Low-energy peak in the one-particle spectral function of the electron gas
at metallic densities

The Hamiltonian $H$ for a 3D homogeneous electron gas is written in second quantization as

	$\displaystyle H=$	$\displaystyle\sum_{{\bm{k}}\sigma}\varepsilon_{{\bm{k}}}c_{{\bm{k}}\sigma}^{+}c_{{\bm{k}}\sigma}$
		$\displaystyle+\frac{1}{2}\sum_{{\bm{q}}\neq{\bm{0}}}\sum_{{\bm{k}}\sigma}\sum_{{\bm{k}^{\prime}}\sigma^{\prime}}V({\bm{q}})c_{{\bm{k}}+{\bm{q}}\sigma}^{+}c_{{\bm{k}^{\prime}}-{\bm{q}}\sigma^{\prime}}^{+}c_{{\bm{k}^{\prime}}\sigma^{\prime}}c_{{\bm{k}}\sigma},$		(2)

where $c_{{\bm{k}}\sigma}$ is the annihilation operator of an electron with momentum ${\bm{k}}$ and spin $\sigma$ whose single-particle energy is given by $\varepsilon_{{\bm{k}}}\!\equiv\!{\bm{k}}^{2}/(2m)\!-\!\varepsilon_{\rm F}$ with $\varepsilon_{\rm F}\!=\!k_{\rm F}^{2}/(2m)$. We will consider the system in unit volume and the total number of electrons $N$ is nothing but the electron density $n$, given in terms of the Fermi momentum $k_{\rm F}$ as $n\!=\!k_{\rm F}^{3}/(3\pi^{2})$.

In this system, the correlation energy per electron $\varepsilon_{c}$ is already known accurately as a function of $r_{s}$ by the Green’s-Function Monte Carlo (GFMC) method [89] and the interpolation formulas to reproduce the GFMC data [90, 91]. With the use of $\varepsilon_{c}(r_{s})$, $\mu_{c}$ the correlation contribution to the chemical potential $\mu$ and the compressibility $\kappa$ are, respectively, obtained as

	$\displaystyle\frac{\mu_{c}}{\varepsilon_{\rm F}}$	$\displaystyle=\alpha^{2}r_{s}^{2}\left(\varepsilon_{c}-\frac{r_{s}}{3}\varepsilon^{\prime}\right),$		(3)
	$\displaystyle\frac{\kappa_{\rm F}}{\kappa}$	$\displaystyle=\frac{d\,\mu}{d\,\varepsilon_{\rm F}}=1-\frac{\alpha r_{s}}{\pi}+\frac{\alpha^{2}r_{s}^{3}}{6}\left(r_{s}\varepsilon_{c}^{\prime\prime}-2\varepsilon_{c}^{\prime}\right),$		(4)

where $\kappa$ is given through the thermodynamic relation of $\kappa\!=\!(d\,n/d\,\mu)/n^{2}$ and $\kappa_{\rm F}$ is the compressibility in the noninteracting electron gas, given by $\kappa_{\rm F}\!=\!D_{\rm F}/n^{2}\!=\!3m/(k_{\rm F}^{2}n)$ with $D_{\rm F}\!=\!d\,n/d\,\varepsilon_{\rm F}\!=\!mk_{\rm F}/\pi^{2}$ the density of states at the Fermi level in the noninteracting system.

The Dyson equation relates the one-particle Green’s function $G(K)$ with the self-energy $\Sigma(K)$ through

$\displaystyle G^{-1}(K)\!=\!i\omega_{n}\!+\!\mu_{x}\!+\!\mu_{c}\!-\!\varepsilon_{{\bm{k}}}-\Sigma(K).$

(5)

Here, $\mu$ is divided into $\mu\!=\!\varepsilon_{\rm F}+\mu_{x}+\mu_{c}$ with $\mu_{x}$ the exchange contribution to $\mu$, given by

$\displaystyle\frac{\mu_{x}}{\varepsilon_{\rm F}}=-\frac{2\alpha r_{s}}{\pi}.$

(6)

Let us divide $\Sigma(K)$ into odd and even parts in $\omega_{n}$ as

$\displaystyle\Sigma(K)=\left[1-Z(K)\right]i\omega_{n}\!+\!\chi(K),$

(7)

where both $Z(K)$ and $\chi(K)$ are not only even functions in $\omega_{n}$ but also real functions, as seen by combining Eq. (7) with the analyticity property $\Sigma({\bm{k}},-i\omega_{n})=\Sigma^{*}({\bm{k}},i\omega_{n})$. Then, $G^{-1}(K)$ is rewritten as

$\displaystyle G^{-1}(K)=Z(K)i\omega_{n}-E(K),$

(8)

with the introduction of $E(K)\!\equiv\!{\bm{k}}^{2}/(2m)\!+\!\chi(K)\!-\!\mu$.

Once $G(K)$ is known, the momentum distribution function $n({\bm{k}})\ (=\langle c_{{\bm{k}}\sigma}^{+}c_{{\bm{k}}\sigma}\rangle)$ is calculated as

$\displaystyle n({\bm{k}})=T\sum_{\omega_{n}}G(K)e^{-i\omega_{n}0^{+}},$

(9)

where the Matsubara sum is taken by the procedure explained in Appendix A. Accuracy of $n({\bm{k}})$ may be checked by the three sum rules related to total electron number, total kinetic energy, and total kinetic-energy fluctuation, as derived in Ref. [54]. Those sum rules are conveniently expressed in terms of the $n$th-power integral $I_{n}$, given as

$\displaystyle I_{n}\equiv\int_{0}^{\infty}dx\,n(x)x^{n},$

(10)

with $x\!=\!|{\bm{k}}|/k_{\rm F}$ and $n(x)\!=\!n({\bm{k}})$. The rigorous values for $I_{n}$ with $n\!=\!2$, $4$, and $6$ are: $I_{2}\!=\!1/3$, $I_{4}\!=\!1/5+\alpha^{2}r_{s}^{2}(-\varepsilon_{c}\!-\!r_{s}\varepsilon^{\prime})/3$, and $I_{6}\!=\!8/105\!+\!5I_{4}^{2}/3\!+\!5\alpha r_{s}[B(r_{s})\!-\!2/3\!+\!2g(0)/3]/(9\pi)$ with $g(0)$ the on-top value of the pair distribution function and $B(r_{s})$ specified in Eq. (51) in Ref. [54]. After several tentative calculations, we find that $n({\bm{k}})$ as obtained from Eq. (9) satisfies the $I_{2}$ sum rule up to five digits or more and $I_{4}$ up to three digits, but $I_{6}$ up to only a single digit in most cases, indicating that $n({\bm{k}})$ for $|{\bm{k}}|\!\gtrsim\!2k_{\rm F}$ is not accurate enough, which reflects the fact that the $k$-mesh (or grid) in ${\bm{k}}$ space in the iterative calculation of $G(K)$ is not dense enough for $|{\bm{k}}|\!\gtrsim\!2k_{\rm F}$.

As a remedy for this problem in $n({\bm{k}})$, we have modified $n({\bm{k}})\ [=\!n(x)]$ to $n_{c}(x)$ by the procedure explained in Appendix B in which the behavior in the region of $x\!\gtrsim\!2$ is rectified by the introduction of $n_{\rm IGZ}(x)$ obtained by the parametrization scheme described in Ref. [54]. In actual calculations, $n_{c}(x)$ always satisfies all the three sum rules up to at least five digits (and mostly seven or eight digits). As an example, the results of $n({\bm{k}})$, $n_{c}(x)$, and $n_{\rm IGZ}(x)$ are shown for $r_{s}=3.93$ in Fig. 10(a) in which we can barely see the difference among those three functions on the scale of the figure.

The charge response function $Q_{c}(Q)$ is related to the polarization function in the charge channel $\Pi(Q)$ through

$\displaystyle Q_{c}(Q)=-\frac{\Pi(Q)}{1+V({\bm{q}})\Pi(Q)},$

(11)

and the formal definition of $\Pi(Q)$ is written as

	$\displaystyle\Pi(Q)$	$\displaystyle=-2\sum_{K}G(K)G(K\!+\!Q)\Gamma(K,K\!+\!Q)$
		$\displaystyle\equiv-2\,T\sum_{\omega_{n}}\sum_{{\bm{k}}}G(K)G(K\!+\!Q)\Gamma(K,K\!+\!Q),$		(12)

where $\Gamma(K,K\!+\!Q)$ is the three-point vertex function in the charge channel.

In the many-body problem, it is often the case that $\Pi(Q)$ is less singular than $G(K)$. In fact, in 1D Luttinger liquids, $\Pi(Q)$ is easily obtained and does not exhibit any NFL-related singularity [12]. In 3D Fermi liquids, it is shown that no nonanalytic corrections are contained in $\Pi(Q)$, in sharp contrast to the polarization function in the spin channel in which a nonanalytic ${\bm{q}}^{2}\ln|{\bm{q}}|$-term exists [52]. Thus we can expect that in the 3D electron gas, even if there were a branch-cut singularity in $G(K)$ at low $T$, it would not induce any singular effects on $\Pi(Q)$, implying that $\Pi(Q)$ will be reliably determined even at $T$ of the order of $0.1\varepsilon_{\rm F}$. On this assumption, we consider $Q_{c}({\bm{q}},0)$ obtained by Monte Carlo simulations [92, 93, 97, 94, 95, 98, 96] as sufficiently accurate data, based on which various parametrized forms for the conventional charge local-field factor $G_{+}(Q)$ have been proposed [99, 100, 101, 102, 103, 104, 105].

In view of this situation, we regard $\Pi(Q)$ as a quantity precisely known from the outset in the self-consistent iteration loop. In actual calculations, $\Pi(Q)$ is given either with $G_{+}(Q)$ as

$\displaystyle\Pi(Q)=\frac{\Pi_{0}(Q)}{1-V({\bm{q}})G_{+}(Q)\Pi_{0}(Q)},$

(13)

or with $G_{s}(Q)$ due to Richardson and Ashcroft [99] as

$\displaystyle\Pi(Q)=\frac{\Pi_{\rm WI}(Q)}{1-V({\bm{q}})G_{s}(Q)\Pi_{\rm WI}(Q)},$

(14)

where the Lindhard function $\Pi_{0}(Q)$ is calculated as [106]

	$\displaystyle\Pi_{0}(Q)$	$\displaystyle=-2\sum_{K}G_{0}(K)G_{0}(K\!+\!Q)$
		$\displaystyle=4\int\frac{d^{3}k}{(2\pi)^{3}}\ n_{0}({\bm{k}})\frac{\varepsilon_{{\bm{k}}+{\bm{q}}}\!-\!\varepsilon_{\bm{k}}}{\omega_{q}^{2}\!+\!(\varepsilon_{{\bm{k}}+{\bm{q}}}\!-\!\varepsilon_{\bm{k}})^{2}},$		(15)

with $n_{0}({\bm{k}})\!=\!\theta(k_{\rm F}\!-\!|{\bm{k}}|)$ the step function and $\Pi_{\rm WI}(Q)$ is given by

$\displaystyle\Pi_{\rm WI}(Q)=4\int\frac{d^{3}k}{(2\pi)^{3}}\ n({\bm{k}})\frac{\varepsilon_{{\bm{k}}+{\bm{q}}}\!-\!\varepsilon_{\bm{k}}}{\omega_{q}^{2}\!+\!(\varepsilon_{{\bm{k}}+{\bm{q}}}\!-\!\varepsilon_{\bm{k}})^{2}}.$

(16)

Note that $G_{s}(Q)$ is obtained from $G_{+}(Q)$ as

$\displaystyle G_{s}(Q)=G_{+}(Q)+\frac{1}{V({\bm{q}})\Pi_{0}(Q)}\frac{\Pi_{0}(Q)\!-\!\Pi_{\rm WI}(Q)}{\Pi_{\rm WI}(Q)},$

(17)

but in this paper, we shall adopt the function form (a slightly modified Richardson-Ashcroft parametrization) prescribed in Eq. (58) in Ref. [54] for $G_{s}(Q)$.

In Eq. (16), we employ $n_{c}(x)$ for $n({\bm{k}})$ to make $\Pi_{\rm WI}({\bm{q}},0)$ correctly behave in the limit of $|{\bm{q}}|\to\infty$. On the other hand, the behaviors of $\Pi_{0}(Q)$ and $\Pi_{\rm WI}(Q)$ in the limit of $Q\to Q_{0}\!\equiv\!\{{\bm{0}},0\}$ can be derived directly from Eqs. (II.4) and (16), respectively; in the $\omega$-limit (i.e., ${\bm{q}}\to{\bm{0}}$ first, and then $\omega_{q}\to 0$), we obtain

$\displaystyle\Pi_{0}(Q)=\Pi_{\rm WI}(Q)=\frac{D_{\rm F}}{3}\frac{v_{\rm F}^{2}{\bm{q}}^{2}}{\omega_{q}^{2}},$

(18)

with $v_{\rm F}=k_{\rm F}/m$ and in the $q$-limit (i.e., $\omega_{q}\to 0$ first, and then ${\bm{q}}\to{\bm{0}}$), we obtain

$\displaystyle\Pi_{0}(Q)=D_{\rm F},\ \ {\rm and}\ \ \Pi_{\rm WI}(Q)=D_{\rm F}I_{0},$

(19)

where $I_{0}$ is defined in Eq. (10) with $n\!=\!0$. Combining the above-mentioned behavior of $\Pi_{0}(Q)$ with the constraints imposed on $G_{+}(Q)$ (or that of $\Pi_{\rm WI}(Q)$ with those on $G_{s}(Q)$) at $Q\to Q_{0}$, we see that in the $\omega$-limit,

$\displaystyle\Pi(Q)=\frac{D_{\rm F}}{3}\frac{v_{\rm F}^{2}{\bm{q}}^{2}}{\omega_{q}^{2}}=\frac{n{\bm{q}}^{2}}{m\omega_{q}^{2}},$

(20)

and in the $q$-limit,

$\displaystyle\Pi(Q)=D_{\rm F}\frac{\kappa}{\kappa_{\rm F}}=\frac{d\,n}{d\,\varepsilon_{\rm F}}\frac{d\,\varepsilon_{\rm F}}{d\,\mu}=\frac{d\,n}{d\,\mu}.$

(21)

The relations in Eqs. (20) and (21) are, respectively, known as the f-sum rule and the compressibility sum rule.

Formally, we can calculate $\Sigma(K)$ rigorously by

$\displaystyle\Sigma(K)\!=\!-\sum_{Q}W(Q)G(K\!+\!Q)\Gamma(K,K\!+\!Q),$

(22)

where $W(Q)$ is the effective interaction, given by

$\displaystyle W(Q)\!=\!\frac{V({\bm{q}})}{1\!+\!V({\bm{q}})\Pi(Q)}.$

(23)

Since we regard $\Pi(Q)$ as a precisely known quantity, $W(Q)$ is already known. As for $\Gamma(K,K\!+\!Q)$, we adopt the improved $GW\Gamma$ scheme described in Ref. [79]. According to Eqs. (54)-(58) in Ref. [79], $\Gamma(K,K\!+\!Q)$ is given in the product of two components as

$\displaystyle\Gamma(K,K\!+\!Q)=\widetilde{\Gamma}_{\rm WI}(K,K\!+\!Q)\Gamma_{\Pi}(K,K\!+\!Q),$

(24)

with

	$\displaystyle\widetilde{\Gamma}_{\rm WI}(K,K\!+\!Q)\!=\!\frac{G^{-1}(K\!+\!Q)\!-\!G^{-1}(K)}{i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{{\bm{k}}}){\tilde{\eta}}_{1}(K\!+\!Q/2)},$		(25a)
	$\displaystyle\Gamma_{\Pi}(K,K\!+\!Q)\!=\!\frac{\Pi(Q)}{\widetilde{\Pi}_{\rm WI}(Q)}$
	$\displaystyle\hskip 59.75095pt+\frac{\Pi(Q)}{D_{\rm F}}\Biggl{\{}\frac{3i\omega_{q}}{v_{\rm F}^{2}{\bm{q}}^{2}}\biggl{[}-i\omega_{q}\left(1\!-\!\frac{\Pi_{1}(Q)}{\widetilde{\Pi}_{\rm WI}(Q)}\right)$
	$\displaystyle\hskip 71.13188pt-\left(\varepsilon_{{\bm{k}}+{\bm{q}}}-\varepsilon_{\bm{k}}\right)\Bigl{(}1-{\tilde{\eta}}_{1}(K\!+\!Q/2)\Bigr{)}\biggr{]}$
	$\displaystyle\hskip 71.13188pt+{\tilde{\eta}}_{2}(K;Q)-\frac{\Pi_{2}(Q)}{\widetilde{\Pi}_{\rm WI}(Q)}\Biggr{\}},$		(25b)

where the functional ${\tilde{\eta}}_{1}(K)$ is introduced as

$\displaystyle{\tilde{\eta}}_{1}(K)$

$\displaystyle\equiv-\frac{\partial G^{-1}(K)}{\partial\varepsilon_{\bm{k}}}\bigg{/}\frac{\partial G^{-1}(K)}{\partial(i\omega_{n})},$

(26)

and three “modified polarization functionals”, $\widetilde{\Pi}_{\rm WI}(Q)$, $\Pi_{1}(Q)$, and $\Pi_{2}(Q)$, are, respectively, defined as

	$\displaystyle\widetilde{\Pi}_{\rm WI}(Q)$	$\displaystyle\!\equiv\!2\!\sum_{K}\frac{G(K\!+\!Q)\!-\!G(K)}{i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{{\bm{k}}}){\tilde{\eta}}_{1}(K\!+\!Q/2)},$		(27a)
	$\displaystyle\Pi_{1}(Q)$	$\displaystyle\!\equiv\!2\!\sum_{K}\frac{[G(K\!+\!Q)\!-\!G(K)](\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{{\bm{k}}})}{[i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{{\bm{k}}}){\tilde{\eta}}_{1}(K\!+\!Q/2)]i\omega_{q}},$		(27b)
	$\displaystyle\Pi_{2}(Q)$	$\displaystyle\!\equiv\!2\!\sum_{K}\frac{[G(K\!+\!Q)\!-\!G(K)]{\tilde{\eta}}_{2}(K;Q)}{i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{{\bm{k}}}){\tilde{\eta}}_{1}(K\!+\!Q/2)}.$		(27c)

The functional ${\tilde{\eta}}_{2}(K;Q)$ will be specified later.

In Ref. [79], this functional form for $\Gamma(K,K\!+\!Q)$ was derived from the perspective of FLT, i.e., with the assumption that $G(K)$ is given in such a form as

$\displaystyle G(K)\!=\!\frac{z_{\bm{k}}}{i\omega_{n}\!-\!\tilde{\varepsilon}_{\bm{k}}\!+\!i\ {\rm sgn}(\omega_{n})(2\tau_{\bm{k}})^{-1}}\!+\!G_{\rm incoh}(K),$

(28)

for $|{\bm{k}}|\!\approx\!k_{\rm F}$ and $|\omega_{n}|\!\ll\!\varepsilon_{\rm F}$, where $z_{\bm{k}}$, $\tilde{\varepsilon}_{\bm{k}}$, and $\tau_{\bm{k}}$ are, respectively, the QP renormalization factor, the QP dispersion, and the QP lifetime. Here, $|\tilde{\varepsilon}_{\bm{k}}|\tau_{\bm{k}}\!\gg\!1$ is assumed and $G_{\rm incoh}(K)$ is a smooth function, corresponding to the incoherent smooth background in $A({\bm{k}},\omega)$. Note that QP appears as a pole in $G(K)$ as long as $z_{\bm{k}}\!\neq\!0$ and this pole-singularity is intimately connected with the condition that $\Sigma(K)$ is smooth enough to be analytically expanded around the Fermi point $K_{\rm F}\!\equiv\!\{{\bm{k}}_{\rm F},0\}$.

The final result in Eq. (24), together with Eqs. (25)-(27), however, does not explicitly contain any FLT-specific parameters such as the Landau interaction and $m^{*}$, implying that this functional form itself can be applied to NFL as well. In fact, ${\tilde{\eta}}_{1}(K)$, a key quantity leading to $m/m^{*}$ at $K\!\to\!K_{\rm F}$ in FLT, appears in this formalism as a direct consequence of the total-current (or total-momentum) conservation law that should be satisfied even in NFL, indicating that ${\tilde{\eta}}_{1}(K)$ must also be a key quantity in NFL.

It is the most important advantage in this formalism that the Ward identity (WI) [81] is automatically satisfied, whatever choice one may make for ${\tilde{\eta}}_{1}(K)$ and ${\tilde{\eta}}_{2}(K;Q)$; if we choose ${\tilde{\eta}}_{1}(K)\!=\!{\tilde{\eta}}_{2}(K;Q)\!=\!1$, the present vertex function is nothing but the one in the original $GW\Gamma$ scheme [80] and $\widetilde{\Gamma}_{\rm WI}(K,K\!+\!Q)$ is reduced to the canonical form appearing in connection to WI as

$\displaystyle\Gamma_{\rm WI}(K,K\!+\!Q)\!=\!\frac{G^{-1}(K\!+\!Q)\!-\!G^{-1}(K)}{i\omega_{q}\!-\!\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!+\!\varepsilon_{{\bm{k}}}}.$

(29)

Thus, we note that by just changing $\Gamma_{\rm WI}(K,K\!+\!Q)$ into $\widetilde{\Gamma}_{\rm WI}(K,K\!+\!Q)$, we can enter into a more advanced stage in which both WI and the total-momentum conservation law are simultaneously fulfilled.

In actual numerical calculations, however, it turns out that it is not easy to proceed at each iteration step with ${\tilde{\eta}}_{1}(K)$ evaluated in accordance with Eq. (26). Because ${\tilde{\eta}}_{1}(K)$ reflects the total-momentum conservation law only in its value at $K\!=\!K_{\rm F}$, we can approximate ${\tilde{\eta}}_{1}(K\!+\!Q/2)$ as $\eta_{1}(Q)$, a function depending only on $Q$ with the condition that $\eta_{1}(Q_{0})\!=\!{\tilde{\eta}}_{1}(K_{\rm F})$ in the $q$-limit. Under this approximation, we can reduce $\Gamma_{\Pi}(K,K\!+\!Q)$ into

	$\displaystyle\Gamma_{\Pi}(K,K\!+\!Q)\!=\!\frac{\Pi(Q)}{\widetilde{\Pi}_{\rm WI}(Q)}\!+\!\frac{\Pi(Q)}{D_{\rm F}}\biggl{\{}\frac{1\!-\!\eta_{1}(Q)}{\eta_{1}(Q)}\frac{3i\omega_{q}}{v_{\rm F}^{2}{\bm{q}}^{2}}$
	$\displaystyle\hskip 1.42271pt\times\!\Bigl{[}i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{\bm{k}})\eta_{1}(Q)\Bigr{]}\!+\!{\tilde{\eta}}_{2}(K;Q)\!-\!\frac{\Pi_{2}(Q)}{\widetilde{\Pi}_{\rm WI}(Q)}\biggr{\}},$		(30)

and $\widetilde{\Pi}_{\rm WI}(Q)$ is easily calculated by

$\displaystyle\widetilde{\Pi}_{\rm WI}(Q)\!\equiv\!\widetilde{\Pi}_{\rm WI}({\bm{q}},i\omega_{q})=\frac{1}{\eta_{1}(Q)}\Pi_{\rm WI}\left({\bm{q}},\frac{i\omega_{q}}{\eta_{1}(Q)}\right),$

(31)

with $\Pi_{\rm WI}({\bm{q}},i\omega_{q})\!\equiv\!\Pi_{\rm WI}(Q)$ in Eq. (16).

Since we may take ${\tilde{\eta}}_{2}(K;Q)$ at our disposal, we choose it so as to make numerical calculations as easy as possible. By carefully inspecting the structure of each term in Eq. (30), we can think of a possible form for ${\tilde{\eta}}_{2}(K;Q)$ as

	$\displaystyle{\tilde{\eta}}_{2}(K;Q)=\eta_{1}(Q)\zeta_{1}(Q)\!+\!\Bigl{[}i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{\bm{k}})\eta_{1}(Q)\Bigr{]}$
	$\displaystyle\hskip 31.2982pt\times\frac{3}{v_{\rm F}^{2}{\bm{q}}^{2}}\Bigl{[}i\omega_{q}\zeta_{2}(Q)-(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{\bm{k}})\zeta_{3}(Q)\Bigr{]},$		(32)

with arbitrary functions $\zeta_{i}(Q)$ ($i\!=\!1,2$, and $3$). By substituting this ${\tilde{\eta}}_{2}(K;Q)$ into Eq. (30), we immediately find the following: (i) $\zeta_{1}(Q)$ is irrelevant, because it is always cancelled by the corresponding term in $\Pi_{2}(Q)/\widetilde{\Pi}_{\rm WI}(Q)$. (ii) By choosing $\zeta_{2}(Q)$ as $[\eta_{1}(Q)-1]/\eta_{1}(Q)$, we can eliminate the first term in the curly brackets in Eq. (30). (iii) Because the term containing $\zeta_{3}(Q)$ is rather difficult to treat, it would be better to approximate $(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{\bm{k}})\zeta_{3}(Q)$ by $v_{\rm F}q\zeta_{3}(Q)$ with $q=|{\bm{q}}|$. As a result, we have arrived at the functional form for $\Gamma(K,K\!+\!Q)$, written as

	$\displaystyle\Gamma(K,K\!+\!Q)\!=\!\frac{G^{-1}(K\!+\!Q)\!-\!G^{-1}(K)}{i\omega_{q}\!-\!(\varepsilon_{{\bm{k}}\!+\!{\bm{q}}}\!-\!\varepsilon_{\bm{k}})\eta_{1}(Q)}\,\frac{\Pi(Q)}{\widetilde{\Pi}_{\rm WI}(Q)}$
	$\displaystyle\hskip 28.45274pt\!-\!\bigl{[}G^{-1}(K\!+\!Q)\!-\!G^{-1}(K)\bigr{]}\frac{\Pi(Q)}{D_{\rm F}}\frac{3\zeta_{3}(Q)}{v_{\rm F}q}.$		(33)

It should be noted that we can rigorously reproduce $\Pi(Q)$ irrespective of $\eta_{1}(Q)$ and $\zeta_{3}(Q)$ in substituting this functional form for $\Gamma(K,K\!+\!Q)$ into Eq. (II.4), verifying the internal consistency of our formulation.

To determine $\eta_{1}(Q)$ and $\zeta_{3}(Q)$, we need to consider several constraints to be imposed on them to make $\Gamma(K,K\!+\!Q)$ behave correctly in various limits. In line with such constraints as discussed in Appendix C, we will use $\eta_{1}(Q)$ in Eq. (C3) and $\zeta_{3}(Q)$ given by

$\displaystyle\zeta_{3}(Q)\!=\!-\frac{1}{3}\,\frac{\Pi_{0}(Q)\!-\!\Pi_{\rm WI}(Q)[1\!-\!\beta_{3}G_{s}(Q)]}{\Pi_{\rm WI}(Q)},$

(34)

in this paper.

In Fig. 2, the self-consistently determined results of $\eta_{1}({\bm{q}},0)$ are plotted at $T\!=\!10^{-4}\varepsilon_{\rm F}$ for the 3D homogeneous electron gas with $2\!<\!r_{s}\!<\!6$. As seen from this figure, $\eta_{1}(Q)$ is smoothly converged to unity for $q\gtrsim 10k_{\rm F}$, but it exhibits a rather rapid change near $q\approx 0$, indicating that $\Sigma(K)$ is not smooth enough for $K\approx K_{\rm F}$, a symptom of possible breakdown of FLT. Incidentally, in obtaining $\eta_{1}(Q)$, we need to know the static physical quantities, $E({\bm{k}},0)$ and $Z({\bm{k}},0)$, which can be obtained by an extrapolation procedure explained in Appendix D.

In Eq. (34), the term involving the parameter $\beta_{3}$ is introduced to rigorously reproduce $\mu_{c}$, the value accurately given in Eq. (3). The appropriately determined values for $\beta_{3}$ providing the correct $\mu_{c}$ with $r_{s}$ in the region of $1<r_{s}<6$ are shown in Fig. 3 in which we take $T$ as $10^{-4}\varepsilon_{\rm F}$. As seen from the figure, the magnitude of $\beta_{3}$ is small, i.e., of the order of 0.02, letting us know that the $\beta_{3}$ term exerts only limited effects on $\Sigma(K)$.

By substituting Eq. (33) with Eq. (34) into Eq. (22), we obtain an expression for the calculation of $\Sigma(K)$ as

$\displaystyle\Sigma(K)=\Sigma_{a}+\Sigma_{b}(K)+\Lambda(K)G^{-1}(K),$

(35)

with

	$\displaystyle\Sigma_{a}\ =-\sum_{Q}W_{\rm WI}(Q)R_{\rm WI}(Q)$
	$\displaystyle\hskip 17.07182pt=-T\sum_{\omega_{q}}\int_{0}^{\infty}\frac{q^{2}dq}{2\pi^{2}}\,W_{\rm WI}(Q)R_{\rm WI}(Q),$		(36a)
	$\displaystyle\Sigma_{b}(K)\!=\!-\sum_{K^{\prime}}\overline{W}_{\rm WI}(Q)G(K^{\prime})\overline{\Gamma}_{\rm WI}(K,K^{\prime})$
	$\displaystyle\hskip 28.45274pt=-T\sum_{\omega_{n^{\prime}}}\frac{1}{k}\int_{0}^{\infty}\frac{qdq}{4\pi^{2}}\,\overline{W}_{\rm WI}({\bm{q}},i\omega_{n^{\prime}}\!-\!i\omega_{n})$
	$\displaystyle\hskip 51.21504pt\times\int_{|k-q|}^{k+q}k^{\prime}dk^{\prime}G(K^{\prime})\overline{\Gamma}_{\rm WI}(K,K^{\prime}),$		(36b)
	$\displaystyle\Lambda(K)\,=\!\sum_{K^{\prime}}W_{\rm WI}(Q)R_{\rm WI}(Q)G(K^{\prime})$
	$\displaystyle\hskip 28.45274pt=T\sum_{\omega_{n^{\prime}}}\frac{1}{k}\int_{0}^{\infty}\frac{qdq}{4\pi^{2}}\,W_{\rm WI}({\bm{q}},i\omega_{n^{\prime}}\!-\!i\omega_{n})$
	$\displaystyle\hskip 42.67912pt\times\!R_{\rm WI}({\bm{q}},i\omega_{n^{\prime}}\!-\!i\omega_{n})\!\int_{|k-q|}^{k+q}\!k^{\prime}dk^{\prime}G(K^{\prime}),$		(36c)

where $W_{\rm WI}(Q)$, $\overline{W}_{\rm WI}(Q)$, $R_{\rm WI}(Q)$, and $\overline{\Gamma}_{\rm WI}(K^{\prime},K)$ are, respectively, defined as

	$\displaystyle W_{\rm WI}(Q)=W(Q)\frac{\Pi(Q)}{\Pi_{\rm WI}(Q)}$
	$\displaystyle\hskip 36.98866pt=\frac{V({\bm{q}})}{1+V({\bm{q}})\Pi_{\rm WI}(Q)[1-G_{s}(Q)]},$		(37a)
	$\displaystyle\overline{W}_{\rm WI}(Q)=W_{\rm WI}(Q)\,\frac{\Pi_{\rm WI}(Q)}{\widetilde{\Pi}_{\rm WI}(Q)},$		(37b)
	$\displaystyle R_{\rm WI}(Q)\ =\,\frac{\Pi_{0}(Q)\!-\!\Pi_{\rm WI}(Q)[1-\beta_{3}G_{s}(Q)]}{D_{\rm F}v_{\rm F}q},$		(37c)
	$\displaystyle\overline{\Gamma}_{\rm WI}(K,K^{\prime})=\frac{G^{-1}(K^{\prime})\!-\!G^{-1}(K)}{i\omega_{q}\!-\!(\varepsilon_{\bm{k}^{\prime}}\!-\!\varepsilon_{\bm{k}})\eta_{1}(Q)}.$		(37d)

Here, $\Sigma_{a}$ is independent of $K$ and directly connected to the chemical potential shift, $\Sigma_{b}(K)$ is the main contribution to the self-energy, and $\Lambda(K)$ partially contributes to the renormalization factor.