Optical conductivity of a metal near an Ising-nematic quantum critical point

The optical conductivity of a correlated electron system contains important information about the strength of electron-electron (ee) interaction. This information is encoded both in the renormalization of the Drude weight and in the frequency and/or temperature scaling of the conductivity. Experimental and theoretical studies of optical conductivity is a very active research area Basov and Timusk (2005); Basov et al. (2011); Maslov and Chubukov (2017); Armitage (2018); Tanner (2019).

Unlike the dc conductivity, which is rendered finite only either by umklapp ee scattering Landau and Pomeranchuk (1936); *Landau:collected; Lifshitz and Pitaevskii (1981) or Baber (electron-hole) scattering in compensated metals Baber (1937); Lifshitz and Pitaevskii (1981), the optical conductivity is relatively free of the constraints imposed by momentum conservation: as long as the electron spectrum is not parabolic, the optical conductivity is rendered finite by ee interaction even in the absence of umklapps/compensation. In the Fermi liquid (FL) regime, the optical conductivity so far has been shown to adhere to two scaling forms. The first one is due to Gurzhi Gurzhi (1959):

$\displaystyle\sigma^{\prime}(\omega,T)\propto\frac{\omega^{2}+4\pi^{2}T^{2}}{\omega^{2}}.$

(1)

This form occurs in the presence of umklapp scattering or compensation both in two dimensions (2D) and three dimensions (3D), but also in 3D even without umklapps/compensation, as long as the electron dispersion contains higher than quadratic terms Pal et al. (2012a). The second scaling form,

	$\displaystyle\sigma^{\prime}(\omega,T)$	$\displaystyle\propto$	$\displaystyle\frac{\omega^{2}+4\pi^{2}T^{2}}{\omega^{2}}(3\omega^{2}+8\pi^{2}T^{2})$		(4)
			$\displaystyle\times\left\{\begin{array}[]{ccc}\ln\max\{|\omega|,T\},\,\mathrm{2D}\\ 1,\,\mathrm{3D}\end{array}\right.,$

is believed to describe systems with isotropic but nonparabolic spectrum, e.g., Dirac metals Sharma et al. (2021); Goyal et al. (2023), in the absence of umklapp scattering/compensation. ¹¹1Although the primary focus of Ref. Sharma et al. (2021) is on Dirac metals, the analysis there is applicable to any nonparabolic spectrum. At $T=0$, Eqs. (1) and (4) are reduced to $\mathrm{const}$ and $\omega^{2}$, respectively (modulo a $\ln|\omega|$ factor in 2D). A suppression of $\sigma^{\prime}(\omega,0)$ in Eq. (4) compared to Eq. (1) reflects partial Galilean invariance of an isotropic system. The limiting forms of Eq. (4), i.e., $\omega^{2}$ and $T^{4}/\omega^{2}$, have been derived by a large number of authors in a variety of contexts Gurzhi et al. (1982, 1987, 1995); Rosch and Howell (2005); Rosch (2006); Briskot et al. (2015); Ledwith et al. (2019). Both scaling forms are valid for $\omega\gg 1/\tau_{\mathrm{j}}(T)$, where $\tau_{\mathrm{j}}(T)$ is the appropriate current relaxation time at finite $T$, which behaves as $1/T^{2}$ under the conditions when Eq. (1) is applicable or as $1/T^{4}$ (modulo a $\ln T$ factor in 2D) under the conditions when Eq. (4) is applicable.

The case of a 2D anisotropic Fermi surface (FS) (again, in the absence of umklapp scattering/compensation) is more complicated. There, the conductivity depends on whether the FS is convex or concave, i.e., on whether it has inflection points. In the dc case, the FL-like $T^{2}$ correction to the residual resistivity vanishes for a convex FS, while the surviving term behaves as $T^{4}\ln T$ Gurzhi et al. (1982, 1987, 1995); Maslov et al. (2011); Pal et al. (2012a). On the other hand, the correction to the residual resistivity for a concave 2D FL is free of such cancellations and behaves as $T^{2}$. A similar cancellation for a convex FS is expected to occur for the optical conductivity Maslov and Chubukov (2017) and, indeed, it has recently been shown to be the case for a non-FL at the Ising-nematic quantum critical point (QCP) Guo et al. (2022), while Ref. Shi et al. (2022) arrived at the same result regardless of the shape (convex vs concave) of the FS. The issues pertaining to the surviving term in the optical conductivity for a convex FS as well of the leading term for a concave FS have not been analyzed even in the FL regime. One of the goals of this paper is to resolve these outstanding issues. We will show explicitly that $\sigma^{\prime}(\omega,T)$ for a 2D FL with convex/concave FS scales as predicted by Eqs. (4) and (1), respectively.

Another goal of this paper is to generalize the results for an electron system at the Ising-nematic QCP, which is an example of the $D=2$, $Z=3$ criticality Hertz (1976); Millis (1993). This subject has a long and somewhat controversial history. The optical conductivity of such a system [in the context of fermions coupled to a $U(1)$ gauge field] was claimed in Ref. Kim et al. (1994) to behave as

$\displaystyle\sigma^{\prime}(\omega,0)\propto|\omega|^{-2/3}.$

(5)

Later, it was realized, however, that the current-relaxing mechanism in Ref. Kim et al. (1994) was not properly specified Maslov and Chubukov (2017); Chubukov and Maslov (2017); Guo et al. (2022); Shi et al. (2022, 2023). As we already mentioned, subsequent studies demonstrated the vanishing of the leading $|\omega|^{-2/3}$ term for a convex FS Guo et al. (2022). In this paper, we revisit the issue of the $-2/3$ scaling of the optical conductivity near an Ising-nematic QCP. We confirm that the $|\omega|^{-2/3}$ term vanishes both for an isotropic but nonparabolic spectrum and for a convex FS. We also show that the surviving term in the FL regime behaves according to Eq. (4) for both cases.²²2The actual form of the scaling function is somewhat different for the nematic system due to another contribution to the charge current, see Sec. II.1. When extrapolated to the immediate vicinity of the Ising-nematic QCP, our result translates into

$\displaystyle\sigma^{\prime}(\omega,T)\propto|\omega|^{2/3}\max\left\{1,(T/|\omega|)^{8/3}\right\}.$

(6)

The $T=0$ limit of our result, $\sigma^{\prime}(\omega,0)\propto|\omega|^{2/3}$, contradicts the $\sigma^{\prime}(\omega,0)=\mathrm{const}$ result of Ref. Guo et al. (2022). With respect to a concave FS, we show explicitly that the conductivity is restored back to the original form, Eq. (5). For finite $T$, we find

$\displaystyle\sigma^{\prime}(\omega,T)\propto|\omega|^{-2/3}\max\{1,(T/|\omega|)^{4/3}\}.$

(7)

The $T=0$ limit of this result was also obtained in Ref. Shi et al. (2023) for a model of fermions coupled to loop current fluctuations. Following Refs. Pal et al. (2012a) and Pal et al. (2012b), we also obtain the scaling form of the conductivity near the convex-to-concave transition. For the reader’s convenience, we summarize the results for the optical conductivity in various regimes in Table 1.

The rest of the paper is organized as follows. In Sec. II, we formulate the model and introduce the modified version of the Kubo formula that offers a practical advantage in calculation of the optical conductivity. In Sec. III, we apply this formalism to an isotropic electron system with a nonparabolic energy spectrum. In Secs. IV and V, we derive the optical conductivity of metals with convex and concave FSs, respectively, and also the near convex-to-concave transition. For each case, we discuss the frequency scaling of the conductivity both in the FL and quantum-critical regions. In Sec. VI we summarize our main finding and provide a broader discussion. Appendices A–C present some technical details of our calculations.

In this section, we consider the case of an isotropic but nonparabolic spectrum, $\epsilon_{\bf k}=\epsilon(k)$, which is encountered, e.g., in Dirac metals. We begin with $\sigma^{\prime}_{1}(\omega,T)$ in Eq. (28a), the analysis of which follows along the same lines as in Ref. Sharma et al. (2021). Relabeling the momenta as ${\bf k}+{\bf q}/2\to{\bf k}$ and ${\bf p}-{\bf q}/2\to{\bf p}$, introducing the energy transfer as $\Omega=\varepsilon_{{\bf k}-{\bf q}}-\varepsilon_{{\bf k}}=\varepsilon_{{\bf p}}-\varepsilon_{{\bf p}+{\bf q}}+\omega$, and restricting the integrals over the fermionic momenta to narrow regions near the FS, we rewrite Eq. (28a) as

	$\displaystyle\sigma^{\prime}_{1}(\omega,T)=$	$\displaystyle e^{2}g^{2}\frac{\pi N_{\mathrm{F}}^{2}}{\omega^{3}}(1-e^{-\omega/T})\int\frac{d^{2}q}{(2\pi)^{2}}\int^{+\infty}_{-\infty}d\varepsilon_{\bf k}\int^{+\infty}_{-\infty}d\varepsilon_{\bf p}\int^{+\infty}_{-\infty}d\Omega\int_{0}^{2\pi}\frac{d\phi_{{\bf k}{\bf q}}}{2\pi}\int_{0}^{2\pi}\frac{d\phi_{{\bf p}{\bf q}}}{2\pi}F^{2}\left({\bf k}-\frac{{\bf q}}{2}\right)F^{2}\left({\bf p}+\frac{{\bf q}}{2}\right)V^{2}({\bf q})$
		$\displaystyle\times\left(\Delta\bm{\varv}\right)^{2}n_{\mathrm{F}}(\varepsilon_{{\bf k}})n_{\mathrm{F}}(\varepsilon_{\bf p})\left[1-n_{\mathrm{F}}(\varepsilon_{{\bf k}}+\Omega)\right]\left[1-n_{\mathrm{F}}(\varepsilon_{\bf p}-\Omega+\omega)\right]\delta(\Omega-\varepsilon_{{\bf k}-{\bf q}}+\varepsilon_{{\bf k}})\delta(\Omega-\omega+\varepsilon_{{\bf p}+{\bf q}}-\varepsilon_{{\bf p}}),$		(31)

where $\phi_{{\bf n}{\bf m}}$ is the angle between vectors ${\bf n}$ and ${\bf m}$, and

$\displaystyle\Delta\bm{\varv}=\bm{\varv}_{\bf k}+\bm{\varv}_{\bf p}-\bm{\varv}_{{\bf k}-{\bf q}}-\bm{\varv}_{{\bf p}+{\bf q}}.$

(32)

Note that $\bm{\varv}_{\mathbf{k}}=\epsilon^{\prime}(k)\mathbf{k}/k$ for an isotropic dispersion. Therefore, if we project all the momenta onto the FS, i.e., put $k=p=|\mathbf{k}-\mathbf{q}|=|{\bf p}+{\bf q}|=k_{F}$ in $\Delta\bm{\varv}$, then $\Delta\bm{\varv}$ vanishes and so does the conductivity. To get a finite result, we need to expand $\Delta\bm{\varv}$ around the FS. Performing such an expansion, we obtain

	$\displaystyle\Delta\bm{\varv}$	$\displaystyle=$	$\displaystyle\frac{w_{\text{np}}}{k_{\mathrm{F}}}\left[\left(\varepsilon_{\bf k}-\varepsilon_{{\bf k}-{\bf q}}\right)\hat{\bf k}+\left(\varepsilon_{\bf p}-\varepsilon_{{\bf p}+{\bf q}}\right)\hat{\bf p}\right.$		(33)
			$\displaystyle\left.+\left(\varepsilon_{{\bf k}-{\bf q}}-\varepsilon_{{\bf p}+{\bf q}}\right)\frac{{\bf q}}{k_{\mathrm{F}}}\right],$

where $\hat{\bf k}$ is the unit vector in the direction of ${\bf k}$, etc., and

$\displaystyle w_{\text{np}}=1-\frac{k_{F}\epsilon^{\prime\prime}(k)}{\epsilon^{\prime}(k)}\Big{|}_{k=k_{F}}$

(34)

is the nonparabolicity coefficient Sharma et al. (2021). A Galilean-invariant system has a parabolic spectrum , in which case $w_{\mathrm{np}}=0$. For a nonparabolic spectrum $w_{\mathrm{np}}\neq 0$; for example, $w_{\mathrm{np}}=1$ for the Dirac spectrum.

Near quantum criticality, momentum transfers are small: $q\lesssim q_{\text{B}}\ll k_{\mathrm{F}}$. Therefore, we can neglect the last term, proportional to ${\bf q}$, in Eq. (33), and also replace $F({\bf k}-{{\bf q}}/{2})\approx F({\bf k})$ and $F({\bf p}+{{\bf q}}/{2})\approx F({\bf p})$ in Eq. (III). Next, we express the dispersions entering Eq. (33) via $\omega$ and $\Omega$, using the constraints imposed by the delta functions in Eq. (III). This yields

	$\displaystyle\left(\Delta\bm{\varv}\right)^{2}=\frac{w^{2}_{\text{np}}}{k_{\mathrm{F}}^{2}}\left[\Omega^{2}+(\Omega+\omega)^{2}-2\Omega(\Omega+\omega)\cos\phi_{{\bf k}{\bf p}}\right].$
			(35)

It is to be expected (and will be shown below to be the case) that typical values of $|\Omega|$ are on the order of $\max\{|\omega|,T\}$. Then $(\Delta\bm{\varv})^{2}\sim w^{2}_{\text{np}}\max\{\omega^{2},T^{2}\}/k_{\mathrm{F}}^{2}$, which implies that the conductivity is suppressed compared to its canonical FL value, Eq. (1). To obtain the leading term in the conductivity, it suffices to neglect $\Omega$ and $\omega$ in the delta functions in Eq. (III). For small $q$, these delta functions are then reduced to the geometric constraints $\cos\phi_{{\bf k}{\bf q}}=0$ and $\cos\phi_{{\bf p}{\bf q}}=0$, which implies that ${\bf k}$ and ${\bf p}$ are either parallel or anti-parallel to each other, i.e., that $\cos\phi_{{\bf k}{\bf p}}=\pm 1$. Recalling the fourfold symmetry of the interaction, we find that the contributions from $\cos\phi_{{\bf k}{\bf p}}=\pm 1$ in Eq. (35) cancel each other, while the integral of the form factors over $\phi_{\bf q}$ gives a constant

$\displaystyle a=\int^{2\pi}_{0}\frac{d\phi_{\bf q}}{2\pi}F^{4}({\bf q}).$

(36)

Next, the triple integral over $\varepsilon_{\bf k}$, $\varepsilon_{\bf p}$, and $\Omega$ is solved as Sharma et al. (2021)

	$\displaystyle\iiint\limits^{+\infty}_{-\infty}d\varepsilon_{\bf k}d\varepsilon_{\bf p}d\Omega\left[(\Omega+\omega)^{2}+\Omega^{2}\right]$
	$\displaystyle\times n_{\mathrm{F}}(\varepsilon_{\bf k})n_{\mathrm{F}}(\varepsilon_{\bf p})\left[1-n_{\mathrm{F}}(\varepsilon_{\bf k}+\Omega)\right]\left[1-n_{\mathrm{F}}(\varepsilon_{\bf p}-\Omega+\omega)\right]$
	$\displaystyle=\frac{\omega^{5}}{30(1-e^{-\omega/T})}\left(1+\frac{4\pi T^{2}}{\omega^{2}}\right)\left(3+\frac{8\pi^{2}T^{2}}{\omega^{2}}\right).$		(37)

Note that typical values of the variables in this integral are $|\varepsilon_{\bf k}|\sim|\varepsilon_{\bf p}|\sim|\Omega|\sim\max\{|\omega|,T\}$, which proves our earlier assertion. Finally, we notice that the integral over $q$ diverges logarithmically at the lower limit because each of the delta functions in Eq. (III) brings in a factor of $1/q$. To regularize the divergence, one needs to return to the dynamic form of the interaction and ask how large $q$ should be for the Landau-damping term to be neglected. The answer is that $q\gg q_{\text{B}}|\omega|/\omega_{\text{FL}}$, where $\omega_{\text{FL}}$ is the upper boundary of the FL region Pimenov et al. (2022) ³³3We thank A. Chubukov for clarifying the cutoff procedure for us and for bringing Ref. Pimenov et al. (2022) to our attention., given by Eq. (13). Therefore,

$\displaystyle\int^{\infty}_{q_{\text{B}}|\omega|/\omega_{\text{FL}}}\frac{dq}{q}V^{2}(q)\approx\frac{1}{q_{\text{B}}^{4}}\ln\frac{\omega_{\text{FL}}}{|\omega|}.$

(38)

Collecting everything together, we arrive at

	$\displaystyle\sigma^{\prime}_{1}(\omega,T)$	$\displaystyle=$	$\displaystyle\frac{e^{2}}{60\pi^{2}}aw^{2}_{\text{np}}\frac{g^{2}N^{2}_{\mathrm{F}}}{q_{\text{B}}^{4}}\left(\frac{\omega}{\varv_{\mathrm{F}}k_{\mathrm{F}}}\right)^{2}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)$		(39)
			$\displaystyle\times\left(3+\frac{8\pi^{2}T^{2}}{\omega^{2}}\right)\ln\left(\frac{\omega_{\text{FL}}}{\max\{\omega,T\}}\right),$

which is the scaling form advertised in Eq. (4).

We now turn to $\sigma^{\prime}_{2}(\omega,T)$ in Eq. (30a). The integrand of $\sigma^{\prime}_{2}(\omega,T)$ can be projected right onto the FS without further expansions. The rest of the calculation is the same as for $\sigma^{\prime}_{1}(\omega,T)$, except for the energy integral, which yields

		$\displaystyle\iiint\limits^{+\infty}_{-\infty}d\varepsilon_{\bf k}d\varepsilon_{\bf p}d\Omega\,\left[1-n_{\mathrm{F}}(\varepsilon_{\bf k}+\Omega)\right]\left[1-n_{\mathrm{F}}(\varepsilon_{\bf p}-\Omega+\omega)\right]$
		$\displaystyle\times n_{\mathrm{F}}(\varepsilon_{\bf k})n_{\mathrm{F}}(\varepsilon_{\bf p})=\frac{\omega^{3}}{6(1-e^{-\omega/T})}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right).$		(40)

A straightforward calculation then leads to

	$\displaystyle\sigma^{\prime}_{2}(\omega,T)=\frac{e^{2}}{24\pi^{2}}\frac{g^{2}N^{2}_{\mathrm{F}}}{q_{\text{B}}^{4}}w_{\text{nem}}^{2}\left(\frac{\omega}{\varv_{\mathrm{F}}k_{\mathrm{F}}}\right)^{2}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)$
	$\displaystyle\times\ln\left(\frac{\omega_{\text{FL}}}{\max\{\omega,T\}}\right),$		(41)

where

$\displaystyle w^{2}_{\text{nem}}=\int^{2\pi}_{0}\frac{d\phi_{\bf q}}{2\pi}\left(\frac{\partial F^{2}({\bf q})}{\partial\phi_{\bf q}}\right)^{2}$

(42)

is the nematicity coefficient.

Finally, we come to the $\sigma^{\prime}_{12}(\omega,T)$ contribution to the conductivity, Eq. (30b). By power counting, it is of the same order as $\sigma^{\prime}_{1}$ and $\sigma^{\prime}_{2}$ because $\Delta\varv\Delta\varepsilon$ scales as $\omega^{2}$ [cf. Eq. (26)]. However $\sigma^{\prime}_{12}(\omega,T)$ vanishes within the approximation of this section. The reason is that, once the last term in Eq. (33) is neglected, $\Delta\bm{\varv}$ becomes a sum of two radial vectors, directed along $\hat{\bf k}$ and $\hat{\bf p}$, respectively. As we already know, the leading contribution to the integral comes from almost collinear momenta ${\bf k}$ and ${\bf p}$. Therefore, $\Delta\bm{\varv}$ is also a radial vector, collinear with ${\bf k}$ (or ${\bf p}$). Now, in Eq. (30b) $\Delta\bm{\varv}$ is dotted into ${\bf w}\equiv(\bm{\nabla}_{\bf k}+\bm{\nabla}_{\bf p})U^{2}({\bf k},{\bf p},{\bf q})$, which is a vector sum of two tangential vectors, proportional to $\hat{\phi}_{\bf k}$ and $\hat{\phi}_{\bf p}$, respectively. Again, due to collinearity of ${\bf k}$ and ${\bf p}$, vector ${\bf w}$ is perpendicular to both. Thus $\Delta\bm{\varv}\cdot{\bf w}=0$, and the leading term in $\sigma^{\prime}_{12}(\omega,T)$ vanishes.

Therefore, the final result for the conductivity is the sum of Eqs. (39) and (41):

	$\displaystyle\sigma^{\prime}(\omega,T)=\frac{e^{2}}{12\pi^{2}}\frac{g^{2}N^{2}_{\mathrm{F}}}{q_{\text{B}}^{4}}\left(\frac{\omega}{\varv_{\mathrm{F}}k_{\mathrm{F}}}\right)^{2}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)$
	$\displaystyle\times\left[\frac{aw^{2}_{\mathrm{np}}}{5}\left(3+\frac{8\pi^{2}T^{2}}{\omega^{2}}\right)+\frac{w_{\mathrm{nem}}^{2}}{2}\right]\ln\left(\frac{\omega_{\text{FL}}}{\max\{|\omega|,T\}}\right).$		(43)

The nonparabolic and nematic contributions to the conductivity are of the same order for $\omega\gtrsim T$, while the nonparabolic contribution is the dominant one for $\omega\ll T$. Schematically,

$\displaystyle\sigma^{\prime}(\omega,T)\propto q_{\text{B}}^{-4}\omega^{2}\max\left\{\ln\frac{\omega_{\text{FL}}}{|\omega|},\frac{T^{4}}{\omega^{4}}\ln\frac{\omega_{\text{FL}}}{T}\right\}.$

(44)

As explained in Sec. II.1, a crossover to the QCP is achieved by replacing $q_{\text{B}}\to\max\{|\omega|^{1/3},T^{1/3}\}$. At the same time, $\omega_{\text{FL}}$ is replaced by $\max\{|\omega|,T\}$, and thus the logarithmic factors are replaced by constants of order one. As a result, Eq. (44) is replaced by Eq. (6).

Next, we turn our attention to anisotropic FSs.

We proceed with the optical conductivity for a generic convex FS. As has been shown by many authors (see Sec. I), the leading term in the conductivity, i.e., $\max\{\text{const},T^{2}/\omega^{2}\}$ in the FL regime, vanishes in this case. Our goal is to derive the surviving term. Our main interest will be in the $\sigma^{\prime}_{1}(\omega,T)$ contribution [Eq. (28a)], because the general form of the $\sigma^{\prime}_{2}(\omega,T)$ contribution [Eq. (30a)] does not depend on the shape of the FS.

We focus on the $T=0$ case first and begin with a brief review of the arguments for the vanishing of the leading term, as given in Refs. Pal et al. (2012a) and Maslov et al. (2011). To leading order in $\max\{\omega,T\}$, we project the integrand in Eq. (III) onto the FS and neglect both frequencies in the delta functions ($\Omega$ and $\omega$). To ensure that the both delta functions are nonzero within the domain of integration, we need to find the solutions of two equations for the initial momenta ${\bf k}$ and ${\bf p}$ at fixed ${\bf q}$:

	$\displaystyle\varepsilon_{{\bf k}-{\bf q}}$	$\displaystyle=$	$\displaystyle\varepsilon_{\bf k}=0,$		(45a)
	$\displaystyle\varepsilon_{{\bf p}+{\bf q}}$	$\displaystyle=$	$\displaystyle\varepsilon_{\bf p}=0.$		(45b)

Excluding umklapps, the solutions of Eqs. (45a) and  (45b) are the points of intersection between the original FS contour and its two copied translated by $\pm{\bf q}$, see Fig. 1(a). Denoting $\bar{\bf p}=-{\bf p}$ and using that $\varepsilon_{-{\bf k}}=\varepsilon_{{\bf k}}$, we rewrite Eq (45b) as $\varepsilon_{\bar{\bf p}-{\bf q}}=\varepsilon_{\bar{\bf p}}=0$, upon which it becomes identical to Eq. (45b). For a convex FS, the first equation has (at most) two solutions, see Fig. 1(a). If we denote one of the solutions by ${\bf k}_{0}$, then the second one is $-{\bf k}_{0}+{\bf q}$. Indeed, $\varepsilon_{-{\bf k}_{0}+{\bf q}-{\bf q}}=\varepsilon_{-{\bf k}_{0}+{\bf q}}\to\varepsilon_{{\bf k}_{0}}=\varepsilon_{{\bf k}_{0}-{\bf q}}$. However, the equation for $\bar{\bf p}$ also has (at most) two solutions: $(\bar{\bf p}_{0},-\bar{\bf p}_{0}+{\bf q})=(-{\bf p}_{0},{\bf p}_{0}+{\bf q})$. Because the equations are the same, the solutions must also coincide, which leaves us with two choices: either ${\bf p}_{0}={\bf k}_{0}-{\bf q}$ or ${\bf p}_{0}=-{\bf k}_{0}$. The first choice corresponds to a “swap” scattering process, in which two electrons swap their initial momenta: ${\bf k}_{0}\to{\bf k}_{0}-{\bf q}={\bf p}_{0}$, ${\bf p}_{0}={\bf k}_{0}-{\bf q}\to{\bf p}_{0}+{\bf q}={\bf k}_{0}$, see Fig. 1 (b). The second choice is a Cooper (or head-on) scattering process: ${\bf k}_{0}\to{\bf k}_{0}-{\bf q}$, ${\bf p}_{0}=-{\bf k}_{0}\to\-{\bf k}_{0}+{\bf q}$; see Fig. 1(c). According to Eqs. (28a)-(28c), a scattering process contributes to the conductivity only if $\Delta\bm{\varv}=\bm{\varv}_{{\bf k}^{\prime}}+\bm{\varv}_{{\bf p}^{\prime}}-\bm{\varv}_{{\bf k}}-\bm{\varv}_{\bf p}\neq 0$. However, in a swap process $\bm{\varv}_{\bf k}^{\prime}=\bm{\varv}_{\bf p}$ and $\bm{\varv}_{\bf p}^{\prime}=\bm{\varv}_{\bf k}$, such that $\Delta\bm{\varv}=0$. Likewise, in a Cooper process $\bm{\varv}_{{\bf p}}=\bm{\varv}_{-{\bf k}}=-\bm{\varv}_{\bf k}$ and $\bm{\varv}_{{\bf p}^{\prime}}=\bm{\varv}_{-{\bf k}^{\prime}}=-\bm{\varv}_{{\bf k}^{\prime}}$, such that $\Delta\bm{\varv}=0$ again. Therefore, the leading term in the conductivity vanishes.

To obtain a nonzero result, we have to consider small deviations from the previously found solutions for the swap and Cooper channels, i.e.,

	$\displaystyle{\bf k}$	$\displaystyle=$	$\displaystyle{\bf k}_{0}+\delta{\bf k},\;{\bf p}={\bf p}_{0}+\delta{\bf p}={\bf k}_{0}-{\bf q}+\delta{\bf p},$		(46a)
	$\displaystyle{\bf k}$	$\displaystyle=$	$\displaystyle{\bf k}_{0}+\delta{\bf k},\;{\bf p}={\bf p}_{0}+\delta{\bf p}=-{\bf k}_{0}+\delta{\bf p}.$		(46b)

Expanding $\Delta\bm{\varv}$ to first order in $\delta{\bf k}$ and $\delta{\bf p}$ around ${\bf k}_{0}$ and ${\bf p}_{0}$, we obtain

	$\displaystyle\Delta\bm{\varv}_{\text{S}}$	$\displaystyle=$	$\displaystyle\left[(\delta{\bf k}-\delta{\bf p})\cdot\bm{\nabla}\right](\bm{\varv}_{{\bf k}_{0}}-\bm{\varv}_{{\bf k}_{0}-{\bf q}}),$		(47a)
	$\displaystyle\Delta\bm{\varv}_{\text{C}}$	$\displaystyle=$	$\displaystyle\left[(\delta{\bf k}+\delta{\bf p})\cdot\bm{\nabla}\right](\bm{\varv}_{{\bf k}_{0}-{\bf q}}-\bm{\varv}_{{\bf k}_{0}})$		(47b)

for the swap (S) and Cooper (C) channels, respectively. Without a loss of generality, we choose $\omega>0$. With this choice and at $T=0$, the $\sigma^{\prime}_{1}(\omega,0)$ contribution to the conductivity is reduced to

	$\displaystyle\sigma^{\prime}_{1}(\omega,0)$	$\displaystyle=$	$\displaystyle\frac{\pi e^{2}g^{2}}{(2\pi)^{6}\omega^{3}}\int d^{2}q\int d^{2}\delta k\int d^{2}\delta p\int^{0}_{-\omega}d\Omega\,F^{2}\left({\bf k}+\frac{{\bf q}}{2}\right)F^{2}\left({\bf p}-\frac{{\bf q}}{2}\right)V^{2}({\bf q})(\Delta\bm{\varv})^{2}$		(48)
			$\displaystyle\times\theta(\varepsilon_{\bf k})\theta(-\varepsilon_{\bf k}-\Omega)\theta(\varepsilon_{\bf p})\theta(-\varepsilon_{\bf p}+\Omega+\omega)\delta(\Omega-\varepsilon_{{\bf k}-{\bf q}}+\varepsilon_{{\bf k}})\delta(\Omega+\omega+\varepsilon_{{\bf p}+{\bf q}}-\varepsilon_{{\bf p}}).$

For any given ${\bf q}$, we need to find solutions ${\bf k}_{0}$ and ${\bf p}_{0}$ of Eq. (45a) and (45b), integrate over $\delta{\bf k}$ and $\delta{\bf p}$ near these solutions, and then integrate over ${\bf q}$.

We start with the swap channel, in which ${\bf p}_{0}={\bf k}_{0}-{\bf q}$. We expand the integrand of Eq. (48) in $\delta{\bf k}$ and $\delta{\bf p}$ and also use the condition $q\ll\langle k_{\mathrm{F}}\rangle$, where $\langle k_{\mathrm{F}}\rangle$ is a characteristic size of the FS. Then

	$\displaystyle\sigma^{\prime}_{1}(\omega,0)$	$\displaystyle=$	$\displaystyle\frac{\pi e^{2}g^{2}}{(2\pi)^{6}\omega^{3}}\int d^{2}qF^{4}({\bf k}_{0})V^{2}({\bf q})\int d^{2}\delta k\int d^{2}\delta p\int^{0}_{-\omega}d\Omega\,(\Delta\bm{\varv}_{\text{S}})^{2}$		(49)
			$\displaystyle\times\theta(\bm{\varv}_{{\bf k}_{0}}\cdot\delta{\bf k})\theta(-\bm{\varv}_{{\bf k}_{0}}\cdot\delta{\bf k}-\Omega)\theta(\bm{\varv}_{{\bf k}_{0}-{\bf q}}\cdot\delta{\bf p})\theta(-\bm{\varv}_{{\bf k}_{0}-{\bf q}}\cdot\delta{\bf p}+\Omega+\omega)\delta(\delta{\bf k}\cdot{\bf u}-\Omega)\delta(\delta{\bf p}\cdot{\bf u}-\Omega-\omega),$

where $\Delta\bm{\varv}_{\mathrm{S}}$ is given by Eq. (47a) and

$\displaystyle{\bf u}=\bm{\varv}_{{\bf k}_{0}-{\bf q}}-\bm{\varv}_{{\bf k}_{0}}\approx-\left({\bf q}\cdot\bm{\nabla}\right)\bm{\varv}_{{\bf k}_{0}}.$

(50)

The delta functions in Eq. (49) imply that the 2D integrals over $\delta{\bf k}$ and $\delta{\bf p}$ are, in fact, one-dimensional integrals along the straight lines:

	$\displaystyle\delta{\bf k}\cdot{\bf u}=\Omega,$		(51a)
	$\displaystyle\delta{\bf p}\cdot{\bf u}=\Omega+\omega.$		(51b)

It is convenient to choose $\delta k_{x}$ and $\delta p_{x}$ as independent integration variables and exclude $\delta k_{y}$ and $\delta p_{y}$ via

	$\displaystyle\delta k_{y}$	$\displaystyle=$	$\displaystyle\frac{\Omega}{u_{y}}-\delta k_{x}\frac{u_{x}}{u_{y}},$		(52a)
	$\displaystyle\delta p_{y}$	$\displaystyle=$	$\displaystyle\frac{\Omega+\omega}{u_{y}}-\delta p_{x}\frac{u_{x}}{u_{y}}.$		(52b)

The Pauli principle (imposed by the theta functions) and energy conservation (imposed by the delta-functions), confine $\delta k_{x}$ and $\delta p_{x}$ to the following intervals:

	$\displaystyle k_{\min}\leq$	$\displaystyle\delta k_{x}$	$\displaystyle\leq k_{\max},$		(53a)
	$\displaystyle p_{\min}\leq$	$\displaystyle\delta p_{x}$	$\displaystyle\leq p_{\max},$		(53b)

where

	$\displaystyle k_{\min}=\theta\left(\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0}-{\bf q},y}}{\nu}\Omega+\theta\left(-\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0},y}}{\nu}\Omega,$		(54a)
	$\displaystyle k_{\max}=\theta\left(\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0},y}}{\nu}\Omega+\theta\left(-\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0}-{\bf q},y}}{\nu}\Omega,$		(54b)
	$\displaystyle p_{\min}=\theta\left(\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0},y}}{\nu}(\Omega+\omega)+\theta\left(-\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0}-{\bf q},y}}{\nu}(\Omega+\omega),$		(54c)
	$\displaystyle p_{\max}=\theta\left(\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0}-{\bf q},y}}{\nu}(\Omega+\omega)+\theta\left(-\frac{\nu}{u_{y}}\right)\frac{\varv_{{\bf k}_{0},y}}{\nu}(\Omega+\omega),$		(54d)

and

	$\displaystyle\nu$	$\displaystyle=$	$\displaystyle\varv_{{\bf k}_{0}-{\bf q},x}\varv_{{\bf k}_{0},y}-\varv_{{\bf k}_{0}-{\bf q},y}\varv_{{\bf k}_{0},x}$		(55)
		$\displaystyle\approx$	$\displaystyle\varv_{{\bf k}_{0},x}\left({\bf q}\cdot\bm{\nabla}\right)\varv_{{\bf k}_{0},y}-\varv_{{\bf k}_{0},y}\left({\bf q}\cdot\bm{\nabla}\right)\varv_{{\bf k}_{0},x},$

with ${\bf u}$ defined by Eq. (50). The last step in Eq. (55) again accounts for the smallness of $q$.