1$ is always satisfied.

Applying the Kramers-Krönig relation (Eq. 2.34), Eq. 2.36 can be rewritten as

$\displaystyle\begin{aligned}
\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{m}{4\pi^{2}k}\int_{0}^{\infty}dq\int_{-\infty}^{\infty}d\omega^{\prime}\tanh\left(\frac{\varepsilon+\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}}{2T}\right)\dfrac{\Theta\left(1-\left|\frac{m\omega^{\prime}}{kq}\right|\right)}{\sqrt{1-\left(\frac{m\omega^{\prime}}{kq}\right)^{2}}}D^{(A)}(\bm{\mathrm{q}},\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m})\\\
+&\frac{m}{4\pi^{3}k}\int_{-\infty}^{\infty}d\omega\coth\left(\frac{\omega}{2T}\right)\int_{0}^{\infty}dq\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\int_{-\frac{kq}{m}}^{\frac{kq}{m}}d\omega^{\prime}\dfrac{1}{\sqrt{1-\left(\frac{m\omega^{\prime}}{kq}\right)^{2}}}\dfrac{1}{\omega^{\prime}-\Delta\varepsilon+\frac{q^{2}}{2m}-\omega-i\eta}.\end{aligned}$
(2.40)

Taking the real part of this equation, one has

$\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{m}{4\pi^{2}k}\int_{-\infty}^{\infty}d\omega\tanh\left(\frac{\varepsilon+\omega}{2T}\right)\int_{q_{-}(\omega)}^{q_{+}(\omega)}dq\dfrac{\operatorname{Re}D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left[\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right]^{2}}}\\\
-&\frac{m}{4\pi^{2}k}\int_{-\infty}^{\infty}d\omega\coth\left(\frac{\omega}{2T}\right)\left(\int_{0}^{q_{-}(\omega)}dq+\int_{q_{+}(\omega)}^{\infty}dq\right)\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\frac{\operatorname{sgn}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)}{\sqrt{\left[\frac{m}{kq}\left(\omega+\Delta\varepsilon-\frac{q^{2}}{2m}\right)\right]^{2}-1}}.\end{aligned}$
(2.41)

Proceeding in a way analogous to the one outlined above for the 2D self-energy
formulas Eqs. 2.38 and 2.41, one can prove that, in 3D, the imaginary and real
parts of electron self-energy are given by the following integrals

$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{1}{8\pi^{2}}\frac{m}{k}\int_{-\infty}^{\infty}d\omega\int_{q_{-}(\omega)}^{q_{+}(\omega)}dqq\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\left[\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\varepsilon+\omega}{2T}\right)\right],\\\
\operatorname{Re}\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\,&\frac{1}{8\pi^{2}}\frac{m}{k}\int_{-\infty}^{\infty}d\omega\int_{q_{-}(\omega)}^{q_{+}(\omega)}dqq\operatorname{Re}D^{(R)}(\bm{\mathrm{q}},\omega)\tanh\left(\frac{\omega+\varepsilon}{2T}\right)\\\
&+\frac{1}{8\pi^{3}}\frac{m}{k}\int_{0}^{\infty}dqq\int_{-\infty}^{\infty}d\omega\operatorname{Im}D^{(R)}(\bm{\mathrm{q}},\omega)\coth\left(\frac{\omega}{2T}\right)\ln\left[\left|\dfrac{\omega-\frac{kq}{m}+\Delta\varepsilon-\frac{q^{2}}{2m}}{\omega+\frac{kq}{m}+\Delta\varepsilon-\frac{q^{2}}{2m}}\right|\right].\end{aligned}$
(2.42)

###  2.2 2D electron self-energy

The self-energy formulas Eqs. 2.38, 2.41 and 2.42 derived in the previous
section are written as two-variables integrals involving the RPA dynamically
screened interaction $D^{(R)}(\bm{\mathrm{q}},\omega)$ which is given by
another integral for the polarization operator
$\Pi^{(R)}(\bm{\mathrm{q}},\omega)$ (Eq. 2.22). In this section, we will use
these formulas to calculate the on-shell ($\varepsilon=\xi_{\bm{\mathrm{k}}}$)
electron self-energy close to the Fermi surface ($k=k_{\mathsf{F}}$) in 2D. We
will work in the regime where $r_{s}^{3/2}\ll\Delta/E_{\mathsf{F}}\ll r_{s}\ll
1$ with $\Delta=\left\\{|\varepsilon|,T\right\\}$, and evaluate the result to
the leading order in $r_{s}$ and to several orders in
$\Delta/(E_{\mathsf{F}}r_{s})$. Note that the theoretical problem is extremely
subtle and challenging since it involves expansions in three distinct ‘small’
parameters: the dimensionless Coulomb coupling $r_{s}$, temperature $T/T_{F}$,
and energy $\varepsilon/E_{F}$. It turns out that the actual small parameters
for temperature (energy) expansions are in fact $T/T_{\mathsf{F}}r_{s}$
($\varepsilon/E_{\mathsf{F}}r_{s}$), but we allow $T$ and $\varepsilon$ to be
comparable in our theory, which considerably complicates the calculations.

Using Eq. 2.38 and 2.41 and setting $\Delta\varepsilon=0$, we find that the
real and imaginary parts of the on-shell self-energy in 2D are given by

$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\varepsilon)=\,&\int_{0}^{\infty}\frac{d\omega}{2\pi}\left[2\coth\left(\frac{\omega}{2T}\right)-\tanh\left(\frac{\omega+\varepsilon}{2T}\right)-\tanh\left(\frac{\omega-\varepsilon}{2T}\right)\right]\operatorname{Im}I(\omega),\end{aligned}$
(2.43a) $\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(\varepsilon)=\,\int_{0}^{\infty}\frac{d\omega}{2\pi}\left[\tanh\left(\frac{\omega+\varepsilon}{2T}\right)-\tanh\left(\frac{\omega-\varepsilon}{2T}\right)\right]\operatorname{Re}I(\omega),\end{aligned}$
(2.43b)

where

$\displaystyle I(\omega)\equiv\frac{m}{2\pi
k}\int_{q_{-}(\omega)}^{q_{+}(\omega)}dq\dfrac{D^{(R)}(\bm{\mathrm{q}},\omega)}{\sqrt{1-\left[\frac{m}{kq}\left(\omega-\frac{q^{2}}{2m}\right)\right]^{2}}}.$
(2.44)

Here we have neglected the second integral in Eq. 2.41 which vanishes to the
leading order in $r_{s}$.

####  2.2.1 Momentum integration

We will now evaluate $I(\omega)$ defined in Eq. 2.44 by performing the
momentum integration. For convenience, we introduce

$\displaystyle\begin{aligned}
\alpha\equiv\frac{r_{s}}{\sqrt{2}},\quad\delta&\equiv\frac{\omega}{4E_{\mathsf{F}}},\quad
x\equiv\frac{q}{2k_{\mathsf{F}}},\quad
x_{\pm}(\delta)\equiv\frac{q_{\pm}(\omega)}{2k_{\mathsf{F}}}.\end{aligned}$
(2.45)

Here $x_{\pm}(\delta)$ is a solution to the equation $(x-\delta/x)^{2}=1$, and
is given by

$\displaystyle x_{-}(\delta)$
$\displaystyle=\frac{1}{2}\Big{|}1-\sqrt{1+4\delta}\,\Big{|},\quad
x_{+}(\delta)=\frac{1}{2}\Big{(}1+\sqrt{1+4\delta}\Big{)}.$ (2.46)

Using the newly introduced variables, Eq. 2.44 can then be rewritten as

$\displaystyle I(\delta)=$
$\displaystyle\frac{m}{\pi}\int_{x_{-}(\delta)}^{x_{+}(\delta)}dx\;D^{(R)}(x,\delta)\;\Bigg{[}1-\left(x-\frac{\delta}{x}\right)^{2}\Bigg{]}^{-1/2}\\!\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\frac{m}{\pi}\int_{x_{-}(\delta)}^{x_{+}(\delta)}dx\;D^{(R)}(x,\delta)\;x\Big{[}\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)\Big{]}^{-1/2}\\!\\!\\!\\!\\!\\!\\!,$
(2.47)

where the retarded RPA interaction $D^{(R)}(x,\delta)$ (Eq. 2.28a) is given by

$D^{(R)}(x,\delta)=\nu^{-1}\dfrac{\left(\frac{x}{\alpha}-\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)\right)+i\left(\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)\right)}{\left(\frac{x}{\alpha}-\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)\right)^{2}+\left(\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)\right)^{2}}.$
(2.48)

Here we have approximated the polarization bubble by the zero-temperature
result $\Pi_{0}$ (Eq. 2.23) whose real and imaginary parts, in terms of the
new variables, take the forms

$\displaystyle\begin{aligned}
\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)=&-1+\frac{1}{2x^{2}}\operatorname{sgn}\bigg{(}1-\frac{\delta}{x^{2}}\bigg{)}\operatorname{Re}\sqrt{-\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)}\\\
&+\frac{1}{2x^{2}}\operatorname{sgn}\bigg{(}1+\frac{\delta}{x^{2}}\bigg{)}\operatorname{Re}\sqrt{4\delta
x^{2}-\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)},\end{aligned}$
(2.49a) $\displaystyle\begin{aligned}
\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)=&-\frac{1}{2x^{2}}\operatorname{Re}\sqrt{\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)}+\frac{1}{2x^{2}}\operatorname{Re}\sqrt{\left(x_{+}^{2}(\delta)-x^{2}\right)\left(x^{2}-x_{-}^{2}(\delta)\right)-4\delta
x^{2}}.\end{aligned}$ (2.49b)

It is clear that the main contribution to the integral in Eq. 2.43 is from
$\omega$ of the order of $|\varepsilon|$ or $T$ due to the presence of the
thermal factor which involves tanh and coth functions. As mentioned earlier,
we consider the regime where $r_{s}^{3/2}\ll\Delta/E_{\mathsf{F}}\ll r_{s}\ll
1$, with $\Delta=\left\\{|\varepsilon|,T\right\\}$, and therefore will
calculate $I(\delta)$ for $\alpha^{3/2}\ll|\delta|\ll\alpha\ll 1$, to the
leading order in $\alpha$ and several orders in $|\delta|/\alpha$.

To proceed, we divide the interval of integration in Eq. 2.47 into three
subintervals: $(x_{-},l_{-})$, $(l_{-},l_{+})$ and $(l_{+},x_{+})$. Here
$l_{\pm}$ can take arbitrary value as long as it satisfies $x_{-}\ll
l_{-}\ll\alpha\ll l_{+}\ll x_{+}$. Within these subregions, we then expand the
integrand using different small parameters, such as $x/\alpha$, $\delta/x$,
$x/x_{+}$ and $\alpha/x$. We note that these variables are small in some
subregions but become comparable to or larger than $1$ in other subregions. In
other words, it is not possible to use a single small parameter for the entire
integration interval. This lack of a single unique small parameter in the
theory has prevented the regime of $\varepsilon\sim T$, while both being
small, being studied in the theoretical literature at all in spite of the long
history of electronic many body field theories [6, 7, 12].

We set the real part of the polarization operator to

$\nu^{-1}\operatorname{Re}\Pi_{0}^{(R)}(x,\delta)=-1,$ (2.50)

which is valid as long as $x$ lies within the region $x\in(a_{-},a_{+})$.
$a_{\pm}$ is defined as $a_{-}=x_{-}[1+O(|\delta|^{1-b})]$ and
$a_{+}=x_{+}[1-O(|\delta|)]$ where $b\in(0,1)$ is arbitrary. Furthermore, for
$x\in(a_{-},l_{+})$, $\operatorname{Im}\Pi_{0}^{(R)}$ (Eq. 2.49b) can be
approximated as

$\displaystyle\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(\sqrt{z^{2}+x_{-}^{2}},\delta)=$
$\displaystyle-\frac{zx_{+}}{2(z^{2}+x_{-}^{2})}\Big{[}1-\sqrt{1-\frac{4\delta(z^{2}+x_{-}^{2})}{z^{2}x_{+}^{2}}}+O\Big{(}\frac{l_{+}^{2}}{x_{+}^{2}}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle-\frac{zx_{+}}{2(z^{2}+x_{-}^{2})}\frac{4\delta(z^{2}+x_{-}^{2})}{2z^{2}x_{+}^{2}}\Big{[}1+O\big{(}|\delta|^{b}\big{)}\Big{]}=-\frac{\delta}{z}\Big{[}1+O\big{(}|\delta|^{b}\big{)}\Big{]},$
(2.51)

where $z^{2}=x^{2}-x_{-}^{2}$. For $x\in(l_{+},a_{+})$, all we need to know is
that

$\nu^{-1}\operatorname{Im}\Pi_{0}^{(R)}(x,\delta)=O\Big{(}|\delta|^{1/2},\frac{\delta}{l_{+}}\Big{)}.$
(2.52)

We will now show that the contribution to $I(\delta)$ from $x$ outside the
region $x\in(a_{-},a_{+})$ is of higher order in $\delta$ and therefore can be
ignored. Let us first calculate $I(x_{-},a_{-})$, i.e., the contribution to
$I(\delta)$ from $x_{-}<x<a_{-}$. Throughout this section, $I(x_{1},x_{2})$ is
used to denote the contribution to $I(\delta)$ from the integration over the
interval $x\in(x_{1},x_{2})$. Applying a change of variables
$z^{2}=x^{2}-x_{-}^{2}$, one obtains

$\displaystyle
I(x_{-},a_{-})=\int_{0}^{O(|\delta|^{\frac{3-b}{2}})}dz\;D^{(R)}(\sqrt{z^{2}+x_{-}^{2}},\delta)\,\Big{(}x_{+}^{2}-z^{2}-x_{-}^{2}\Big{)}^{-1/2}=O(|\delta|^{\frac{3-b}{2}}).$
(2.53)

Here we have used the fact that
$|D^{(R)}(x,\delta)|\leq\nu^{-1}\alpha\,O(\alpha^{-1})$ inside the region
$x\in(x_{-},a_{-})$. Similarly, for the region $a_{+}<x<x_{+}$,
$|\nu^{-1}\Pi_{0}^{(R)}(x,\delta)|=O(|\delta|^{1/2})$ which leads to
$|D^{(R)}(x,\delta)|=\nu^{-1}O(\alpha)$. Applying the transformation
$u^{2}=x_{+}^{2}-x^{2}$, we find the contribution to $I(\delta)$ from this
region

$\displaystyle
I(a_{+},x_{+})=\int_{0}^{O(|\delta|^{1/2})}du\;D^{(R)}(\sqrt{x_{+}^{2}-u^{2}},\delta)\,\Big{(}x_{+}^{2}-u^{2}-x_{-}^{2}\Big{)}^{-1/2}=\alpha\;O(|\delta|^{1/2}).$
(2.54)

We have proved that the contribution from outside the regime
$x\in(a_{-},a_{+})$ can be ignored, and will now evaluate
$\operatorname{Im}I(\delta)$ by carrying out the integration in Eq. 2.47
separately for the three subintervals: $(a_{-},l_{-})$, $(l_{-},l_{+})$, and
$(l_{+},a_{+})$. Within the first region $x\in(a_{-},l_{-})$, the integrand
can be expanded in terms of the small parameter $x/\alpha\ll 1$. Employing the
transformation $z^{2}=x^{2}-\delta^{2}$, and inserting Eqs. 2.50 and 2.2.1
into Eq. 2.47, one obtains

$\displaystyle\operatorname{Im}I(a_{-},l_{-})=-\int_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}dz\;\frac{\delta/z}{(\frac{\sqrt{z^{2}+\delta^{2}}}{\alpha}+1)^{2}+(\delta/z)^{2}}$
$\displaystyle=-\int_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}dz\;\frac{\delta}{z}\bigg{(}1+\Big{(}\frac{\delta}{z}\Big{)}^{2}\bigg{)}^{-1}\bigg{[}1-\frac{2\sqrt{z^{2}+\delta^{2}}}{\alpha}\bigg{(}1+\Big{(}\frac{\delta}{z}\Big{)}^{2}\bigg{)}^{-1}\bigg{]}$
$\displaystyle=-{\alpha}\bigg{[}\frac{\delta}{\alpha}\ln\left(\frac{l_{-}}{|\delta|}\right)+4\frac{\delta^{2}}{\alpha^{2}}\operatorname{sgn}\delta-2\frac{l_{-}\delta}{\alpha^{2}}\bigg{]}.$
(2.55)

For $x$ lying inside the region $(l_{-},l_{+})$, we instead expand in terms of
$|\delta|/x\ll 1$ and $x\ll 1$, which leads to

$\displaystyle\operatorname{Im}I(l_{-},l_{+})=-\int_{l_{-}}^{l_{+}}dx\;\frac{\delta\alpha^{2}}{x(x+\alpha)^{2}}=-\alpha\bigg{[}-\frac{\delta}{\alpha}+\frac{\delta}{\alpha}\ln\left(\frac{\alpha}{l_{-}}\right)+2\frac{l_{-}\delta}{\alpha^{2}}\bigg{]}.$
(2.56)

Finally, the contribution from $x\in(l_{+},a_{+})$, i.e.,
$\operatorname{Im}I(l_{+},a_{+})$, is negligible as the integrand is of the
order of $\alpha$. Combining contributions from all subregions, we find the
imaginary part of $I(\delta)$

$\operatorname{Im}I(\delta)=-\alpha\left\\{\frac{\delta}{\alpha}\left[-\\!1+\ln\left(\frac{\alpha}{|\delta|}\right)\right]+4\frac{\delta^{2}}{\alpha^{2}}\operatorname{sgn}\delta\right\\}.$
(2.57)

We then turn to the calculation of the real part of $I(\delta)$ by considering
the three subregions separately as in the case of
$\operatorname{Im}I(\delta)$. For $a_{-}\leq x\leq l_{-}$, we use $x/\alpha\ll
1$ as the small expansion parameter, and approximate
$\operatorname{Re}D^{(R)}$ by

$\displaystyle\operatorname{Re}D^{(R)}(x,\delta)=\nu^{-1}\frac{1+\frac{x}{\alpha}}{(1+\frac{x}{\alpha})^{2}+\frac{\delta^{2}}{z^{2}}+O(|\delta|^{\frac{3b-1}{2}})}=\nu^{-1}\frac{1+\frac{x}{\alpha}}{1+\frac{\delta^{2}}{z^{2}}}\bigg{[}1-\frac{\frac{2x}{\alpha}+\frac{x^{2}}{\alpha^{2}}}{1+\frac{\delta^{2}}{z^{2}}}+\bigg{(}\frac{\frac{2x}{\alpha}}{1+\frac{\delta^{2}}{z^{2}}}\bigg{)}^{2}+O\Big{(}\frac{x^{3}}{\alpha^{3}}\Big{)}\bigg{]},$
(2.58)

where $z^{2}=x^{2}-x_{-}^{2}$. This leads to

$\displaystyle\operatorname{Re}I(a_{-},l_{-})=\nu\int_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}dz\;\operatorname{Re}D^{(R)}(\sqrt{z^{2}+x_{-}^{2}},\delta)\,\Big{[}1+O(l_{-}^{2})\Big{]}$
$\displaystyle=\alpha\bigg{[}\frac{z}{\alpha}-\frac{z(5\delta^{2}+z^{2})}{2\alpha^{2}\sqrt{\delta^{2}+z^{2}}}-\frac{\delta}{\alpha}\arctan\left(\frac{z}{\delta}\right)+\frac{5\delta^{2}}{2\alpha^{2}}\ln\big{(}z+\sqrt{\delta^{2}+z^{2}}\big{)}\bigg{]}\bigg{|}_{O(|\delta|^{\frac{3-b}{2}})}^{l_{-}-\delta^{2}\\!/(2l_{-})}$
$\displaystyle=\alpha\bigg{[}-\frac{\pi|\delta|}{2\alpha}+\frac{l_{-}}{\alpha}-\frac{7\delta^{2}}{4\alpha^{2}}+\frac{5\delta^{2}}{2\alpha^{2}}\ln\left(\frac{2l_{-}}{|\delta|}\right)+\frac{\delta^{2}}{2\alpha
l_{-}}-\frac{l_{-}^{2}}{2\alpha^{2}}+O\Big{(}\frac{l_{-}^{3}}{\alpha^{3}},\;\frac{\delta^{3}}{l_{-}^{3}},\;\frac{|\delta|^{\frac{3-b}{2}}}{\alpha},\;|\delta|^{\frac{3b-1}{2}}\Big{)}\bigg{]}.$
(2.59)

On the other hand, for the region $x\in(l_{-},l_{+})$, we expand in terms of
$|\delta|/x\ll 1$ and $x\ll 1$ instead of $x/\alpha$. Substituting

$\operatorname{Re}D^{(R)}(x,\delta)=\nu^{-1}\frac{1+\frac{x}{\alpha}}{(1+\frac{x}{\alpha})^{2}+\frac{\delta^{2}}{x^{2}}+O(|\delta|^{\frac{3b-1}{2}},\,\frac{\delta^{4}}{x^{4}})},$
(2.60)

into Eq. 2.47, one obtains

$\displaystyle\operatorname{Re}I(l_{-},l_{+})=\nu\int_{l_{-}}^{l_{+}}dx\;\operatorname{Re}D^{(R)}(x,\delta)\,\bigg{[}1+\frac{\delta^{2}}{2x^{2}}+O\Big{(}\frac{\delta^{3}}{x^{3}},x\Big{)}\bigg{]}$
$\displaystyle=\alpha\bigg{[}-\frac{l_{-}}{\alpha}-\frac{5\delta^{2}}{2\alpha^{2}}+\frac{5\delta^{2}}{2\alpha^{2}}\ln\left(\frac{\alpha}{l_{-}}\right)-\frac{\delta^{2}}{2\alpha
l_{-}}+\frac{l_{-}^{2}}{2\alpha^{2}}+\ln\left(\frac{l_{+}}{\alpha}\right)+O\Big{(}\frac{l_{-}^{3}}{\alpha^{3}},\;\frac{\delta^{3}}{l_{-}^{3}},\;\frac{\alpha}{l_{+}},\;l_{+}\Big{)}\bigg{]}.$
(2.61)

For the last region $x\in(l_{+},a_{+})$, $\alpha/x\ll 1$ now becomes the small
expansion parameter. Keeping only the leading order term in $\alpha/x\ll 1$,
we have

$D^{(R)}(x,\delta)=\nu^{-1}\Big{[}\frac{\alpha}{x}+O\Big{(}\frac{\alpha^{2}}{x^{2}}\Big{)}\Big{]},$
(2.62)

which leads to

$\displaystyle\operatorname{Re}I(l_{+},a_{+})=\int_{l_{+}}^{1+O(\delta)}dx\;\Big{[}\frac{\alpha}{x}+O\Big{(}\frac{\alpha^{2}}{x^{2}}\Big{)}\Big{]}(1-x^{2})^{-1/2}=\alpha\Big{[}\ln\left(\frac{2}{l_{+}}\right)+O\Big{(}\frac{\alpha}{l_{+}},\;l_{+}\Big{)}\Big{]}.$
(2.63)

Combining everything, we arrive at

$\displaystyle\operatorname{Re}I(\delta)=\alpha\bigg{[}\ln\left(\frac{2}{\alpha}\right)-\frac{\pi|\delta|}{2\alpha}+\frac{\delta^{2}}{\alpha^{2}}\Big{(}-\frac{17}{4}+\frac{5}{2}\ln\left(\frac{2\alpha}{|\delta|}\right)\Big{)}+O\Big{(}\frac{\delta^{3}}{\alpha^{3}},\;\alpha,\;|\delta|^{\frac{1-b}{2}},\;|\delta|^{\frac{3b-1}{2}}\Big{)}\bigg{]}.$
(2.64)

####  2.2.2 Frequency integration

The calculation presented in the previous section shows that

$\displaystyle\begin{aligned}
\operatorname{Im}I(\omega)=\,&-\left\\{\frac{1}{4}\frac{|\omega|}{E_{\mathsf{F}}}\left[\ln\left(\frac{2\sqrt{2}r_{s}E_{\mathsf{F}}}{|\omega|}\right)-1\right]+\frac{1}{2\sqrt{2}r_{s}}\left(\frac{\omega}{E_{\mathsf{F}}}\right)^{2}\right\\}\operatorname{sgn}\omega,\end{aligned}$
(2.65a) $\displaystyle\begin{aligned}
\operatorname{Re}I(\omega)=\frac{r_{s}}{\sqrt{2}}\left[\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)-\frac{\pi}{4\sqrt{2}r_{s}}\frac{|\omega|}{E_{\mathsf{F}}}+\frac{5}{16r_{s}^{2}}\frac{\omega^{2}}{E_{\mathsf{F}}^{2}}\ln\left(\frac{4\sqrt{2}r_{s}E_{\mathsf{F}}}{|\omega|}\right)-\frac{17}{32r_{s}^{2}}\frac{\omega^{2}}{E_{\mathsf{F}}^{2}}\right],\end{aligned}$
(2.65b)

where we have transformed back to the original variables. These expressions
can be used to extract the electron self-energy on the mass shell by
substitution using Eq. 2.43 and carrying out the frequency integrations.

The frequency integrations in Eq. 2.43 involve hyperbolic functions $\tanh(x)$
and $\coth(x)$, and are of the form

$\displaystyle\begin{aligned}
I_{1}(a)=&\int_{0}^{\infty}dxf(x)\left[2\coth(x)-\tanh(x+a)-\tanh(x-a)\right],\\\
I_{2}(a)=&\int_{0}^{\infty}dxf(x)\left[\tanh(x+a)-\tanh(x-a)\right].\end{aligned}$
(2.66)

To carry out such integrals, one may take advantage of the following equations
which express $\tanh(x)$ and $\coth(x)$ as infinite exponential series:

$\displaystyle\begin{aligned}
&\tanh(x)=\,1+2\sum_{k=1}^{\infty}(-1)^{k}e^{-2kx},\qquad\coth(x)=\,1+2\sum_{k=1}^{\infty}e^{-2kx},\qquad
x>0.\end{aligned}$ (2.67)

In Appendix. B, we use the equations above and provide the analytical results
for integrals of the form Eq. 2.66 for various functions $f(x)$.

Using Eq. B.2 in Appendix B, we obtain the analytical expression for the
imaginary part of the 2D self-energy on the mass shell:

$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)=&-\frac{1}{8\pi}\left(\pi^{2}+\frac{\varepsilon^{2}}{T^{2}}\right)\frac{T^{2}}{E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{T}\right)\\\
&-\left\\{-\frac{\pi}{24}\left(6-\gamma_{E}-\ln\left(\frac{2}{\pi^{2}}\right)-24\ln\mathrm{A}\right)-\frac{\left(2-\gamma_{E}-\ln
2\right)}{8\pi}\frac{\varepsilon^{2}}{T^{2}}\right.\\\
&\left.\qquad+\frac{1}{4\pi}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\varepsilon/T})+\partial_{s}\operatorname{Li}_{s}(-e^{\varepsilon/T})\right]\bigg{\lvert}_{s=2}\right\\}\frac{T^{2}}{E_{\mathsf{F}}}\\\
&-\frac{\sqrt{2}}{\pi}\left[\zeta(3)-\frac{1}{2}\operatorname{Li}_{3}(-e^{\varepsilon/T})-\frac{1}{2}\operatorname{Li}_{3}(-e^{-\varepsilon/T})\right]\frac{T^{3}}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.68)

Here $\gamma_{E}\approx 0.577216$ is the Euler’s constant, and
$\mathrm{A}\approx 1.28243$ is the Glaisher’s constant.
$\operatorname{Li}_{s}(z)=\sum_{k=1}^{\infty}{z^{k}}/{k^{s}}$ stands for the
polylogarithm function, and $\zeta(z)=\sum_{k=1}^{\infty}{1}/{k^{z}}$
represents the Riemann zeta function. We emphasize that this result is valid
for arbitrary value of $\varepsilon/T$, as long as $|\varepsilon|,T\ll
E_{\mathsf{F}}$ condition is satisfied.

One can derive from Eq. 2.68 the asymptotic behaviors of
$\operatorname{Im}\Sigma^{(R)}$ in the low-energy limit $|\varepsilon|\ll T$
as well as the low-temperature limit $T\ll|\varepsilon|$. With the help of

$\displaystyle\begin{aligned}
\partial_{s}\operatorname{Li}_{s}(-1)|_{s=2}=\,-\frac{\pi^{2}}{12}\left(\gamma_{E}+\ln\left(4\pi\right)-12\ln\mathrm{A}\right),\qquad\operatorname{Li}_{3}(-1)=-\frac{3}{4}\zeta(3),\end{aligned}$
(2.69)

we find that, for $|\varepsilon|\ll T$, $\operatorname{Im}\Sigma^{(R)}$
assumes the form

$\displaystyle\begin{aligned} &\operatorname{Im}\Sigma^{(R)}(|\varepsilon|\ll
T)=\,-\frac{\pi}{8}\frac{T^{2}}{E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{T}\right)+\frac{\pi}{24}\left(6+\ln\left(2\pi^{3}\right)-36\ln\mathrm{A}\right)\frac{T^{2}}{E_{\mathsf{F}}}-\frac{7\zeta(3)}{2\sqrt{2}\pi}\frac{T^{3}}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.70)

Similarly for $T\ll|\varepsilon|$, we have

$\displaystyle\begin{aligned} &\operatorname{Im}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,-\frac{\varepsilon^{2}}{8\pi
E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{|\varepsilon|}\right)-\left(\ln
4-1\right)\frac{\varepsilon^{2}}{16\pi
E_{\mathsf{F}}}-\frac{1}{6\sqrt{2}\pi}\frac{|\varepsilon|^{3}}{r_{s}E_{\mathsf{F}}^{2}},\end{aligned}$
(2.71)

where we have used

$\displaystyle\begin{aligned}
&\lim\limits_{x\rightarrow\infty}\operatorname{Li}_{s}(-e^{x})=-\frac{x^{s}}{\Gamma\left(s+1\right)},\quad\lim\limits_{x\rightarrow\infty}\partial_{s}\operatorname{Li}_{s}(-e^{x})=-\frac{x^{s}\ln
x}{\Gamma\left(s+1\right)}+\frac{\Gamma^{\prime}\left(s+1\right)}{\Gamma^{2}\left(s+1\right)}x^{s}.\end{aligned}$
(2.72)

In a way analogous to the one that leads to the expression of the imaginary
part of self-energy, we evaluate the integration in Eq. 2.43 using Eq. B.2 in
Appendix B, and derive the real part of the on-shell self-energy for arbitrary
$\varepsilon/T$ in 2D:

$\displaystyle\begin{aligned}
&\operatorname{Re}\Sigma^{(R)}(\varepsilon,T)=\,\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)\varepsilon-\frac{1}{8}\frac{T}{\varepsilon}\left[\operatorname{Li}_{2}(-e^{-\frac{\varepsilon}{T}})-\operatorname{Li}_{2}(-e^{\frac{\varepsilon}{T}})\right]\frac{T\varepsilon}{E_{\mathsf{F}}}+\frac{5}{48\sqrt{2}\pi}\left(\pi^{2}+\frac{\varepsilon^{2}}{T^{2}}\right)\frac{T^{2}\varepsilon}{r_{s}E_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}E_{\mathsf{F}}}{T}\right)\\\
&-\left\\{\frac{1}{96\sqrt{2}\pi}\left(32-10\gamma_{E}-25\ln
2\right)\left(\frac{\varepsilon^{2}}{T^{2}}+\pi^{2}\right)+\frac{5}{8\sqrt{2}\pi}\frac{T}{\varepsilon}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\frac{\varepsilon}{T}})-\partial_{s}\operatorname{Li}_{s}(-e^{\frac{\varepsilon}{T}})\right]\bigg{\lvert}_{s=3}\right\\}\frac{T^{2}\varepsilon}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.73)

In the limit of $|\varepsilon|/T\ll 1$, one has

$\displaystyle\begin{aligned}
&\operatorname{Li}_{2}(-e^{-\frac{\varepsilon}{T}})-\operatorname{Li}_{2}(-e^{\frac{\varepsilon}{T}})=2\ln
2\frac{\varepsilon}{T}+O\left((\frac{\varepsilon}{T})^{2}\right),\\\
&\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\frac{\varepsilon}{T}})-\partial_{s}\operatorname{Li}_{s}(-e^{\frac{\varepsilon}{T}})\right]\bigg{\lvert}_{s=3}=2\partial_{s}\partial_{z}\operatorname{Li}_{s}(z)|_{s=3,z=-1}\frac{\varepsilon}{T}+O\left((\frac{\varepsilon}{T})^{2}\right),\end{aligned}$
(2.74)

where $\partial_{s}\partial_{z}\operatorname{Li}_{s}(z)|_{s=3,z=-1}$ can be
further simplified to $(\pi^{2}\ln 2+6\zeta^{\prime}(2))/12$. Substitution of
Eq. 2.74 into Eq. 2.73 leads to the asymptotic expression for the real part of
self-energy in the limit of $|\varepsilon|\ll T$,

$\displaystyle\begin{aligned} \operatorname{Re}\Sigma^{(R)}(|\varepsilon|\ll
T)=\,&\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)\varepsilon-\frac{\ln
2}{4}\frac{T\varepsilon}{E_{\mathsf{F}}}+\frac{5\pi}{48\sqrt{2}r_{s}}\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}E_{\mathsf{F}}}{T}\right)\\\
&+\left[-\frac{\pi}{96\sqrt{2}}\left(32-10\gamma_{E}-25\ln
2\right)-\frac{5}{8\sqrt{2}\pi}\left(\zeta^{\prime}(2)+\frac{\pi^{2}}{6}\ln
2\right)\right]\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}r_{s}}.\end{aligned}$
(2.75)

The expression for $\operatorname{Re}\Sigma^{(R)}$ in the low-temperature
limit $T\ll|\varepsilon|$ can also be extracted from Eq. 2.73. With the help
of Eq. 2.72, we arrive at the result

$\displaystyle\begin{aligned} &\operatorname{Re}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)\varepsilon-\frac{1}{16}\frac{\varepsilon|\varepsilon|}{E_{\mathsf{F}}}+\frac{5}{48\sqrt{2}\pi}\frac{\varepsilon^{3}}{r_{s}E_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}E_{\mathsf{F}}}{|\varepsilon|}\right)+\frac{-41+75\ln
2}{288\sqrt{2}\pi}\frac{\varepsilon^{3}}{r_{s}E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.76)

###  2.3 3D electron self-energy

Starting from the general formulas Eq. 2.42, one can derive the 3D electron
self-energy in a way analogous to the one presented in Sec. 2.2 in the case of
2D. In this section, without giving the details of the calculation, we present
directly the explicit expressions for the 3D electron self-energy on the mass
shell, which are valid to the leading order in $r_{s}$, and to several orders
in $\varepsilon/E_{\mathsf{F}}$ and $T/E_{\mathsf{F}}$. The 3D calculations
follow the general procedure given above in depth for the corresponding 2D
theory. To simplify the calculation, we consider a range of temperature
(energy) slightly different from the one in the 2D case:
$r_{s}\ll{T}/{T_{\mathsf{F}}}\ll\sqrt{r_{s}}$
($r_{s}\ll{\varepsilon}/{E_{\mathsf{F}}}\ll\sqrt{r_{s}}$).

In 3D, for arbitrary energy-to-temperature ratio, the imaginary part of the
on-shell self-energy acquires the form of

$\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)=\,&-\frac{1}{32}(\frac{\pi^{4}}{12})^{1/3}\sqrt{r_{s}}\frac{1}{E_{\mathsf{F}}}\left(\pi^{2}T^{2}+\varepsilon^{2}\right)-\frac{\pi}{8}C_{0}\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left[\zeta(3)-\frac{1}{2}\left(\operatorname{Li}_{3}(-e^{-\frac{\varepsilon}{T}})+\operatorname{Li}_{3}(-e^{\frac{\varepsilon}{T}})\right)\right],\end{aligned}$
(2.77)

where $C_{0}$ is a constant defined as

$\displaystyle
C_{0}=\int_{0}^{1}dy\frac{1}{y^{2}}\left[\dfrac{1}{\left(1-\frac{y}{2}\ln\left(\dfrac{1+y}{1-y}\right)\right)^{2}+\left(\frac{\pi}{2}y\right)^{2}}-1\right]-1\approx-1.5326.$
(2.78)

From the equation above, one can find $\operatorname{Im}\Sigma^{(R)}$ in both
the low-energy limit and the low-temperature limit:

$\displaystyle\begin{aligned} \operatorname{Im}\Sigma^{(R)}(|\varepsilon|\ll
T)=\,&-\frac{1}{32}(\frac{\pi^{4}}{12})^{1/3}\pi^{2}\sqrt{r_{s}}\frac{T^{2}}{E_{\mathsf{F}}}-\frac{7\pi}{32}\zeta(3)C_{0}\frac{T^{3}}{E_{\mathsf{F}}^{2}},\end{aligned}$
(2.79a) $\displaystyle\begin{aligned}
\operatorname{Im}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,&-\frac{1}{32}(\frac{\pi^{4}}{12})^{1/3}\sqrt{r_{s}}\frac{\varepsilon^{2}}{E_{\mathsf{F}}}-\frac{\pi}{96}C_{0}\frac{|\varepsilon|^{3}}{E_{\mathsf{F}}^{2}}.\end{aligned}$
(2.79b)

Similarly, we evaluate the real part of the 3D on-shell self-energy in the
case where $\varepsilon$ and $T$ are arbitrary with respect to each other, and
find

$\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(\varepsilon,T)=\,&\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)\varepsilon-\frac{4-\pi^{2}}{192}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{T}{E_{\mathsf{F}}}\right)\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left(\frac{\varepsilon^{3}}{T^{3}}+\pi^{2}\frac{\varepsilon}{T}\right)\\\
&-\left[\frac{4-\pi^{2}}{384}\left(3-2\gamma_{E}-2\ln
2\right)-\frac{2}{3}C_{1}\right]\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left(\frac{\varepsilon^{3}}{T^{3}}+\pi^{2}\frac{\varepsilon}{T}\right)\\\
&-\frac{4-\pi^{2}}{32}\frac{T^{3}}{E_{\mathsf{F}}^{2}}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-\frac{\varepsilon}{T}})-\partial_{s}\operatorname{Li}_{s}(-e^{\frac{\varepsilon}{T}})\right]\bigg{|}_{s=3},\end{aligned}$
(2.80)

where

$\displaystyle\begin{aligned}
C_{1}\equiv\frac{1}{32}\int_{0}^{1}dy\frac{1}{y^{3}}\left[\dfrac{1-\frac{y}{2}\ln\left(\dfrac{1+y}{1-y}\right)}{\left(1-\frac{y}{2}\ln\left(\dfrac{1+y}{1-y}\right)\right)^{2}+\left(\frac{\pi}{2}y\right)^{2}}-\left(1+\frac{4-\pi^{2}}{4}y^{2}\right)\right]+\frac{-16+3\pi^{2}-4(4-\pi^{2})\ln
2}{512}\approx 0.059.\end{aligned}$ (2.81)

In the limits where $|\varepsilon|/T\ll 1$ and $|\varepsilon|/T\gg 1$, Eq.
2.80 reduces to, respectively,

$\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(T\gg|\varepsilon|)=\,&\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)\varepsilon-\frac{4-\pi^{2}}{192}\pi^{2}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{T}{E_{\mathsf{F}}}\right)\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}}\\\
&-\left[\frac{4-\pi^{2}}{384}\left(3-2\gamma_{E}-2\ln
2\right)-\frac{2}{3}C_{1}+\frac{4-\pi^{2}}{32\pi^{2}}\left(\frac{\pi^{2}}{6}\ln
2+\zeta^{\prime}(2)\right)\right]\pi^{2}\frac{T^{2}\varepsilon}{E_{\mathsf{F}}^{2}},\end{aligned}$
(2.82a) $\displaystyle\begin{aligned}
\operatorname{Re}\Sigma^{(R)}(|\varepsilon|\gg
T)=\,&\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)\varepsilon-\frac{4-\pi^{2}}{192}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{|\varepsilon|}{E_{\mathsf{F}}}\right)\frac{\varepsilon^{3}}{E_{\mathsf{F}}^{2}}\\\
&+\left[\frac{4-\pi^{2}}{576}\left(1+3\ln
2\right)+\frac{2}{3}C_{1}\right]\frac{\varepsilon^{3}}{E_{\mathsf{F}}^{2}}.\\\
\end{aligned}$ (2.82b)

###  2.4 Discussion of the analytical results

It may be worthwhile to summarize our analytical findings for the self-energy
at low temperatures and energies (and to the leading-order in $r_{s}$).

Both in 2D and 3D, $\operatorname{Im}\Sigma^{(R)}$ goes as $T^{2}$ for
$\varepsilon=0$ and as $\varepsilon^{2}$ for $T=0$ (with an additional log
correction in 2D) in the leading order, as is already well-known. This
establishes the perturbative stability of the Fermi surface, implying that
both 3D and 2D interacting systems are Fermi liquids, in contrast to
interacting 1D fermions. The subleading terms in the 3D
$\operatorname{Im}\Sigma^{(R)}$ are $O(T^{3})$ for $\varepsilon=0$ and
$O(\varepsilon^{3})$ for $T=0$. In 2D, however, the corresponding subleading
terms are $O(T^{2})$ and $O(\varepsilon^{2})$, respectively since the leading
order terms go as $O(T^{2}\ln T)$ and $O(\varepsilon^{2}\ln\varepsilon)$. The
next order terms in 2D are cubic, as expected.

When energy and temperature are comparable, the results for
$\operatorname{Im}\Sigma^{(R)}$ are complicated, involving logarithmic
integrals in $\exp(-\varepsilon/T)$ in addition to powers of $\varepsilon$ and
$T$ along with log factors in 2D. In 3D (Eq. 2.77),
$\operatorname{Im}\Sigma^{(R)}$ goes as $(\varepsilon^{2}+\pi^{2}T^{2})$ plus
term involving logarithmic integrals. In 2D (Eq. 2.69),
$\operatorname{Im}\Sigma^{(R)}$ for small, but comparable, $\varepsilon$ and
$T$, goes as $(\varepsilon^{2}+\pi^{2}T^{2})\ln(T/E_{\mathsf{F}})$ with
additional terms involving $O(T^{2})$, $O(\varepsilon^{2})$, and logarithmic
integrals.

The analytical behavior of $\operatorname{Re}\Sigma^{(R)}$ is as follows.

In 3D, for $\varepsilon\ll T$, $\operatorname{Re}\Sigma^{(R)}$ goes as
$O(\varepsilon)+O(\varepsilon T^{2}\ln T)+O(\varepsilon T^{2})$, whereas for
$\varepsilon\gg T$, it goes as
$O(\varepsilon)+O(\varepsilon^{3}\ln\varepsilon)+O(\varepsilon^{3})$. In 2D,
for $\varepsilon\ll T$, $\operatorname{Re}\Sigma^{(R)}$ goes as
$O(\varepsilon)+O(\varepsilon T)+O(\varepsilon T^{2}\ln T)+O(\varepsilon
T^{2})$, whereas for $\varepsilon\gg T$, it goes as
$O(\varepsilon)+O(\varepsilon^{2})+O(\varepsilon^{3}\ln\varepsilon)+O(\varepsilon^{3})$.

When energy and temperature are comparable, but both small, the behavior of
$\operatorname{Re}\Sigma^{(R)}$ is complicated with the appearance of
logarithmic integrals similar to the situation for
$\operatorname{Im}\Sigma^{(R)}$ discussed above. In 3D,
$\operatorname{Re}\Sigma^{(R)}$ then goes as
$O(\varepsilon)+O([\varepsilon^{3}+\pi^{2}\varepsilon T^{2}]\ln
T)+O(\varepsilon^{3}+\pi^{2}\varepsilon T^{2})$ plus terms involving
logarithmic integrals as in Eq. 2.80. In 2D, for general small values of
energy and temperature, $\operatorname{Re}\Sigma^{(R)}$ behaves as
$O(\varepsilon)+O([\pi^{2}T^{2}\varepsilon+\varepsilon^{3}]\ln
T)+O(\pi^{2}T^{2}\varepsilon+\varepsilon^{3})$ plus several terms involving
logarithmic integrals as shown in Eq. 2.73.

The presence of various logarithmic terms and combinations of powers of $T$
and $\varepsilon$ along with logarithmic integrals made the calculation of the
self-energy a challenge for arbitrary (but small) energy and temperature even
in the $r_{s}\ll 1$ limit for the last 60 years [6], which we finally managed
to resolve [9].

The $O(\varepsilon^{2})$ or $O(\varepsilon^{2}\ln\varepsilon)$ asymptotic
behavior of $\operatorname{Im}\Sigma^{(R)}$ in 3D or 2D respectively assures
the existence of a Fermi surface at $\varepsilon=0$ since
$\operatorname{Re}\Sigma^{(R)}$ always goes as $O(\varepsilon)$. The question
we address in the rest of this paper is what happens at finite temperature and
energy where $\operatorname{Im}\Sigma^{(R)}$ in principle could be larger than
energy or temperature itself, indicating that the quasiparticle picture fails
there.

Our goal is to ascertain the domain of validity of the Fermi liquid theory and
the associated quasiparticle picture by comparing the quasiparticle energy
$\varepsilon$ with the quasiparticle damping defined by the magnitude of
$\operatorname{Im}\Sigma^{(R)}(\varepsilon)$, both in 3D and 2D, contrasting
the two cases.

##  3 Results

###  3.1 Applicability of the FL theory

The Fermi liquid theory describes interacting Fermi systems at low
temperatures and excitation energies in terms of quasiparticles - long-lived
elementary excitations which are adiabatically connected to the excitations of
noninteracting systems. Therefore it relies on the existence of well-defined
quasiparticles, whose damping rates, determined by the imaginary part of the
self-energy, should be small compared with their energies. More specifically,
the quasiparticle description is valid when the magnitude of the imaginary
part of the self-energy $|\operatorname{Im}\Sigma^{(R)}(\varepsilon)|$ is
small compared with $\varepsilon$. To the leading order,
$\operatorname{Im}\Sigma^{(R)}(\varepsilon)$ scales as
$\varepsilon^{2}\ln\varepsilon$ and $\varepsilon^{2}$, respectively, in 2D and
3D (see Refs. [8, 17, 18, 19, 20, 21], as well as Eqs. 2.71 and 2.79b).
Therefore, at sufficiently low energy $\varepsilon$, the criterion
$|\operatorname{Im}\Sigma^{(R)}(\varepsilon)|<\varepsilon$ is satisfied and
the quasiparticle is well defined. However, at higher energy or temperature,
this might no longer be the case. In this section, we use the previously
obtained analytical expressions for the electron self-energy to determine the
regime in which the quasiparticle description is applicable (invalid) - called
Fermi-liquid (FL) or non-Fermi liquid (NFL) regime in the following. We
compute the on-shell $\operatorname{Im}\Sigma^{(R)}(\varepsilon)/\varepsilon$
at zero temperature as well as $\operatorname{Im}\Sigma^{(R)}(T)/T$ at the
Fermi level (i.e. $\varepsilon=0$), in an attempt to determine the crossover
energy $\varepsilon_{c}$ and crossover temperature $T_{c}$ below which the
quasiparticles are well defined. $\varepsilon_{c}$ and $T_{c}$ separate the FL
and NFL regimes, and are determined by the conditions:

$\displaystyle\begin{aligned}
-\operatorname{Im}\Sigma^{(R)}(\varepsilon_{c},T=0)/\varepsilon_{c}=1,\qquad-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T_{c})/T_{c}=1.\end{aligned}$
(3.1)

Of course, the precise magnitudes of $\varepsilon_{c}$ and $T_{c}$ will depend
on the expressions for the self-energy we use, and the leading-order and sub-
leading-order theories may give different results, but our interest is in
understanding the qualitative trends being mindful of the fact that the
analytical theory is meaningful only up to energy/temperature where the
subleading terms are smaller than the leading terms and the constraint
$\varepsilon,T<E_{\mathsf{F}}$ applies. Strictly speaking, the analytical
expressions for the electron self-energy presented in the previous section are
derived within RPA, and are valid in the high-density, low-energy and low-
temperature regime. However, it has been found that the RPA approximation
works reasonably well even outside the high density regime. For example, in
Ref. [10], it has been shown that, for an electron gas at metallic densities,
the effective mass, Pauli spin susceptibility and compressibility obtained
from RPA are in good agreement with the experiment. We note that
$\varepsilon/E_{\mathsf{F}}r_{s}$ and $T/E_{\mathsf{F}}r_{s}$
($\varepsilon/E_{\mathsf{F}}\sqrt{r_{s}}$ and $T/E_{\mathsf{F}}\sqrt{r_{s}}$)
are used, apart from $r_{s}$, as the small expansion parameters in the
calculation of the 2D (3D) self-energy. Larger $r_{s}$ therefore means wider
energy and temperature ranges of applicability (for not too large value of
$r_{s}$). For these reasons, in this section, we assume that the previously
obtained analytical expressions for the self-energy apply to arbitrary
$r_{s}$, $\varepsilon/E_{\mathsf{F}}$ and $T/E_{\mathsf{F}}$, and use them to
estimate when the quasiparticle description breaks down. Our goal is to
determine the energy/temperature regime where the Fermi liquid theory applies
within our approximations.

Figure 2: (a) $-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon$ as a
function of $\varepsilon/E_{\mathsf{F}}$ at $T=0$ in 2D; (b)
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/T$ as a function of
$T/E_{\mathsf{F}}$ at $\varepsilon=0$ in 2D; (c) and (d) are same as (a) and
(b), respectively, but in $d=3$ dimensions. (a) [(c)] is obtained using the
zero temperature self-energy expression Eq. 2.71 (Eq. 2.79b) up to the
$\varepsilon^{2}$ term, while (b) [(d)] is from zero energy self-energy
expression Eq. 2.70 (Eq. 2.79a) up to the $T^{2}$ term. In all four panels, we
consider $r_{s}=0.1,0.5,1.0,2.0,5.0,10.0$ represented by solid lines with
different colors. The horizontal dashed lines take the value of $1$ and
separate the FL and NFL regimes where the quasiparticle description is
applicable and invalid, respectively.

In Fig. 2(a) (Fig. 2(c)), the ratio
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon$ at $T=0$ is
plotted as a function of the dimensionless energy $\varepsilon/E_{\mathsf{F}}$
for 2D (3D) electron systems for various interaction parameter $r_{s}$ values.
For Fig. 2(a), we apply Eq. 2.71, the analytical expression for zero
temperature $\operatorname{Im}\Sigma^{(R)}$ in 2D, and retain the leading
order $\varepsilon^{2}\ln\varepsilon$ term as well as the subleading
$\varepsilon^{2}$ term, while for Fig. 2(c), only the leading order
$\varepsilon^{2}$ term in the zero temperature expression for the imaginary
part of the 3D self-energy (Eq. 2.79b) is used. Figs. 2(b) and 2(d) are
plotted in a similar way. Instead of
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$, we plot
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ as a function of
$T/E_{\mathsf{F}}$ for the same set of $r_{s}$ values for both 2D (Fig. 2(b))
and 3D (Fig. 2(d)) systems. For these two figures, we apply Eq. 2.70 and Eq.
2.79a, which give the zero-energy form of $\operatorname{Im}\Sigma^{(R)}$ for
$d=2$ and $d=3$, respectively, and neglect the highest order $T^{3}$ terms in
both equations.

In Figs. 2(a) and 2(b), which depict the 2D case, almost all curves lie below
the horizontal dashed line with the value of $1$ (except for the one
corresponds to $r_{s}=10$ in Fig. 2(b)). This means that the conditions
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon<1$ and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T<1$ always hold in the
observed regime. However, as $\varepsilon/E_{\mathsf{F}}$ $(T/E_{\mathsf{F}})$
increases, the self-energy expression used to plot Fig. 2(a) (Fig. 2(b)) is no
longer valid, and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
$\left(-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T\right)$ becomes
negative for large enough $\varepsilon$ ($T$) for almost all $r_{s}$
considered. This implies that the corresponding approximation breaks down at
high energy, but quasiparticles remain well-defined up to the energy cut off
where the theory is valid. In Fig. 2(b), the purple curve which represents the
case of $r_{s}=10$ intersects with $-\operatorname{Im}\Sigma^{(R)}/T=1$ line
(dashed line) at $T_{c}(r_{s}=10)\approx 1.25E_{\mathsf{F}}$, above which the
quasiparticle is no longer well defined (for this value of $r_{s}$). If we
smoothly extrapolate the 2D results in Figs. 2(a) and 2(b) from their low
energy monotonic behavior (before the pathological maxima induced by the
failure of the expansion), then the results for different $r_{s}$ values all
smoothly cross the dashed line (i.e. unity) for increasing
$\varepsilon/E_{\mathsf{F}}$ and $T/T_{\mathsf{F}}$ with decreasing $r_{s}$,
implying that the Fermi liquid regime is larger for lower $r_{s}$, which is
understandable since lower $r_{s}$ indicates weaker interactions.

In 3D (Figs. 2(c) and 2(d)),
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
($-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$) from the leading order
zero temperature (zero energy) self-energy expression remains positive for all
$\varepsilon$ ($T$). Furthermore, for each $r_{s}$, there exists a
$\varepsilon_{c}$ ($T_{c}$) above which the ratio
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
$\left(-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T\right)$ exceeds $1$,
signaling the break down of the quasiparticle description. Substituting the
leading order terms in Eq. 2.79b and 2.79a into Eq. 3.1, we have

$\displaystyle\begin{aligned}
\frac{\varepsilon_{c}^{(3D)}}{E_{\mathsf{F}}}=\,32(\frac{12}{\pi^{4}})^{1/3}\frac{1}{\sqrt{r_{s}}},\qquad\frac{T_{c}^{(3D)}}{E_{\mathsf{F}}}=\,32(\frac{12}{\pi^{10}})^{1/3}\frac{1}{\sqrt{r_{s}}}.\end{aligned}$
(3.2)

From this result, one can see that $\varepsilon_{c}^{(3D)}/E_{\mathsf{F}}$ and
$T_{c}^{(3D)}/E_{\mathsf{F}}$ have a $r_{s}$-dependence of $1/\sqrt{r_{s}}$
and decay with increasing $r_{s}$, in accordance with Figs. 2(c) and 2(d). The
regime of Fermi liquid validity shrinks in energy with increasing $r_{s}$
consistent with increasing interaction strength in the system. We emphasize
that this result is only an approximation from the leading order 3D self-
energy expressions valid for a small range of temperatures and energies.

Figure 3: Plots of
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$ in the
$(r_{s},\varepsilon/E_{\mathsf{F}})$-plane (left panels), and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ in the
$(r_{s},T/E_{\mathsf{F}})$-plane (right panels) in 2D (upper panels) and 3D
(lower panels). For panels (a)-(d), Eqs. 2.71, 2.70, 2.79b and 2.79a are used,
in respective order, and the highest order $T^{3}$ or $\varepsilon^{3}$ terms
in these equations are neglected. In the left (right) panels, the FL regime
where $0<-\operatorname{Im}\Sigma(\varepsilon,T=0)/\varepsilon<1$
($0<-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T<1$) is indicated by the
blue area, whereas the NFL regime where
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon>1$
($-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T>1$) is represented by the
yellow area. Their boundary is indicated by red contour line which corresponds
to $\varepsilon_{c}$ ($T_{c}$) defined in Eq. 3.1. In the region colored in
orange, which is separated from the rest by green contour with the value of
zero, the approximated self-energy expressions are no longer valid and lead to
a positive value of $\operatorname{Im}\Sigma^{(R)}$.

Using the same formulas for the imaginary part of the self-energy (given by an
expansion to order $\varepsilon^{2}$ for the zero temperature case and to
order $T^{2}$ for the zero energy case), in Fig 3, we show contour plots of
$\operatorname{Im}\Sigma^{(R)}/\varepsilon$ at $T=0$ in the
$(r_{s},\varepsilon/E_{\mathsf{F}})$-plane (left panels), as well as
$-\operatorname{Im}\Sigma^{(R)}/T$ at $\varepsilon=0$ in the
$(r_{s},T/E_{\mathsf{F}})$-plane (right panels) for $d=2$ (upper panels) and
$d=3$ (lower panels). In the left (right) panels, the blue areas are used to
indicate the FL regimes where $0<-\operatorname{Im}\Sigma^{(R)}/\varepsilon<1$
($0<-\operatorname{Im}\Sigma^{(R)}/T<1$), whereas the yellow areas correspond
to the NFL regimes where $-\operatorname{Im}\Sigma^{(R)}/\varepsilon>1$
($-\operatorname{Im}\Sigma^{(R)}/T>1$). They are separated by the red contour
lines which take the value of one and correspond to $\varepsilon_{c}$
($T_{c}$) defined by Eq. 3.1. As shown in Figs. 3(c) and 3(d),
$\varepsilon_{c}^{(3D)}/E_{\mathsf{F}}$ and $T_{c}^{(3D)}/E_{\mathsf{F}}$
decreases monotonically with increasing $r_{s}$ in 3D, consistent with Eq.
3.2. In Figs. 3(a) and 3(b), there exist regimes where the self-energy
expressions are no longer valid and the resulting 2D
$\operatorname{Im}\Sigma^{(R)}$ is positive. We use the orange areas to
indicate such regimes, where the current self-energy expressions are unable to
tell if the quasiparticle is well defined or not. Furthermore, using the
current approximation for $\operatorname{Im}\Sigma^{(R)}$, the solution to Eq.
3.1 \- $\varepsilon_{c}^{(2D)}$ (or $T_{c}^{(2D)}$ for small $r_{s}$) does not
exist (see Figs. 3(a) and 3(b)) . In this case,
$-\operatorname{Im}\Sigma^{(R)}/\varepsilon<1$
($-\operatorname{Im}\Sigma^{(R)}/T<1$) always holds in the region where this
formula is applicable. We mention that if we use extrapolations of the 2D
self-energy from their low energy behavior so that the pathological behavior
of the imaginary self-energy decreasing at higher energies and temperatures is
eliminated (so that the orange region in the figures disappears), then the 2D
results in Fig. 3 look qualitatively the same as the 3D results with the
orange regions in Fig. 3(a) and 3(b) mostly becoming blue.

Figure 4:  The regimes (green areas) where the applicable criterion -
$E^{3}(T^{3})$ term must be smaller than $E^{2}(T^{2})$ term - is satisfied
for self-energy expressions Eq. 2.71 (panel a), Eq. 2.70 (panel b), Eq. 2.79b
(panel c) and Eq. 2.79a (panel d).

As explained earlier, the self-energy expressions used to obtain Figs. 2 and 3
are applicable to a small range of energies and temperatures because of the
leading order nature of the analytical expansion. We now retain the $T^{3}$
and $\varepsilon^{3}$ terms in the self-energy formulas Eqs. 2.71, 2.70, 2.79b
and 2.79a to evaluate $\operatorname{Im}\Sigma^{(R)}/\varepsilon$ and
$\operatorname{Im}\Sigma^{(R)}/T$ and to analyze the applicable regime of the
FL quasiparticle description. We note that these expressions have a wider
range of applicability compared with the one used in Figs. 2 and 3, but are
only valid when the $T^{3}$ ($\varepsilon^{3}$) term is smaller than the
$T^{2}$ ($\varepsilon^{2}$) term. We find that the leading upper bounds for
the regimes satisfying this condition for the 2D and 3D cases are given by,
respectively,

$\displaystyle\begin{aligned}
\frac{\varepsilon_{3}^{(2D)}}{E_{\mathsf{F}}}=\,&\frac{3\sqrt{2}\left(\ln
4-1\right)}{8}r_{s},\qquad&\frac{T_{3}^{(2D)}}{E_{\mathsf{F}}}=\,&\frac{\pi^{2}\left(6+\ln
2\pi^{3}-36\ln\mathrm{A}\right)}{42\sqrt{2}\zeta(3)}r_{s},\\\
\frac{\varepsilon_{3}^{(3D)}}{E_{\mathsf{F}}}=\,&\frac{(\frac{9\pi}{4})^{1/3}}{C_{0}}\sqrt{r_{s}},\qquad&\frac{T_{3}^{(3D)}}{E_{\mathsf{F}}}=\,&\frac{(\frac{\pi^{7}}{12})^{1/3}}{7\zeta(3)C_{0}}\sqrt{r_{s}}.\end{aligned}$
(3.3)

We note that, in 2D, within the applicability range
$\varepsilon<\varepsilon_{3}^{(2D)}$ ($T<T_{3}^{(2D)}$), the leading order
$\varepsilon^{2}\ln\varepsilon$ ($T^{2}\ln T$) term is also larger than
$\varepsilon^{2}$ ($T^{2}$) term. $\varepsilon_{3}$ (left panels) and $T_{3}$
(right panels) are plotted by the dashed green curves in Fig. 4 for 2D (upper
panels) and 3D (lower panels) systems. The green areas in this figure
represent the regimes where $E^{3}(T^{3})$ term in the self-energy expressions
is always smaller than $E^{2}(T^{2})$ term.

Figure 5: The same as Fig. 2 but retaining the highest order $T^{3}$ and
$\varepsilon^{3}$ terms in Eqs. 2.71, 2.70, 2.79b and 2.79a. Figure 6: The
same as Fig. 3 but using the self-energy expressions which are valid up to
$\varepsilon^{3}$ and $T^{3}$ (i.e. keeping all terms in Eqs. 2.71, 2.70,
2.79b and 2.79a).

In Figs. 5 and 6, we replot
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$ and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ as in Figs. 2 and 3, but
with the self-energy expressions which include the $T^{3}(\varepsilon^{3})$
term. As in Fig. 2, different curves in Fig. 6 correspond to different values
of $r_{s}$. For each curve, we use a vertical dotted line of the same color to
indicate the upper limit $\varepsilon_{3}$ or $T_{3}$ (Eq. 3.3) of the
applicable range for the corresponding self-energy expression. We notice that,
for the 2D case, these curves exhibit nonmonotonic dependence on $r_{s}$,
which will disappear if we incorporate the $r_{s}$ dependence of the Fermi
energy ($E\sim r_{s}^{-2}$) and plot in fixed units instead (see Figs. 7(a)
and 7(b)). In the observed regime, for the 3D case, the ratios
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$ and
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$ are always smaller than
one and become negative (similar to what happens for 2D results in Fig. 2)
outside the applicable ranges of the self-energy expressions (see Figs. 5(c)
and 5(d)), whereas for the 2D case these ratios exceed one at higher energies
and temperatures (see Figs. 5(a) and 5(b)).

In Fig. 6, the blue, yellow and orange areas correspond to, respectively, the
regimes where the ratio
$-\operatorname{Im}\Sigma^{(R)}(\varepsilon,T=0)/\varepsilon$
($-\operatorname{Im}\Sigma^{(R)}(\varepsilon=0,T)/T$) lies within $(0,1)$,
stays above one, and remains negative (i.e. theoretically inapplicable), as in
Fig. 3. We also add a green area in each panel to indicate the region where
the corresponding self-energy expression is applicable (see Eq. 3.3). In Figs.
6(a) and 6(b) which are associated with the 2D case, we find both FL and NFL
regimes, indicated by the blue and yellow areas, respectively. The red contour
lines take the value of one and represent $\varepsilon_{c}^{(2D)}$ (Fig. 6
(a)) or $T_{c}^{(2D)}$ (Fig. 6 (b)). As one can see from these figures, over
some range of $r_{s}$ values, $\varepsilon_{c}^{(2D)}/E_{\mathsf{F}}$
($T_{c}^{(2D)}/E_{\mathsf{F}}$) grows with increasing $r_{s}$ which seems
contrary to a naive expectation that the quasiparticle description breaks down
at lower energy and temperature for larger interaction strength. However,
taking into account the $r_{s}$ dependence of $E_{\mathsf{F}}$, we find that
$\varepsilon_{c}^{(2D)}$ and $T_{c}^{(2D)}$ in fixed units decay with
increasing $r_{s}$ as expected. See Figs. 7(c) and 7(d) where we replot Figs.
6(a) and 6(b) in fixed units. We also note that $\varepsilon_{c}^{(2D)}$ and
$T_{c}^{(2D)}$ stay outside the applicable regime of the self-energy
expression (green area), and are therefore not reliable. This result shows
that within the applicable regime, the FL quasiparticle description is valid.
In the lower panels of Fig. 6 which depict the 3D cases, the yellow areas
disappear. Instead, one finds orange areas which correspond to the regimes
where the associated self-energy expressions are invalid and yield
$\operatorname{Im}\Sigma^{(R)}>0$. As before, we have no information about
whether the quasiparticle is well defined or not in the orange areas. The
current analytical theory is simply inapplicable in the orange areas. It is,
in principle, possible that the Fermi liquid theory remains valid with well-
defined quasiparticles for all energies and temperatures since our analytical
theory cannot access arbitrary quasiparticle energy and temperature well above
$E_{\mathsf{F}}$.

Figure 7: The same as Figs. 5(a)-(b) and 6(a)-(b) but plotted in fixed units.
Note that the nonmonotonic $r_{s}$ dependence in Fig. 5(a)-(b) disappears
after using the fixed units.

One compelling conclusion of the results presented in Figs. 2 -7 is that the
Fermi liquid theory and the quasiparticle picture are extremely robust in both
2D and 3D interacting systems, generically remaining valid at least up to an
energy $E_{\mathsf{F}}$ above the Fermi level. In general, the regime of this
robustness decreases with increasing $r_{s}$, but even for $r_{s}\sim 10$,
where our RPA theory is suspect, the quasiparticles remain well-defined (i.e.,
the magnitude of $\operatorname{Im}\Sigma^{(R)}$ is less than $\varepsilon$
and/or $T$) up to an energy $E_{\mathsf{F}}$ above the Fermi level. The
stability is quantitatively slightly weaker in 2D than in 3D, but the
difference is not significant enough to draw any conclusion.

###  3.2 Effective mass

As mentioned earlier, in the Fermi liquid theory, an interacting electron
system is composed of quasiparticles whose effective mass is different from
the bare mass of electron but is renormalized by the electron-electron
interactions. The effective mass of the quasiparticle is a fundamental
parameter in the Fermi liquid theory, and has been extensively studied before.
See for example, Refs. [8, 10, 22, 23, 24, 25, 26] for electron systems with
Coulomb interactions and Refs. [11, 12] for the case of the short-range
interactions. In this section, we use the analytical expressions for the real
part of the self-energy provided in Sec. 2 to rederive the effective mass for
Coulomb interactions. We also provide a higher order $T^{2}$ correction to the
previous result [25] in both 2D and 3D.

To the leading order in dynamically screened interactions, the effective mass
can be obtained from the real part of the self-energy using the following
formula [25]:

$\displaystyle\begin{aligned}
\frac{m^{*}(T)}{m}=\dfrac{1-\frac{\partial}{\partial\varepsilon}\operatorname{Re}\Sigma^{(R)}(\varepsilon,\xi_{\bm{\mathrm{k}}})}{1+\frac{\partial}{\partial\xi_{\bm{\mathrm{k}}}}\operatorname{Re}\Sigma^{(R)}(\varepsilon,\xi_{\bm{\mathrm{k}}})}\bigg{|}_{\varepsilon=\xi_{\bm{\mathrm{k}}}=0}\approx
1-\left(\frac{\partial}{\partial\varepsilon}+\frac{\partial}{\partial\xi_{\bm{\mathrm{k}}}}\right)\operatorname{Re}\Sigma^{(R)}(\varepsilon,\xi_{\bm{\mathrm{k}}})\bigg{|}_{\varepsilon=\xi_{\bm{\mathrm{k}}}=0}.\end{aligned}$
(3.4)

Here $m^{*}$ and $m$ denote the effective mass and bare mass, respectively.

Inserting the low energy limit ($\varepsilon\ll T$) expression for the real
part of the on-shell ($\varepsilon=\xi_{\bm{\mathrm{k}}}$) self-energy (Eq.
2.75) into the equation above, we obtain the effective mass for 2D systems

$\displaystyle\begin{aligned}
\frac{m^{*}(T)}{m}=\,&1-\frac{r_{s}}{\sqrt{2}\pi}\ln\left(\frac{2\sqrt{2}}{r_{s}}\right)+\frac{\ln
2}{4}\frac{T}{T_{\mathsf{F}}}-\frac{5\pi}{48\sqrt{2}r_{s}}\frac{T^{2}}{T_{\mathsf{F}}^{2}}\ln\left(\frac{r_{s}T_{\mathsf{F}}}{T}\right)\\\
&-\left[-\frac{\pi}{96\sqrt{2}}\left(32-10\gamma_{E}-25\ln
2\right)-\frac{5}{8\sqrt{2}\pi}\left(\zeta^{\prime}(2)+\frac{\pi^{2}}{6}\ln
2\right)\right]\frac{T^{2}}{T_{\mathsf{F}}^{2}r_{s}}.\end{aligned}$ (3.5)

This expression is valid for quasiparticles with $\varepsilon\ll T$ in the
regime of $r_{s}\ll 1$ and $r_{s}^{3/2}\ll T/E_{\mathsf{F}}\ll r_{s}$.
Similarly, applying Eq. 2.82a, we find that the 3D effective mass takes the
following form for $r_{s}\ll{T}/{T_{\mathsf{F}}}\ll\sqrt{r_{s}}$:

$\displaystyle\begin{aligned}
\frac{m^{*}(T)}{m}=\,&1-\frac{1}{3\pi^{4/3}}(\frac{2}{3})^{2/3}r_{s}\ln\left(\frac{3\pi^{2}}{2r_{s}^{3/2}}\right)+\frac{4-\pi^{2}}{192}\pi^{2}\frac{T^{2}}{E_{\mathsf{F}}^{2}}\ln\left(\frac{1}{2(\frac{16}{3\pi^{2}})^{1/3}\sqrt{r_{s}}}\frac{T}{E_{\mathsf{F}}}\right)\\\
&+\left[\frac{4-\pi^{2}}{384}\left(3-2\gamma_{E}-2\ln
2\right)-\frac{2}{3}C_{1}+\frac{4-\pi^{2}}{32\pi^{2}}\left(\frac{\pi^{2}}{6}\ln
2+\zeta^{\prime}(2)\right)\right]\pi^{2}\frac{T^{2}}{E_{\mathsf{F}}^{2}}.\end{aligned}$
(3.6)

We note that the applicable temperature range of this result is different from
the one in Ref. [25]. Our current results include both leading and subleading
temperature corrections to the renormalized quasiparticle effective mass in
both 2D and 3D.

We emphasize that, in 2D, the leading order temperature correction to the
effective mass is linear in $T$ and independent of $r_{s}$ [24, 25]. This
linear-in-$T$ correction has also been found for 2D systems with short-range
interactions [11, 12]. By contrast, for 3D systems, the leading order
temperature correction is of the order of $T^{2}\ln T$, much smaller compared
with its 2D counterpart. The corresponding coefficient is also
$r_{s}$-independent. The appearance of the linear-in-$T$ leading order
effective mass renormalization in 2D compared with the leading order $T^{2}\ln
T$ renormalization in 3D implies much stronger interaction effects in 2D
compared with 3D. But, this stronger effective 2D interaction does not imply
any failure of the Fermi liquid theory.

###  3.3 Hydrodynamic and ballistic regimes

Hydrodynamics is a useful approach to describe physical processes of
interacting systems at time and length scales that are much larger compared
with the ones associated with local equilibration, provided that the dominant
microscopic inter-particle scattering process is momentum-conserving. Electron
liquids can be described hydrodynamically if the dominant scattering process
is the electron-electron collision which is responsible for local
equilibration. In this case, the electron system reaches local equilibrium at
the time scale of the electron-electron inelastic scattering time, which is
much shorter than the other time scales in the problem, and therefore obeys
hydrodynamics. By contrast, if electron-impurity and/or electron-phonon
scattering is stronger than the electron-electron scattering so that momentum
conservation is not preserved, hydrodynamics does not apply. We assume here
that the electron-impurity and electron-phonon scattering are negligible in
the system under consideration.

The main condition for hydrodynamics to be applicable to an electron liquid
[27, 28, 29] is that the electron-electron scattering time
$\tau_{\varepsilon}$ is much smaller than the time scale or inverse frequency
of the physical process being studied ($\varepsilon^{-1}$):

$\displaystyle\begin{aligned}
\tau_{\varepsilon}\ll\varepsilon^{-1}.\end{aligned}$ (3.7)

Equivalently, the mean free path $l=v_{\mathsf{F}}\tau_{\varepsilon}$
associated with electron-electron collisions should be much smaller compared
with the external length scale $q^{-1}$:

$\displaystyle l\ll q^{-1}.$ (3.8)

Here energy $\varepsilon$ can be considered as an external frequency for an
experiment probing the response of the system to an applied field, and $q$ is
the corresponding external wavevector which is related to external energy
$\varepsilon$ by $\varepsilon=qv_{\mathsf{F}}$. The collision-dominated regime
where Eq. 3.7 is satisfied is known as the hydrodynamic regime. On the other
hand, the collisionless regime where the hydrodynamics condition Eq. 3.7 no
longer applies is usually called the ballistic regime.

As mentioned earlier, the electron-electron inelastic scattering rate can be
extracted from the imaginary part of the self-energy:

$\displaystyle\begin{aligned}
\tau^{-1}_{\varepsilon}=-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T).\end{aligned}$
(3.9)

We then insert the previously obtained formulas for
$\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)$ for arbitrary $\varepsilon/T$
to investigate the hydrodynamic and ballistic regimes. As in Sec. 3.1, we
assume the self-energy formulas are applicable even for low densities, high
temperatures and high energies.

Figure 8: Boundaries separating the hydrodynamic regimes where
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon>1$ and the
ballistic regimes where
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon<1$ for 2D (upper
panels) and 3D (lower panels) electron systems for
$r_{s}=0.1,0.5,1.0,2.0,5.0,10.0$. The left panels are obtained using all terms
in the self-energy expressions Eqs. 2.68 and 2.77 (which are applicable to
arbitrary $\varepsilon/T$), while for the right panels, the highest order
terms in these equations are neglected. In the
$(\varepsilon/E_{\mathsf{F}},T/E_{\mathsf{F}})$-plane, the hydrodynamics
regimes lie in the low energy (high temperature) part, whereas the ballistic
regime correspond to the high energy (low temperature) part. The inset of
panel (b) zooms in the lower-left (low energy and temperature) part to clearly
show the behavior of $r_{s}=0.1$ case.

Using Eq. 3.7 and Eq. 3.9, it is straightforward to see that the hydrodynamic
regime should follow
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon>1$, whereas the
ballistic regime satisfies
$-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon<1$. In Fig. 8, in
the $(\varepsilon/E_{\mathsf{F}},T/E_{\mathsf{F}})$-plane, we depict contours
given by the following condition for various $r_{s}$

$\displaystyle\begin{aligned}
-2\operatorname{Im}\Sigma^{(R)}(\varepsilon,T)/\varepsilon=1.\end{aligned}$
(3.10)

These contours divide the plane into two parts: the low energy (high
temperature) part corresponds to the hydrodynamic regime, while the high
energy (low temperature) part is associated with the ballistic regime. We
consider both 2D and 3D cases which are shown in the upper and lower panels of
Fig. 8, respectively. For Fig. 8(a) (Fig. 8(c)), all terms in the 2D (3D)
self-energy expression Eq. 2.68 (Eq. 2.77) are used, whereas for Fig. 8(b)
(Fig. 8(d)), the highest order term in that equation is ignored. We note that
the boundaries of the hydrodynamic and ballistic regimes obtained from
different self-energy approximations are quite distinct from each other. Both
approximations are in fact only applicable to sufficiently low energies and
temperatures, and the results for $\varepsilon/E_{\mathsf{F}}>1$ or
$T/E_{\mathsf{F}}>1$ in Fig. 8 are therefore not reliable.

We emphasize that if electron-impurity and/or electron-phonon scattering
effects are present, then one must also compare the electron-electron
scattering rates with these other scattering rates, and hydrodynamics would
apply only if electron-electron scattering is the dominant scattering process.
In real materials it is a challenge to create conditions for the hydrodynamic
regime since electron-impurity and electron-phonon scattering tend to dominate
at low and high temperatures, respectively.

##  4 Wiedemann-Franz (WF) and Kadowaki-Woods (KW) relations for 2D
interacting systems

Given our analytical results for the 2D inelastic scattering rates for
electron-electron interactions, we can briefly comment on the repercussions
for the well-known Wiedemann-Franz (WF) and Kadowaki-Woods (KW) relations in
2D Fermi liquids.

The WF law [30] states that the ratio $L=\kappa/(\sigma T)$, where $\sigma$
$(\kappa)$ is the electronic electrical (thermal) conductivity, is universal
in metals (i.e. Fermi liquids), and the KW law [31] states that the ratio
$K=\mathcal{A}/\gamma^{2}$ is universal, where $\mathcal{A}$ $(\gamma)$ is the
coefficient of the $T^{2}$ term (the linear $T$ term) in the temperature
dependence of the resistivity (specific heat). Both of these universalities
are obeyed rather widely in metals (for the KW law, the metal must show a
dominant $T^{2}$ dependence in its resistivity limiting it to the so-called
strongly correlated materials such as transition metals, heavy fermion
compounds and metallic oxides).

Both of these quantities involve the electronic dc conductivity $(\sigma)$ or
the resistivity ($\rho=1/\sigma$), which is operationally defined by the
following Drude formula:

$\displaystyle\sigma=\frac{ne^{2}\tau}{m^{*}}.$ (4.1)

Here, $n$ is the 2D carrier density and $m^{*}$ is the effective mass, and the
key quantity is the transport relaxation time $\tau$ which depends on the
details of the scattering process. In ordinary metals and doped
semiconductors, transport is mostly limited by electron-impurity (at lower
temperatures) and electron-phonon (at higher temperatures) scattering of the
carriers. This is consistent with the observed temperature-independence
(arising from impurity scattering) and linear-in-$T$ temperature dependence
(arising from phonon scattering in the equipartition regime) of the electrical
resistivity in simple metals and doped semiconductors at low and high
temperatures, respectively. Consideration of electron-impurity and electron-
phonon interactions is beyond the scope of the current work - see, e.g., Refs.
[32, 33, 34].

Our focus in the current work is electron-electron interaction, which, by
virtue of the Galilean invariance in a continuum system, cannot affect the
resistivity since electron-electron scattering usually conserves total
momentum of the system (and transport is a momentum relaxation phenomenon). It
is, however, often claimed in the literature that a hallmark of a Fermi liquid
is the manifestation of a dominant $T^{2}$ temperature dependence of the
resistivity, with the temperature-dependent part of the resistivity going
primarily as

$\displaystyle\rho(T)=1/\sigma(T)=\mathcal{A}T^{2}.$ (4.2)

Although no simple metal or 2D doped semiconductor shows such a $T^{2}$
temperature dependence in the resistivity, many strongly correlated metals
(e.g. transition metals, heavy fermion compounds, various metallic oxides) do.
Since the intraband electron-electron scattering cannot contribute to the
resistivity by virtue of its explicit momentum conserving nature, the
widespread assumption is that the $T^{2}$ resistivity manifesting in many
narrow band metals arises from electron-electron scattering involving umklapp
or interband scattering processes in a Fermi liquid which do not conserve
total momentum. We accept this assumption uncritically, and consider the
consequences for such a $T^{2}$ resistivity arising from electron-electron
scattering for the WF and KW relations using our theory for the inelastic
electron-electron interaction induced scattering rate.

This will involve the assumption that our calculated imaginary part of the on-
shell electron self-energy at the Fermi energy is indeed the appropriate
scattering for the resistivity so that we can make the identification that our
calculated imaginary part of the temperature-dependent electron self-energy at
the Fermi energy defines the transport scattering relaxation time $\tau$
entering the electrical resistivity through:

$\displaystyle 1/\tau=-2\operatorname{Im}\Sigma^{(R)}(0,T).$ (4.3)

Using our analytical results for $\operatorname{Im}\Sigma^{(R)}$ (Eq. 2.70),
we get:

$\displaystyle\frac{1}{2\tau}=\frac{\pi}{8}\frac{T^{2}}{E_{\mathsf{F}}}\ln\left(\frac{\sqrt{2}r_{s}E_{\mathsf{F}}}{T}\right)-\frac{\pi}{24}\left(6+\ln
2\pi^{3}-36\ln\mathrm{A}\right)\frac{T^{2}}{E_{\mathsf{F}}}+\frac{7\zeta(3)}{2\sqrt{2}\pi}\frac{T^{3}}{r_{s}E_{\mathsf{F}}^{2}}.$
(4.4)

With this expression for $\tau$ along with the Drude formula for the dc
resistivity, we now discuss below the WF and KW relations individually for 2D
systems.

###  4.1 WF Law

For the WF law, which is a statement on the ratio of the electrical and
thermal conductivity, we can write [35]

$\displaystyle L=\frac{\kappa}{\sigma
T}=\frac{\pi^{2}}{3}\frac{\tau}{\tau+\tau_{i}}.$ (4.5)

Here, $\tau_{i}$ is the temperature-independent momentum relaxation time due
to electron-impurity elastic scattering which must dominate at $T=0$, where
$\tau(T)$ above, arising from inelastic electron-electron scattering, becomes
infinite. Since $1/\tau\sim-T^{2}\ln(T/T_{F})$, we conclude that the relevant
WF law for 2D strongly correlated systems becomes:

$\displaystyle L\sim\dfrac{L_{0}}{1+B(T/T_{F})^{2}\ln(T/T_{F})}.$ (4.6)

Here the nonuniversal materials and sample dependent constant $B$ [36] depends
both on the strength of the impurity scattering and the prefactor of the
$T^{2}\ln T$ term in our imaginary self-energy, and $L_{0}$ is the ideal
Lorenz number for the WF law which must be recovered at $T=0$ where the
electron-electron scattering vanishes. The important qualitative result,
however, is that the WF law will be strongly violated if the electron-electron
scattering induced inelastic scattering is strong in the system, and at low
temperatures this violation, or equivalently the suppression of the Lorenz
number from its ideal WF value, will follow a
$[1+B(T/T_{F})^{2}\ln(T/T_{F})]^{-1}$ dependence in a Fermi liquid 2D metal
(with $B$ being a materials-dependent nonuniversal number). This violation of
the WF law in 2D metals should be observable for strongly interacting systems
at low enough temperatures, where the electron-phonon interaction is
negligible (but electron-electron interaction is still significant). Note that
we can further refine the violation of the WF law by adding the next-to-
leading-order temperature dependent terms in the imaginary part of the self-
energy (Eq. 2.70). We note that the $\ln T$ term in above 2D equations is most
likely absent in the transport coefficients, but this is beyond the scope of
the current work.

###  4.2 KW law

For the KW law, we need an expression for the 2D electronic specific heat,
which in the leading-order Fermi liquid theory can be written as:

$\displaystyle C_{e}=C(m^{*}T),$ (4.7)

where $C$ depends only on universal constants, and $m^{*}$ is the carrier
effective mass. Thus, $\gamma=Cm^{*}$ for the 2D Fermi liquid. Note that in
2D, carrier density or Fermi energy does not enter the expression for the
electronic specific heat by virtue of the constancy of the 2D density of
states. For the KW relation, we need the ratio $K$ of the $T^{2}$ term in the
resistivity with the linear-in-$T$ term in the specific heat, which is given
by:

$\displaystyle K=\mathcal{A}/(Cm^{*})^{2}$ (4.8)

with $\mathcal{A}$ given by:

$\displaystyle\mathcal{A}=\frac{m^{*}}{ne^{2}}\frac{1}{\tau}\frac{1}{T^{2}}=\mathcal{A}^{\prime}\frac{m^{*}}{n}\frac{1}{E_{F}}\ln\left(\frac{T_{F}}{T}\right),$
(4.9)

where $\mathcal{A}^{\prime}$ depends on universal constants. Remembering that
in 2D Fermi systems, $E_{\mathsf{F}}\sim(n/m^{*})$, we get, combining
everything, for the KW ratio:

$\displaystyle K=\frac{B^{\prime}}{n^{2}}\ln\left(\frac{T_{F}}{T}\right),$
(4.10)

where $B^{\prime}$ is universal within the free electron type Fermi liquid
theories. (Again, the $\ln T$ is most likely absent, but this is beyond the
scope of the current work.) We note that within the free electron type band
structure model for Fermi liquids, the Kadowaki-Woods ratio goes as $n^{-2}$
in 2D in contrast to the corresponding $n^{-7/3}$ dependence in 3D systems.
This precise difference by a factor of $n^{-1/3}$ can be understood as arising
from the dimensional difference between 2D and 3D resistivity definitions: in
2D the resistivity is simply measured in ohms whereas in 3D it is measured in
ohm.cm, so the two units must differ by a unit of length, which is precisely
what $n^{-1/3}$ is for a 3D carrier density $n$. It may be worthwhile to
emphasize that the KW relation is by no means universal either in 2D or in 3D
by virtue of the strong density dependence inherent in the KW ratio. It is
only when different materials have similar effective carrier densities, one
can talk about a universal KW relation, and even then it is rather a dubious
universality since the origin of the $T^{2}$ resistivity term may differ from
system to system.

##  5 Conclusion

We have analytically investigated the domain of validity of the Fermi liquid
theory and the quasiparticle picture in Coulomb interacting continuum 2D and
3D electron liquids. Using the leading-order dynamical screening
approximation, which is exact in the high-density limit, we calculate exact
expressions for the real and imaginary parts of the electron self-energy in
expansions in temperature and energy measured from the Fermi surface. Using
the calculated imaginary part of the self-energy, we estimate the
$r_{s}$-dependent crossover energy and temperature scales above which well-
defined quasiparticles no longer exist. We find that in both 2D and 3D systems
the quasiparticle picture remains valid at energies and temperatures
comparable to Fermi energies or above, implying a robust validity of Fermi
liquid theory, not only in 3D, but also in 2D. In general, the energy scales
for the validity of the Fermi liquid theory decreases with increasing $r_{s}$,
but the theory being exact only for small $r_{s}$, this finding, although
intuitively appealing, is only tentative. Recent work, which calculates both
the on-shell and the off-shell self-energy numerically without making
$\varepsilon\ll E_{\mathsf{F}}$ approximation, has also concluded that the
regime of validity of the 2D Fermi liquid theory and quasiparticle concept is
wide and applies to very high energies [37].

We emphasize that the applicability of the theory for $r_{s}>1$ is irrelevant
to our work since we are theoretically addressing a matter of principle: how
high in $\varepsilon$ ($T$) does the theory still remain valid? All we need is
the controlled nature of the theory, which is true for $r_{s}\ll 1$, in the
weak coupling perturbative RG sense. We mention that RPA itself is a
controlled approximation giving exact results for small $r_{s}$ although it is
known to work very well empirically at metallic densities (with $r_{s}\sim
4-6$). The reason is a likely cancellation of higher-order diagrams, but no
systematic theory exists for $r_{s}>1$ because of the failure of the
$r_{s}$-expansion. Quantum Monte Carlo calculations can be used in some
situations to study the ground state properties, but are inapplicable to the
question of the validity of the Fermi liquid theory (we are asking in this
work) which must study the quasiparticle excitations that cannot be done by
Monte Carlo techniques. What we show here is that the Fermi liquid theory
applies at very high energies away from the Fermi surface as well as at very
high temperatures not only at small $r_{s}$, but also at intermediate to large
$r_{s}$. Our results for $r_{s}>1$ are obviously not exact, but as long as
there is no ground state strong-coupling quantum phase transition, the theory
should remain valid qualitatively at higher $r_{s}$.

We also provide 2D and 3D ‘phase diagrams’ for temperature and frequency,
where the electron liquid crosses over from the collisionless ballistic regime
to the collision dominated hydrodynamic regime neglecting effects of impurity
and phonon scattering, and obtained general expressions for 2D Wiedemann-Franz
and Kadowaki-Woods relations, using our calculated self-energy expressions. We
have also provided effective mass renormalization results for 2D and 3D
systems including leading and subleading temperature corrections, extending
existing results in the literature.

## Acknowledgement

This work is supported by the Laboratory for Physical Sciences (SDS) and the
Simons Foundation “Ultra-Quantum Matter” Research Collaboration (YL).

##  Appendix A Derivation of self-energy formulas using Matsubara technique

For the sake of completeness, in this Appendix we briefly review an
alternative derivation of the self-energy formula Eq. 2.33 using the Matsubara
instead of Keldysh technique. For more details, see for example Refs. [6, 7].

As explained in earlier sections, for electrons interacting via Coulomb
interactions, in the high density limit $r_{s}\ll 1$, the electron self-energy
is given by the RPA self-energy represented by the diagram in Fig. 1(b). In
the Matsubara formalism, the black sold line represents the noninteracting
Matsubara Green’s function for electrons, which acquires the form

$\displaystyle
G_{0}(\bm{\mathrm{k}},i\varepsilon_{n})=\left(i\varepsilon_{n}-\xi_{\bm{\mathrm{k}}}\right)^{-1}.$
(A.1)

Here $\varepsilon_{n}=2\pi(n+1/2)T$ is the fermionic Matsubara frequency. The
red wavy line with a solid dot stands for the dynamically screened RPA
interaction $D$ defined diagrammatically by the Dyson equation shown in Fig.
1(a). In particular, $D$ is related to the bare interaction $V$ (red wavy line
with a open dot) and the polarization bubble $\Pi$ (black bubble) through the
following equation:

$\displaystyle\begin{aligned}
D(\bm{\mathrm{q}},i\omega_{m})=\left[V^{-1}(\bm{\mathrm{q}})-\Pi(\bm{\mathrm{q}},i\omega_{m})\right]^{-1},\end{aligned}$
(A.2)

with $\omega_{m}=2\pi mT$ being the bosonic Matsubara frequency.
$\Pi(\bm{\mathrm{q}},i\omega_{m})$ is given by a product of two bare electron
Green’s functions:

$\displaystyle\begin{aligned}
\Pi(\bm{\mathrm{q}},i\omega_{m})=2T\sum_{\varepsilon_{n}}\int\frac{d^{d}\bm{\mathrm{k}}}{(2\pi)^{d}}G_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},i\varepsilon_{n}+i\omega_{m})G_{0}(\bm{\mathrm{k}},i\varepsilon_{n}),\end{aligned}$
(A.3)

where the overall factor of two arises from the summation over spin indices.
It is then straightforward to see that the RPA self-energy depicted in Fig.
1(b) evaluates to

$\displaystyle\begin{aligned}
\Sigma(\bm{\mathrm{k}},i\varepsilon_{n})=-T\sum_{\omega_{m}}\int\frac{d^{d}\bm{\mathrm{q}}}{(2\pi)^{d}}D(\bm{\mathrm{q}},i\omega_{m})G_{0}(\bm{\mathrm{k}}+\bm{\mathrm{q}},i\varepsilon_{n}+i\omega_{m}).\end{aligned}$
(A.4)

After analytical continuation, the equation above then leads to the retarded
self-energy

$\displaystyle\begin{aligned}
\Sigma^{(R)}(\bm{\mathrm{k}},\varepsilon)=\frac{i}{2}\int\frac{d^{d}\bm{\mathrm{q}}}{(2\pi)^{d}}\int\frac{d\omega}{2\pi}&\left\\{\left[D^{(R)}(\bm{\mathrm{q}},\omega)-D^{(A)}(\bm{\mathrm{q}},\omega)\right]G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\coth\left(\frac{\omega}{2T}\right)\right.\\\
&\left.+D^{(A)}(\bm{\mathrm{q}},\omega)\left[G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)-G_{0}^{(A)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right]\tanh\left(\frac{\omega+\varepsilon}{2T}\right)\right\\}.\end{aligned}$
(A.5)

With the help of the FDT Eqs. 2.9 and 2.28b, one can prove that this equation
is equivalent to the self-energy formula Eq. 2.33 derived in the Keldysh
formalism. Similarly, by analytical continuation, Eq. A.3 can be transformed
to

$\displaystyle\begin{aligned}
\Pi^{(R)}(\bm{\mathrm{q}},\omega)=-i\int\frac{d^{d}\bm{\mathrm{k}}}{(2\pi)^{d}}\int\frac{d\varepsilon}{2\pi}&\left\\{G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\left[G_{0}^{(R)}(\bm{\mathrm{k}},\varepsilon)-G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\right]\tanh\left(\frac{\varepsilon}{2T}\right)\right.\\\
&\left.+\left[G_{0}^{(R)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)-G_{0}^{(A)}(\bm{\mathrm{k}}+\bm{\mathrm{q}},\varepsilon+\omega)\right]G_{0}^{(A)}(\bm{\mathrm{k}},\varepsilon)\tanh\left(\frac{\varepsilon+\omega}{2T}\right)\right\\},\end{aligned}$
(A.6)

which is equivalent to the retarded polarization operator formula Eq. 2.21
derived in Sec. 2.1.

##  Appendix B Integrals involving hyperbolic functions

In this Appendix, we provide analytical results for integrals of the form
$I_{1,2}(a)$ in Eq. 2.66, which are needed for the calculation of electron
self-energy with arbitrary value of $\varepsilon/T$. The results are obtained
with the help of the exponential series expansion of the hyperbolic functions
(Eq. 2.67). In particular, $I_{1,2}(a)$ can be rewritten as

$\displaystyle\begin{aligned}
I_{1}(a)=&2\sum_{k=1}^{\infty}\left[2-(-1)^{k}e^{-2ka}-(-1)^{k}e^{2ka}\right]\int_{0}^{\infty}dxf(x)e^{-2kx}\\\
&+4\sum_{k=1}^{\infty}\int_{0}^{a}dxf(x)(-1)^{k}\cosh\left({2kx-2ka}\right)+2\int_{0}^{a}dxf(x),\end{aligned}$
(B.1a) $\displaystyle\begin{aligned}
I_{2}(a)=\,&2\sum_{k=1}^{\infty}(-1)^{k}\left[e^{-2ka}-e^{2ka}\right]\int_{0}^{\infty}dxf(x)e^{-2kx}\\\
&+4\sum_{k=1}^{\infty}\int_{0}^{a}dxf(x)(-1)^{k}\cosh\left({2kx-2ka}\right)+2\int_{0}^{a}dxf(x).\end{aligned}$
(B.1b)

For both $I_{1}(a)$ and $I_{2}(a)$, the last two terms cancel with each other
due to the fact that $\sum_{k=1}^{\infty}(-1)^{k}\cosh(2kx-2ka)=-1/2$, and we
are left with the first terms in both Eq. B.1a and Eq. B.1b. By inserting the
result of the integral $\int_{0}^{\infty}dxf(x)e^{-2kx}$ into Eq. B.1, one is
able to evaluate $I_{1,2}(a)$ for various $f(x)$:

$\displaystyle\begin{aligned}
&\int_{0}^{\infty}dx\left[2\coth\left(x\right)-\tanh\left(x+a\right)-\tanh\left(x-a\right)\right]x=\,\frac{\pi^{2}}{4}+a^{2},\\\
&\int_{0}^{\infty}dx\left[2\coth\left(x\right)-\tanh\left(x+a\right)-\tanh\left(x-a\right)\right]x\ln
x\\\ &=\,\left(1-\gamma_{E}-\ln
2\right)a^{2}+\frac{\pi^{2}}{12}\left(3-\gamma_{E}-\ln\frac{2}{\pi^{2}}-24\ln\mathrm{A}\right)-\frac{1}{2}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-2a})+\partial_{s}\operatorname{Li}_{s}(-e^{2a})\right]\bigg{\lvert}_{s=2},\\\
&\int_{0}^{\infty}dx\left[2\coth\left(x\right)-\tanh\left(x+a\right)-\tanh\left(x-a\right)\right]x^{2}=\,\zeta(3)-\frac{1}{2}\left[\operatorname{Li}_{3}(-e^{-2a})+\operatorname{Li}_{3}(-e^{2a})\right],\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]=\,2a,\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]x=\,\frac{1}{2}\left[\operatorname{Li}_{2}(-e^{-2a})-\operatorname{Li}_{2}(-e^{2a})\right],\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]x^{2}=\,\frac{2}{3}a^{3}+\frac{\pi^{2}}{6}a,\\\
&\int_{0}^{\infty}dx\left[\tanh(x+a)-\tanh(x-a)\right]x^{2}\ln
x=\,\left(3-2\gamma_{E}-\ln
4\right)\left(\frac{1}{3}a^{3}+\frac{\pi^{2}}{12}a\right)+\frac{1}{2}\left[\partial_{s}\operatorname{Li}_{s}(-e^{-2a})-\partial_{s}\operatorname{Li}_{s}(-e^{2a})\right]\bigg{\lvert}_{s=3}.\end{aligned}$
(B.2)

## References

  * [1] P. W. Anderson, in: L. Yu, J. C. Pati, Q. Shafi, S. R. Wadia (Eds.), Current Trends in Condensed Matter, Particle Physics and Cosmology, in Kathmandu Summer School Lecture Notes, World Scientific, 1990.
  * [2] J. Feldman, H. Knörrer, E. Trubowitz, A two dimensional Fermi liquid. Part 1: Overview, Commun. Math. Phys. 247 (1) (2004) 1–47.
  * [3] J. Feldman, H. Knörrer, E. Trubowitz, A two dimensional Fermi liquid. Part 2: Convergence, Commun. Math. Phys. 247 (1) (2004) 49–111.
  * [4] J. Feldman, H. Knörrer, E. Trubowitz, A two dimensional Fermi liquid. Part 3: The Fermi surface, Commun. Math. Phys. 247 (1) (2004) 113–177.
  * [5] B.-K. Hu, S. Das Sarma, Many-body exchange-correlation effects in the lowest subband of semiconductor quantum wires, Phys. Rev. B 48 (1993) 5469–5504.
  * [6] A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of quantum field theory in statistical physics, Dover, New York, 1975.
  * [7] A. Fetter, J. D. Walecka, Quantum theory of many-particle systems, McGraw-Hill, New York, 1971.
  * [8] V. M. Galitskii, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 151-162 (1958) [Sov. Phys. JETP, 34(7), 104 (1958)].
  * [9] Y. Liao, D. Buterakos, M. Schecter, S. Das Sarma, Two-dimensional electron self-energy: Long-range Coulomb interaction, Phys. Rev. B 102 (2020) 085145\.
  * [10] T. Rice, The effects of electron-electron interaction on the properties of metals, Ann. Phys. 31 (1) (1965) 100 – 129.
  * [11] A. V. Chubukov, D. L. Maslov, Nonanalytic corrections to the Fermi-liquid behavior, Phys. Rev. B 68 (2003) 155113.
  * [12] A. V. Chubukov, D. L. Maslov, Singular corrections to the Fermi-liquid theory, Phys. Rev. B 69 (2004) 121102(R).
  * [13] A. V. Chubukov, D. L. Maslov, First-Matsubara-frequency rule in a Fermi liquid. I. Fermionic self-energy, Phys. Rev. B 86 (15) (2012) 155136.
  * [14] A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press, 2011.
  * [15] F. Stern, Polarizability of a two-dimensional electron gas, Phys. Rev. Lett. 18 (14) (1967) 546.
  * [16] J. Lindhard, On the properties of a gas of charged particles, Dan. Vid. Selsk Mat.-Fys. Medd. 28 (1954) 8.
  * [17] J. J. Quinn, R. A. Ferrell, Electron self-energy approach to correlation in a degenerate electron gas, Phys. Rev. 112 (1958) 812–827.
  * [18] A. Chaplik, Energy spectrum and electron scattering processes in inversion layers, Sov. Phys. JETP 33 (1971) 997–1000.
  * [19] G. F. Giuliani, J. J. Quinn, Lifetime of a quasiparticle in a two-dimensional electron gas, Phys. Rev. B 26 (8) (1982) 4421.
  * [20] L. Zheng, S. Das Sarma, Coulomb scattering lifetime of a two-dimensional electron gas, Phys. Rev. B 53 (15) (1996) 9964.
  * [21] Q. Li, S. Das Sarma, Finite temperature inelastic mean free path and quasiparticle lifetime in graphene, Phys. Rev. B 87 (8) (2013) 085406.
  * [22] M. Gell-Mann, Specific heat of a degenerate electron gas at high density, Phys. Rev. 106 (1957) 369–372.
  * [23] B. Vinter, Correlation energy and effective mass of electrons in an inversion layer, Phys. Rev. Lett. 35 (15) (1975) 1044.
  * [24] S. Das Sarma, V. M. Galitski, Y. Zhang, Temperature-dependent effective-mass renormalization in two-dimensional electron systems, Phys. Rev. B 69 (12) (2004) 125334.
  * [25] V. M. Galitski, S. Das Sarma, Universal temperature corrections to Fermi liquid theory in an interacting electron system, Phys. Rev. B 70 (2004) 035111\.
  * [26] Y. Zhang, S. Das Sarma, Temperature-dependent effective mass renormalization in a Coulomb Fermi liquid, Phys. Rev. B 70 (3) (2004) 035104.
  * [27] D. Pines, P. Nozieres, The theory of quantum liquids, CRC Press, 1989.
  * [28] A. Lucas, S. Das Sarma, Electronic sound modes and plasmons in hydrodynamic two-dimensional metals.
  * [29] S.-K. Jian, S. Das Sarma, Hydrodynamic sound and plasmons in three dimensions, Phys. Rev. B 103 (2021) 155101.
  * [30] R. Franz, G. Wiedemann, Ueber die Wärme-Leitungsfähigkeit der Metalle, Ann. Phys. 165 (8) (1853) 497–531.
  * [31] K. Kadowaki, S. Woods, Universal relationship of the resistivity and specific heat in heavy-Fermion compounds, Solid State Commun. 58 (8) (1986) 507 – 509\.
  * [32] A. Lavasani, D. Bulmash, S. Das Sarma, Wiedemann-Franz law and Fermi liquids, Phys. Rev. B 99 (2019) 085104.
  * [33] E. H. Hwang, S. Das Sarma, Linear-in-$T$ resistivity in dilute metals: A Fermi liquid perspective, Phys. Rev. B 99 (2019) 085105.
  * [34] D. Buterakos, S. Das Sarma, Coupled electron-impurity and electron-phonon systems as trivial non-Fermi liquids, Phys. Rev. B 100 (2019) 235149.
  * [35] A. Lucas, S. Das Sarma, Electronic hydrodynamics and the breakdown of the Wiedemann-Franz and Mott laws in interacting metals, Phys. Rev. B 97 (2018) 245128.
  * [36] G. Catelani, Interaction corrections: Temperature and parallel field dependencies of the Lorentz number in two-dimensional disordered metals, Phys. Rev. B 75 (2007) 024208.
  * [37] S. Ahn, S. Das Sarma, Fragile versus stable two-dimensional fermionic quasiparticles, Phys. Rev. B 104 (2021) 125118.