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# Quantum criticality and optical conductivity in a two-valley system

Yasha Gindikin<EMAIL_ADDRESS>W.I. Fine Theoretical Physics Institute
and School of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota 55455, USA  Songci Li  Department of Physics and Tianjin Key
Laboratory of Low-Dimensional Materials Physics and Preparation Technology,
Tianjin University, Tianjin 300354, China  Alex Levchenko  Department of
Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA  Alex
Kamenev  Andrey V. Chubukov  W.I. Fine Theoretical Physics Institute and
School of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota 55455, USA  Dmitrii L. Maslov  Department of Physics, University of
Florida, Gainesville, FL 32611-8440, USA

###### Abstract

We demonstrate that the optical conductivity of a Fermi liquid (FL) in the
absence of umklapp scattering is dramatically affected by the topology of the
Fermi surface (FS). Specifically, electron-electron (_ee_) scattering leads to
rapid current relaxation in systems with multiple, or multiply connected, FSs,
provided the valleys have different effective masses. This effect results from
intervalley drag. We microscopically derive the optical conductivity of a two-
valley system, both within the FL regime and near a quantum critical point
(QCP) of the Ising-nematic type. In the FL regime, intervalley drag restores
the Gurzhi-like scaling of the conductivity,
$\operatorname{\mathrm{Re}}\sigma(\omega)\sim\omega^{0}$. This dependence
contrasts sharply with the previously identified sub-leading contribution to
the conductivity of a two-dimensional FL with a single convex FS, where
$\operatorname{\mathrm{Re}}\sigma(\omega)\sim\omega^{2}\ln|\omega|$. The
vanishing of the leading term in the optical conductivity is a signature of
geometric constraints on _ee_ scattering channels, which are lifted for a
multiply connected FS. A large differential response,
$d\operatorname{\mathrm{Re}}\sigma/d\mu$ with $\mu$ being the chemical
potential, is predicted at the Lifshitz transition from a single-valley to a
multi-valley FS, which should be observable within the experimentally
accessible frequency range. Near a QCP, intervalley drag leads to a
$|\omega|^{-2/3}$ scaling of $\operatorname{\mathrm{Re}}\sigma(\omega)$ in 2D,
thus providing a specific current-relaxing process for this long-standing
conjecture.

##  I Introduction

In recent years there has been a revival of interest to optical studies of
various materials with a non-trivial band structure, including graphene in
different incarnations [1, 2], transition metal dichalcogenides (TMD) [3, 4],
and Weyl and Dirac semimetals [5, 6]. Latest advancements in optical
spectroscopy, encompassing a broad frequency spectrum, have provided a unique
probe of intra-band electron dynamics, demonstrating how electron-electron
(ee) interactions influence current relaxation and manifest themselves in
distinct patterns of frequency scaling of the conductivity [7, 8, 9, 10, 11,
12, 13]. To a large degree, these studies are driven by an enduring interest
in the formation of novel quantum states.

The dissipative part of the optical conductivity of a non-Galilean-invariant
Fermi liquid (FL) at frequencies $\omega\tau\gg 1$, where $\tau\propto T^{-2}$
is the quasiparticle transport time, is believed to be described by the Gurzhi
formula [14],

$\operatorname{\mathrm{Re}}\sigma(\omega,T)=\sigma_{0}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\,,$
(1)

with $\sigma_{0}$ being the material-dependent constant. However, recent
studies have demonstrated that, in the absence of umklapp scattering, the
validity of the Gurzhi scaling is critically dependent on the Fermi surface
(FS) geometry and topology [15, 16, 11, 17, 18, 19, 20, 21]. Specifically, an
isotropic FS both in two-dimensional (2D) and three-dimensional (3D) systems,
and a convex 2D FS gives rise to a dramatic suppression of the conductivity
due to a slow current relaxation.111In 2D, a slow current relaxation is the
manifestation of a more general effect: a long lifetime of odd harmonics of
the electron distribution function [22, 23, 24, 25, 26]. This unfolds as
follows.

Regardless of the method employed — be it equations of motion for the current
operator [15, 16, 17, 19], Fermi Golden Rule (FGR) [27, 28], or diagrammatic
expansion of the Kubo formula [11, 21] — the optical conductivity is
ultimately proportional to $(\Delta\bm{v})^{2}$, averaged over the FS with the
scattering probability, where

$\Delta\bm{v}\equiv\bm{v}_{\bm{p}}+\bm{v}_{\bm{p}^{\prime}}-\bm{v}_{\bm{k}}-\bm{v}_{\bm{k^{\prime}}}$
(2)

is a change of the total group velocity (proportional to the total current)
due to a collision of two electrons with initial momenta $\bm{k}$ and
$\bm{k}^{\prime}$, and final momenta $\bm{p}$ and $\bm{p}^{\prime}$. In a
Galilean-invariant system, $\bm{v}=\bm{k}/m$, and $\Delta\bm{v}$ is equal to
zero identically by momentum conservation. Same happens for an isotropic FS
both in 2D and 3D, except for in this case $\Delta\bm{v}$ vanishes only on the
FS rather than identically [15, 16, 17]. For a convex FS in 2D, the only
allowed scattering channels are the Cooper channel, in which
${\bm{k}}=-{\bm{k}}^{\prime}$ and ${\bm{p}}=-{\bm{p}}^{\prime}$, and the swap
channel, in which ${\bm{k}}^{\prime}={\bm{p}}$ and
${\bm{p}}^{\prime}={\bm{k}}$, and $\Delta{\bm{v}}=0$ for both these channels
[22, 23, 24, 25, 29, 30]. As long as $\Delta{\bm{v}}=0$, the conductivity
vanishes to leading order for all the cases described above and, to obtain a
finite result, one needs to expand $\Delta{\bm{v}}$ either around the FS (for
an isotropic FS) or near the Cooper and swap solutions (for a convex FS). This
leads to a strong suppression of the conductivity compared to the Gurzhi
formula:
$\operatorname{\mathrm{Re}}\sigma(\omega,T)\propto\max\\{\omega^{2}\ln|\omega|^{-1},T^{4}\ln
T^{-1}/\omega^{2}\\}$ in 2D and
$\operatorname{\mathrm{Re}}\sigma(\omega,T)\propto\max\\{\omega^{2},T^{4}/\omega^{2}\\}$
in 3D.

The Gurzhi result for the conductivity is restored under specific conditions.
In 3D, this occurs when anisotropic terms in the electron dispersion beyond
the quadratic level are taken into account. In 2D, the Gurzhi result is valid
for a concave FS, which allows for scattering channels beyond the Cooper and
swap ones with $\Delta{\bm{v}}\neq 0$ [29, 30, 19, 21].

In this work, we consider another mechanism of fast, Gurzhi-like current
relaxation: _ee_ scattering in systems with multiple, or multiply connected,
FSs with different effective masses. At the phenomenological level, the case
of a multivalley conductor with parabolic dispersions in each valley can be
analyzed via the classical equations of motion, in which the intervalley
interaction is accounted for by the drag force, proportional to the difference
of the drift velocities in different valleys. The drag force renders the ee
contribution to the conductivity finite, both in the optical case and in the
_dc_ case in the presence of disorder [31, 11]. The _dc_ case was further
analysed in Refs. [29] and [30] via the semiclassical Boltzmann equation. The
goal of the present paper is to derive the optical conductivity of a two-
valley system microscopically, both in the FL regime and in the situation when
the _ee_ interaction drives the system to the quantum critical point (QCP) of
the Ising-nematic type [32].

We start with a simpler problem of a two-valley FL with Coulomb interaction
within and between two nonequivalent valleys. Our model spans a wide range of
2D systems and materials, from systems with non-parabolic spectrum and valley
imbalance, typical for TMDs, to systems with annular FS, including biased
bilayer and rhombohedral trilayer graphene. We calculate the optical
conductivity via the FGR and show that it exhibits the Gurzhi scaling of Eq.
(1). The interest to such systems lies in the ability to manipulate the
scattering channels by electrical means, which promises large differential
responses at the threshold of the channel opening, thus crafting a unique
experimental signature within the experimentally accessible frequency range.

We proceed with a more intricate example of a two-valley system, in which one
of the valleys is tuned to an Ising-nematic QCP, while the other valley plays
the role of a momentum sink for the first one. We demonstrate that the
intervalley conductivity scales with frequency as $\lvert\omega\rvert^{-2/3}$
at the QCP in 2D, which is the same as conjectured long time ago in Ref. [33]
in the context of fermions interacting with a $U(1)$ gauge field.

##  II Two-valley Fermi liquid away from criticality

###  II.1 The model

We start with an effective Hamiltonian of a two-valley electron
system222Extending our approach to accommodate an arbitrary number of valleys
is conceptually straightforward.

$H=\sum_{\bm{k},s}(\varepsilon_{\bm{k},s}-\mu)c_{\bm{k},s}^{\dagger}c_{\bm{k},s}^{\vphantom{\dagger}}+\frac{1}{2}\sum_{\bm{q}ss^{\prime}}u_{ss^{\prime}}(\bm{q})\rho_{\bm{q},s}\rho_{-\bm{q},s^{\prime}}\,.$
(3)

Here, $c_{\bm{k},s}$ and $\rho_{\bm{q},s}$ represent the electron annihilation
operator and the density operator, respectively, $\varepsilon_{\bm{k},s}$ is
the electron dispersion, $\mu$ is the chemical potential, and indices
$s,s^{\prime}=1,2$ label distinct valleys. We ignore the electron spin if the
valleys do not result from spin splitting; if they do, $s$ and $s^{\prime}$
label the spin projection. The Hamiltonian captures two types of _ee_
interactions involving small momentum transfers: intravalley interaction,
denoted by $u_{ss}(\bm{q})$, and intervalley drag, denoted by
$u_{ss^{\prime}}(\bm{q})$, but neglects the intervalley swaps of electrons.
The last assumption is justified if the valley centers are located far away
from each other or, for the case of concentric valleys, if their Fermi momenta
are widely different. For simplicity, we will consider the case of isotropic
but otherwise arbitrary $\varepsilon_{{\bm{k}},s}$; adding anisotropy of the
FS does not change the scaling form of
$\operatorname{\mathrm{Re}}\sigma(\omega,T)$. We focus on the 2D case first
and discuss the 3D case at the end of this section.

The distinction between $u_{11}$, $u_{22}$ and $u_{12}$ becomes significant
near the QCP, a topic explored in detail in the Section III. Here, away from
criticality, we simplify the model by assuming uniformity of the interactions:
$u_{ss}(\bm{q})=u_{ss^{\prime}}(\bm{q})=\mathcal{U}_{\bm{q}}$. In the static
limit, we consider the 2D screened Coulomb interaction in the Thomas-Fermi
approximation,

$\mathcal{U}(\bm{q})=\frac{2\pi
e^{2}}{\lvert\bm{q}\rvert+\kappa},\;\kappa=2\pi
e^{2}(N_{\mathrm{F},1}+N_{\mathrm{F},2})\,,$ (4)

with $N_{\mathrm{F},s}$ being the density of states at the Fermi energy in
valley $s$. We assume that the interaction is sufficiently long-ranged, such
that the condition to neglect intervalley swaps is satisfied.

The gradient part of the current operator is given by

$\bm{J}=\sum_{\bm{k},s}\bm{v}_{\bm{k},s}c_{\bm{k},s}^{\dagger}c_{\bm{k},s}^{\vphantom{\dagger}},\quad\bm{v}_{\bm{k},s}=\grad_{\bm{k}}\varepsilon_{\bm{k},s}=\hat{k}\,v_{\bm{k},s}\,,$
(5)

where $\hat{k}\equiv\bm{k}/k$.

The real part of the conductivity, $\sigma(\omega)$, can be derived via the
FGR (see Refs. [27, 28] and Appendix A) up to the two-loop order to give

$\displaystyle\operatorname{\mathrm{Re}}\sigma(\omega,T)={}$
$\displaystyle\frac{\pi e^{2}}{2\omega
D}\left(1-e^{-\frac{\omega}{T}}\right)\sum_{\stackrel{{\scriptstyle\bm{k}\bm{k^{\prime}}\bm{q}}}{{ss^{\prime}}}}\int\\!d\Omega\,\lvert\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})\rvert^{2}n\left(\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}-\Omega\right)n\left(\varepsilon_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}-\omega+\Omega\right)\left(1-n\left(\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}\right)\right)\left(1-n\left(\varepsilon_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}\right)\right)$
$\displaystyle{}\times\delta\left(\Omega+\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s}-\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}\right)\delta\left(\omega-\Omega+\varepsilon_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}-\varepsilon_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}\right)\,,$
(6)

where $\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})$ is the
matrix element of the Coulomb interaction and
$n(\varepsilon_{\bm{k},s})={\left(e^{\frac{\varepsilon_{\bm{k}s}-\mu}{T}}+1\right)}^{-1}$
is the Fermi function, and $D$ is the spatial dimensionality. Note that the
conductivity represents the sum of intravalley, $\sigma_{ss}$, and
intervalley, $\sigma_{s\neq s^{\prime}}$, contributions, of which we are
mostly interested in the latter.

For our case of a long-range interaction, the exchange part of the matrix
element can be neglected, and then

$\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})=\frac{\mathcal{U}(\bm{q})\Delta\bm{v}_{ss^{\prime}}}{\omega}\,,$
(7)

where

$\Delta\bm{v}_{ss^{\prime}}\equiv\bm{v}_{\bm{k}+\tfrac{\bm{q}}{2},s}+\bm{v}_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}-\bm{v}_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}-\bm{v}_{\bm{k}-\tfrac{\bm{q}}{2},s}$
(8)

is the velocity change due to a two-electron scattering process in a two-
valley system. In contrast to the single-valley case,
$\Delta\bm{v}_{ss^{\prime}}\neq 0$ even for the parabolic dispersions
$\varepsilon_{{\bm{k}},s}=k^{2}/2m_{s}$, as long as $m_{1}\neq m_{2}$.

###  II.2 Fermi surface geometry

The velocity change in Eq. (8) can be expanded, to the leading order in the
momentum transfer $q$, into the sum of two qualitatively different terms,
$\Delta\bm{v}_{ss^{\prime}}=\Delta\mathfrak{u}_{ss^{\prime}}+\Delta\mathfrak{w}_{ss^{\prime}}$,
where

$\Delta\mathfrak{u}_{ss^{\prime}}=\bm{q}\left(\frac{1}{\bar{m}_{s}(k)}-\frac{1}{\bar{m}_{s^{\prime}}(k^{\prime})}\right)\,,$
(9)

and

$\begin{split}\Delta\mathfrak{w}_{ss^{\prime}}={}&\left(\frac{1}{m^{*}_{s}(k)}-\frac{1}{\bar{m}_{s}(k)}\right)\hat{k}(\bm{q}\cdot\hat{k})\\\
&-\left(\frac{1}{m^{*}_{s^{\prime}}(k^{\prime})}-\frac{1}{\bar{m}_{s^{\prime}}(k^{\prime})}\right)\hat{k}^{\prime}(\bm{q}\cdot\hat{k}^{\prime})\,,\end{split}$
(10)

with $1/\bar{m}_{s}(k)=(1/k)\partial\varepsilon_{{\bm{k}},s}/\partial k$ and
$1/m^{*}_{s}(k)=\partial^{2}\varepsilon_{{\bm{k}},s}/\partial k^{2}$ being the
(inverse) density-of-states and band masses, respectively.

We call two valleys “equivalent”, if they have equal density-of-states masses
on the corresponding FSs, i.e.,
$\bar{m}_{1}\equiv\bar{m}(k_{\mathrm{F},1})=\bar{m}_{2}\equiv\bar{m}(k_{\mathrm{F},2})$.
For equivalent valleys, $\Delta\mathfrak{u}_{ss^{\prime}}$ gives zero when
projecting $k$ and $k^{\prime}$ onto the corresponding FSs. In this case, the
main contribution to the conductivity comes from
$\Delta\mathfrak{w}_{ss^{\prime}}$ but, similarly to the single-valley case,
this contribution is suppressed as compared to the Gurzhi result of Eq.
(1).333Note that $\Delta\mathfrak{w}_{ss^{\prime}}=0$ for a parabolic
dispersion.

For nonequivalent valleys the situation is quite the opposite — the dominant
contribution to the conductivity is due to $\Delta\mathfrak{u}_{ss^{\prime}}$,
and this dramatically affects the scaling form of the conductivity.

###  II.3 Optical conductivity

The dissipative conductivity, to the leading order of the interaction
strength, is determined by all possible _ee_ scattering processes accompanied
by the creation of two electron-hole pairs.

Energy conservation in a process of _ee_ scattering is ensured by two
$\delta$-functions in Eq. (II.1):

$\begin{split}&\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}-\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s}=\Omega\,,\\\
&\varepsilon_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}-\varepsilon_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}=\omega-\Omega\,.\end{split}$
(11)

To leading order in $\max\\{\omega,T\\}$, one can set the right-hand sides of
the last two equations to zero. Then, at fixed ${\bm{q}}$, the allowed
electron momenta $\bm{k}$ satisfy the equation
$\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2}}=\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2}}$.
Geometrically, the solutions correspond to the intersection points between two
FSs, shifted by the momentum transfer $\bm{q}$.

In what follows we assume that the electron dispersion is even in ${\bm{k}}$,
i.e., $\varepsilon_{{\bm{k}},s}=\varepsilon_{-{\bm{k}},s}$. The time-reversal
symmetry guarantees this to be the case if spin-orbit interaction can be
neglected, regardless of whether the inversion symmetry is present or not
[34]. If so, the intersection points arise in pairs at ${\bm{k}}$ and
$-{\bm{k}}$, like at points $A$ and $A^{\prime}$ in Fig. 1.

Figure 1: The intersection points between two Fermi surfaces, shifted by the
momentum transfer ${\bm{q}}$.

The dominant contribution to the conductivity at the smallest energy transfer
$\Omega$ (which is of the same order as the photon frequency $\omega$) comes
from _ee_ scattering in a close vicinity of the intersection points. A channel
of electron-electron scattering is defined by any pair of the intersection
points. The contributions from distinct channels are being summed up in the
expression for the conductivity.

For a single circular (and, more general, convex) FS in 2D, there are at most
two intersections between the shifted FSs. In this case, the only scattering
channels for electrons are the swap channel with
$\bm{k}\approx\bm{k}^{\prime}$ (channels $\\{A,A\\}$ and
$\\{A^{\prime},A^{\prime}\\}$ in Fig. 1) and Cooper channel with
$\bm{k}\approx-\bm{k}^{\prime}$ (channels $\\{A,A^{\prime}\\}$ and
$\\{A^{\prime},A\\}$ ibid.)

The situation changes qualitatively for electrons residing in nonequivalent
valleys. The new scattering channels arise for two electron situated near the
intersection points of distinct FSs: $\\{A,B\\}$, $\\{A,B^{\prime}\\}$, etc.
When two electrons are scattered close to these points, their momenta align in
the same or opposing directions, $\hat{k}\approx\hat{k}^{\prime}$ or
$\hat{k}\approx-\hat{k}^{\prime}$, but the absolute values of their momenta
are no longer equal, $\lvert\bm{k}\rvert\neq\lvert\bm{k}^{\prime}\rvert$, in a
distinction from the swap and Cooper channels.

The intervalley ($s^{\prime}\neq s$) contribution to the conductivity of Eq.
(II.1) is determined by the factor $\Delta\bm{v}_{ss^{\prime}}$, evaluated for
$\bm{k}$ and $\bm{k}^{\prime}$ on the corresponding FSs. In this case, the
leading contribution to $\Delta\bm{v}$ comes from
$\Delta\mathfrak{u}_{ss^{\prime}}=\bm{q}\delta$ of Eq. (9), with the mass
mismatch

$\delta=\frac{v_{Fs}}{k_{Fs}}-\frac{v_{Fs^{\prime}}}{k_{Fs^{\prime}}}=\frac{1}{\bar{m}_{s}}-\frac{1}{\bar{m}_{s^{\prime}}}\,.$
(12)

For nonequivalent valleys, $\delta\neq 0$.

It is crucially important that, in contrast to
$\delta\mathfrak{w}_{ss^{\prime}}$ of Eq. (10),
$\Delta\mathfrak{u}_{ss^{\prime}}$ is not small when electron momenta are
close to their respective intersection points. This smallness, inherent in Eq.
(10), reflects the small-angle nature of the ee scattering, and is enforced by
the dot product of $\bm{q}$ and $\hat{k}$,

$(\bm{q}\cdot\hat{k})\approx
v_{ks}^{-1}\left(\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}-\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s}\right)\,,$
(13)

which, according to Eq. (11), gives an extra power of frequency to
conductivity (see Eq. (21) below).

Hence, the intervalley conductivity is given by

$\begin{split}\operatorname{\mathrm{Re}}\sigma_{s\neq
s^{\prime}}(\omega)={}&\frac{e^{2}}{4\pi\omega^{3}}(1-e^{-\frac{\omega}{T}})\sum_{\bm{q}}\int
d\Omega\,\mathcal{U}_{\bm{q}}^{2}{(\Delta\mathfrak{u}_{ss^{\prime}})}^{2}\\\
&\times{\mathrm{Im}}\mathfrak{P}_{s}(\Omega,\bm{q}){\mathrm{Im}}\mathfrak{P}_{s^{\prime}}(\omega-\Omega,\bm{q})\,,\end{split}$
(14)

with

$\mathfrak{P}_{s}(\Omega,\bm{q})=\sum_{\bm{k}}\frac{n(\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}-\Omega)(1-n(\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}))}{\Omega+\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s}-\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}+i0}\,.$
(15)

Note that $\mathfrak{P}_{s}$ is distinct from the electron polarization
$\Pi_{s}$, which arises from the interplay of “absorption” and “emission”
processes, represented as $n(1-n^{\prime})-n^{\prime}(1-n)$. In contrast,
$\mathfrak{P}_{s}$ only involves the “absorption” component. Consequently, its
imaginary part is not an odd function of the frequency.

In the limit of $v_{\mathrm{F},s}q\gg\lvert\Omega\rvert$, one gets

${\mathrm{Im}}\mathfrak{P}_{s}(\Omega,\bm{q})=\frac{N_{\mathrm{F,s}}}{e^{-\frac{\Omega}{T}}-1}\frac{\Omega}{v_{\mathrm{F},s}q}\,.$
(16)

A subsequent integration over $\Omega$ yields

$\int_{-\infty}^{\infty}\frac{\Omega(\omega-\Omega)\,d\Omega}{\left(e^{-\frac{\Omega}{T}}-1\right)\left(e^{-\frac{\omega-\Omega}{T}}-1\right)}=\frac{\omega^{3}}{6\left(1-e^{-\frac{\omega}{T}}\right)}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\,.$
(17)

In 2D, the integral over momentum transfer is regularized by imposing the
ultra-violet cutoff $\lambda=\min\\{k_{Fs},k_{Fs^{\prime}}\\}$ to give

$\int q\mathcal{U}^{2}_{\bm{q}}\,dq={(2\pi
e^{2})}^{2}\left[\ln\left(1+\frac{\lambda}{\kappa}\right)-\frac{\lambda}{\lambda+\kappa}\right]\,,$
(18)

where the expression in brackets, $\mathcal{C}$, interpolates between
$\mathcal{C}=\ln{(\lambda/\kappa)}$ for $\lambda\gg\kappa$, and
$\mathcal{C}=\lambda^{2}/(2\kappa^{2})$ for $\lambda\ll\kappa$.

This way, we obtain the intervalley conductivity in 2D,

$\sigma_{s\neq
s^{\prime}}(\omega)=\frac{\mathcal{C}\delta^{2}}{12}e^{2}\frac{e^{2}N_{\mathrm{F,s}}}{v_{\mathrm{F},s}}\frac{e^{2}N_{\mathrm{F,s^{\prime}}}}{v_{\mathrm{F},s^{\prime}}}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\,.$
(19)

Remarkably, the signature of the FL state — the Gurzhi scaling of Eq. (1) — is
reinstated due to the drag between nonequivalent valleys.

The 3D case can be considered along the same lines. Although a generic 3D FS
allows for infinitely many channels of current relaxation, the intravalley
contribution to the conductivity still vanishes to leading order for the
special case of an isotropic FS. Indeed, the group velocity in this case is
given by ${\bm{v}}_{{\bm{k}},s}={\bm{k}}/\bar{m}_{s}(k)$. Projecting all
momenta in Eq. (8) onto the FS, we obtain
$\Delta{\bm{v}}_{ss}|_{k=k_{\mathrm{F},s}}=0$. A finite result is obtained by
expanding $\Delta{\bm{v}}_{ss}$ near the FS, and the resultant conductivity is
suppressed by a factor of $\max\\{\omega^{2},T^{4}/\omega^{2}\\}$ [17, 35].
The intervalley contribution differs from the 2D result (19) only by a
numerical coefficient due to a different form of the screened Coulomb
interaction.

###  II.4 Applications

Consider a two-valley Dirac metal with the dispersion of
$\varepsilon_{\bm{k}s}=v\lvert\bm{k}\rvert$ and valley imbalance — be it
interaction-driven or invoked by the Zeeman splitting — characterized by
$\zeta=\Delta\varepsilon_{F}/\mu=|\varepsilon_{\mathrm{F},1}-\varepsilon_{\mathrm{F},2}|/\mu$.
In this case the total intervalley conductivity
$\sigma=\sigma_{12}+\sigma_{21}$ becomes

$\operatorname{\mathrm{Re}}\sigma(\omega)=e^{2}\frac{\alpha^{2}\lvert\ln{\alpha}\rvert}{24\pi^{2}}\zeta^{2}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\,,$
(20)

with the effective fine structure constant $\alpha=e^{2}/v$.

The intravalley contribution [17, 35]

$\operatorname{\mathrm{Re}}\sigma_{ss}(\omega)=\frac{e^{2}}{960\pi^{2}}\frac{\omega^{2}}{\varepsilon_{\mathrm{F},s}^{2}}\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\left(3+\frac{8\pi^{2}T^{2}}{\omega^{2}}\right)\ln\frac{\alpha
k_{\mathrm{F},s}}{\Lambda}\,,$ (21)

with infrared cutoff $\Lambda=\max\left\\{\omega/v,T/v\right\\}$, is
suppressed by a factor of $\max\\{\omega^{2}\ln\omega,T^{4}\ln
T/\omega^{2}\\}$, as compared to Eq. (20).

The next example is given by a biased bilayer graphene, which features the
moat-band dispersion of the form [36]

$\varepsilon_{\bm{k}}=\frac{V}{2}-V\frac{v^{2}}{t_{\perp}^{2}}k^{2}+\frac{v^{4}}{t_{\perp}^{2}V}k^{4}\,,$
(22)

where $t_{\perp}$ the interlayer hopping parameter and $V$ the voltage drop
across the layers. Provided that a single band is occupied, a multiply-
connected annular FS, depicted in Fig. 1, arises if $V>2\mu$.

Here, the mass mismatch between the inner and outer components of the FS
equals

$\delta=\frac{4v^{4}}{t_{\perp}^{2}V}(k_{Fs}^{2}+k_{Fs^{\prime}}^{2})\approx\frac{4v^{2}V}{t_{\perp}^{2}}\,,$
(23)

and $\mathcal{C}\approx(V-2\mu)t_{\perp}^{2}/4\kappa^{2}v^{2}V$, for $\mu$
close to $2V$. This gives rise to the conductivity exhibiting a threshold
behavior, described by the following expression:

$\operatorname{\mathrm{Re}}\sigma(\omega)=\mathcal{D}\,\theta(V-2\mu)(V-2\mu)\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\,,$
(24)

where $\mathcal{D}$ represents a combination of the system parameters. This
threshold behavior marks the Lifshitz transition, at the voltage $V$ equal to
$2\mu$. 444While the model predicts a sharp transition at this threshold, it
is important to note that at finite temperature, this transition is not
perfectly abrupt. Specifically, we assume that the temperature $T$ is much
smaller than the Fermi energies
$T\ll\min\\{\varepsilon_{F,1},\varepsilon_{F,2}\\}$, yet in the vicinity of
the transition it becomes large enough to introduce a smearing around the
threshold on the scale of $T$.

The valence band of biased rhombohedral trilayer graphene also features a
moat-band dispersion. While the low-energy Hamiltonian fully accounting for
the moat-band structure is yet to be derived from the many-band model [37],
for current purposes it is sufficient to approximate it by the dispersion
similar to Eq. (22), again neglecting the trigonal warping effect. Then we
obtain

$\operatorname{\mathrm{Re}}\sigma(\omega)=\tilde{\mathcal{D}}\,\theta(2\mu-V)(2\mu-V)\left(1+\frac{4\pi^{2}T^{2}}{\omega^{2}}\right)\,.$
(25)

In modern double-gate experimental setups, it’s possible to independently
adjust the voltage across layers without changing the electron concentration.
This allows for the electrical control of conductivity, according to Eqs.
(24–25), at fixed electron density. Opening a new scattering channel within
the small FS by adjusting the chemical potential to cross the peak of the
“Mexican hat” gives rise to the jump in the differential conductivity, which
could be directly observed through modulation spectroscopy [38].

##  III Quantum-critical two-valley system

###  III.1 The Model

####  III.1.1 Charge currents

In this section, we consider a two-valley system near an Ising-nematic quantum
critical point. The interaction part of the Hamiltonian is the same as in Eq.
(3), except for the form-factors that project the interaction onto a channel
with a given angular momentum. Accordingly, $u_{ss^{\prime}}({\bm{q}})\to
F_{s}({\bm{k}})F_{s^{\prime}}({\bm{p}})u_{ss^{\prime}}({\bm{q}})$. Due to
these form-factors, the electric field couples not only to the free but also
to the interaction part of the Hamiltonian. Correspondingly, the current
contains the regular part, given by Eq. (5), and the “anomalous” part [19,
21]:

$\displaystyle\bm{J}_{\text{an}}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{k}},{\bm{p}},{\bm{q}},s,s^{\prime}}\left[\left(\grad_{{\bm{k}}}+\grad_{\bm{p}}\right)F_{s}({\bm{k}})F_{s^{\prime}}({\bm{p}})\right]u_{ss^{\prime}}({\bm{q}})$
(26) $\displaystyle\times
c^{\dagger}_{{\bm{k}}+\frac{{\bm{q}}}{2},s}c^{\dagger}_{{\bm{p}}-\frac{{\bm{q}}}{2},s^{\prime}}c^{\phantom{\dagger}}_{{\bm{p}}+\frac{{\bm{q}}}{2},s^{\prime}}c^{\phantom{\dagger}}_{{\bm{k}}-\frac{{\bm{q}}}{2},s}.$

The time derivatives of the current operators in Eqs. (5) and (26) show
directly which processes contribute to current relaxation. For the two-valley
case, we obtain

$\displaystyle\partial_{t}\bm{J}=-i[H,\bm{J}]=-i[H_{\text{int}},\bm{J}]=-\frac{i}{2}\sum_{{\bm{k}},{\bm{p}},{\bm{q}},s,s^{\prime}}$
$\displaystyle({\bm{v}}_{{\bm{k}}+{\bm{q}}/2,s}+{\bm{v}}_{{\bm{p}}-{\bm{q}}/2,s^{\prime}}-{\bm{v}}_{{\bm{k}}-{\bm{q}}/2,s}-{\bm{v}}_{{\bm{p}}+{\bm{q}}/2,s^{\prime}})F_{s}({\bm{k}})F_{s^{\prime}}({\bm{p}})u_{ss^{\prime}}({\bm{q}})$
(27a) $\displaystyle\times
c^{\dagger}_{{\bm{k}}+{\bm{q}}/2,s}c^{\dagger}_{{\bm{p}}-{\bm{q}}/2,s^{\prime}}c^{\phantom{\dagger}}_{{\bm{p}}+{\bm{q}}/2,s^{\prime}}c^{\phantom{\dagger}}_{{\bm{k}}-{\bm{q}}/2,s},$
and
$\displaystyle\partial_{t}\bm{J}_{\text{an}}=-i[H,\bm{J}_{\text{an}}]=-i[H_{0},\bm{J}_{\text{an}}]=i\sum_{{\bm{k}},{\bm{p}},{\bm{q}},s,s^{\prime}}$
$\displaystyle\left(\varepsilon_{{\bm{k}}+{\bm{q}}/2,s}+\varepsilon_{{\bm{p}}-{\bm{q}}/2,s^{\prime}}-\varepsilon_{{\bm{k}}-{\bm{q}}/2,s}-\varepsilon_{{\bm{p}}+{\bm{q}}/2,s^{\prime}}\right)(\grad_{{\bm{k}}}+\grad_{{\bm{p}}})F_{s}({\bm{k}})F_{s^{\prime}}({\bm{p}})u_{ss^{\prime}}({\bm{q}})$
(27b) $\displaystyle\times
c^{\dagger}_{{\bm{k}}+{\bm{q}}/2,s}c^{\dagger}_{{\bm{p}}-{\bm{q}}/2,s^{\prime}}c^{\phantom{\dagger}}_{{\bm{p}}+{\bm{q}}/2,s^{\prime}}c^{\phantom{\dagger}}_{{\bm{k}}-{\bm{q}}/2,s}.$

We see that $\partial_{t}\bm{J}_{\text{an}}$ in Eq. (27b) contains the
difference between the energies of two electrons before and after a collision,
which is equal to the photon energy, $\omega$. Therefore, the corresponding
contribution to the conductivity is suppressed by a factor of $\omega^{2}$
[19, 21]. On the contrary, $\partial_{t}\bm{J}$ in Eq. (27a) contains the
difference of the velocities of the initial and final states. This difference
is non-zero for non-degenerate valleys, even if the electron spectrum in each
of the valleys is parabolic, and the corresponding contribution to the
conductivity is not suppressed. For this reason, we will neglect the anomalous
part of the current. For the normal part of the current, the form-factors do
not bring any qualitative changes, and we will ignore them in what follows.
Thus, the interaction part of the Hamiltonian is reduced back to Eq. (3).

A FL near a QCP is strongly interacting, i.e., its $Z$-factor is much smaller
than one (or, equivalently, the renormalized mass is much larger than the bare
one). To keep track of the Z-factor renormalization, we employ the
diagrammatic treatment of the Kubo formula in this section, as opposed to the
FGR scheme employed in Sec. II.

####  III.1.2 Quantum criticality in a two-valley system

As in the previous part of the paper, we assume that valleys are located
sufficiently far from each other, such that the exchange of fermions between
the valleys can be neglected. We adopt a model of Hubbard-like interaction
($u_{ss^{\prime}}=\mathrm{const})$ and, for simplicity, assume that fermions
in valley 2 do not interact with each other ($u_{22}=0$). Then the only bare
interactions in the model are the intravalley interaction in valley 1
$(u_{11})$ and intervalley drag ($u_{12}$). As before, we also assume that the
electron spectrum in each valley is isotropic but not necessarily parabolic,
but consider both the 2D and 3D cases on the same footing.

Within the Random Phase Approximation (RPA), the matrix of dressed interaction
is given by555RPA is controllable if the number of fermionic flavors in each
valley, $N$, is large. We assume this to be the case but, for brevity, do not
display $N$ in the formulas.

$\displaystyle\hat{U}=\hat{u}\left(\hat{\mathbb{1}}-\hat{\Pi}\hat{u}\right)^{-1},$
(28)

where

$\displaystyle\Pi_{ss^{\prime}}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle
T\sum_{\nu_{m}}\int\frac{d^{D}k}{(2\pi)^{2}}G_{s}({\bm{k}}+{\bm{q}}/2,\nu_{m}+\Omega_{m}/2)$
(29) $\displaystyle\times
G_{s^{\prime}}({\bm{k}}-{\bm{q}}/2,\nu_{m}-\Omega_{m}/2)$

and $G_{s}({\bm{k}},\nu_{m})$ is the Green’s function of valley $s$.

The solution of Eq. (28) for our case of $u_{22}=0$ is given by

$\displaystyle U_{11}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{u_{11}+\Pi_{22}({\bm{q}},\Omega_{m})}{R}u^{2}_{12}$ (30a)
$\displaystyle=$
$\displaystyle\frac{u_{11}}{R_{1}({\bm{q}},\Omega_{m})}+\frac{\Pi_{22}({\bm{q}},\Omega_{m})u^{2}_{12}}{R_{1}({\bm{q}},\Omega_{m})R({\bm{q}},\Omega_{m})},$
$\displaystyle U_{12}({\bm{q}},\Omega_{m})$ $\displaystyle=$ $\displaystyle
U_{21}({\bm{q}},\Omega_{m})=\frac{u_{12}}{R({\bm{q}},\Omega_{m})},$ (30b)
$\displaystyle U_{22}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{\Pi_{11}({\bm{q}},\Omega_{m})u_{12}^{2}}{R({\bm{q}},\Omega_{m})},$
(30c)

where

$\displaystyle R_{1}({\bm{q}},\Omega_{m})$ $\displaystyle=$ $\displaystyle
1-u_{11}\Pi_{11}({\bm{q}},\Omega_{m}),$ (31a) $\displaystyle
R({\bm{q}},\Omega_{m})$ $\displaystyle=$ $\displaystyle
R_{1}({\bm{q}},\Omega_{m})-\Pi_{11}({\bm{q}},\Omega_{m})\Pi_{22}({\bm{q}},\Omega_{m})u_{12}^{2}.$

The QCP in a two-valley system occurs when $R({\bm{q}}\to 0,\Omega_{m}=0)=0$,
which corresponds to the condition

$\displaystyle 1+\lambda_{11}-\lambda_{12}=0,$ (32)

where

$\displaystyle\lambda_{11}$ $\displaystyle=$ $\displaystyle
N_{\mathrm{F},1}u_{11}$ (33a) $\displaystyle\mathrm{and}\;\lambda_{12}$
$\displaystyle=$ $\displaystyle N_{\mathrm{F},1}N_{\mathrm{F},2}u^{2}_{12}$
(33b)

are the dimensionless couplings of the intra- and intervalley interaction,
respectively, and $N_{\mathrm{F},s}$ is the density of states in valley $s$ at
the corresponding Fermi energy. For future convenience, we singled out the
first term in Eq. (30a), which is the intravalley interaction in the absence
of coupling between the valleys.

In what follows, we will need the asymptotic forms of the polarization bubbles
for

$\displaystyle|\Omega_{m}|\ll\varv_{\text{F},s}q\ll k_{\mathrm{F},s},$ (34)

which are given by

$\displaystyle\Pi_{ss}({\bm{q}},\Omega_{m})=-N_{F,s}\left(1-\frac{q^{2}}{q^{2}_{\mathrm{F},s}}-C_{D}\frac{|\Omega_{m}|}{\varv_{\text{F},s}q}\right),$
(35)

where $C_{2}=1$, $C_{3}=\pi/2$, and $q_{\text{F},s}\sim k_{\text{F},s}$.666A
$q^{2}$ term in the free-fermion polarization bubble is present as long as the
band dispersion is neither parabolic nor linear. In addition, the interaction
between low-energy fermions generate extra $q^{2}$ terms [39]. It is assumed
that the $q^{2}$ term in Eq. (35) includes both the free-fermion and
interaction-induced parts.

Now we focus on the immediate vicinity of the QCP, where $R({\bm{q}},\Omega)$
in Eq. (31a) is finite but small. According to Eqs. (31a) and (LABEL:R12), the
factor $1/R_{1}$ in Eq. (30a) can be then replaced by
$1/\Pi_{11}({\bm{q}},\Omega_{m})\Pi_{22}u_{12}^{2}$, which in non-singular.
Therefore, the first term in this equation can be neglected. Keeping only the
most singular terms in the rest of the interactions in Eq. (30a)-(30c), we
cast them into a familiar $Z=3$ critical form [40]:

$\displaystyle U_{11}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{1}{N_{\mathrm{F},1}R({\bm{q}},\Omega_{m})}$ (36a)
$\displaystyle U_{12}({\bm{q}},\Omega_{m})$ $\displaystyle=$ $\displaystyle
U_{21}({\bm{q}},\Omega_{m})=\frac{u_{12}}{R({\bm{q}},\Omega_{m})}$ (36b)
$\displaystyle U_{22}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{N_{\mathrm{F},1}u_{12}^{2}}{R({\bm{q}},\Omega_{m})}$
(36c)

with

$\displaystyle
R({\bm{q}},\Omega_{m})=\frac{1}{q^{*2}}\left(q^{2}+q_{\text{B}}^{2}+\gamma\frac{|\Omega_{m}|}{q}\right).$
(37)

Here, the parameters of the critical interaction are

$\displaystyle\frac{1}{q^{*2}}$ $\displaystyle=$
$\displaystyle\frac{1}{q_{\text{F},1}^{2}}+\frac{\lambda_{12}}{q_{\text{F},2}^{2}},\;\frac{1}{\varv^{*}_{\mathrm{F}}}=\frac{1}{\varv_{\mathrm{F},1}}+\frac{\lambda_{12}}{\varv_{\mathrm{F},2}}$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
C_{D}\frac{q^{*2}}{\varv^{*}_{\mathrm{F}}},\;q_{\text{B}}^{2}=(1+\lambda_{11}-\lambda_{12})q^{*2},$
(38)

where $q_{\text{B}}$ is the mass (the inverse correlation length) of order
parameter fluctuations. At the QCP, $q_{\text{B}}=0$. As long as
$q_{\text{B}}>0$, the system is in the FL regime for
$\omega\ll\omega_{\text{FL}}$ and in the NFL regime for
$\omega\gg\omega_{\text{FL}}$, where

$\displaystyle\omega_{\text{FL}}\equiv\varv^{*}_{\mathrm{F}}q_{\text{B}}^{3}/q^{*2}.$
(39)

In the FL regime, the fermionic self-energy of the $s^{\text{th}}$ valley
behaves as $\Sigma_{s}(\nu_{m})\propto
i\nu_{m}/q_{\text{B}}+iq_{\text{B}}^{-4}\nu_{m}|\nu_{m}|\ln(\omega_{\text{FL}}/|\nu_{m}|)$
in 2D [41], and as $\Sigma_{s}(\nu_{m})\propto
i\nu_{m}\ln(\omega_{\text{FL}}/|\nu_{m}|)+iq_{\text{B}}^{-3}\nu_{m}|\nu_{m}|$
in 3D. A crossover to the NFL regime can be achieved by replacing
$q_{\text{B}}\to|\nu_{m}|^{1/3}$, which yields familiar results:
$\Sigma_{s}(\nu_{m})\propto i\mathrm{sgn}\nu_{m}|\nu_{m}|^{2/3}$ in 2D and
$\Sigma_{s}(\nu_{m})\propto i\nu_{m}\ln|\nu_{m}|$ in 3D. In the next section,
we present a calculation of the optical conductivity in the FL regime near a
QCP.

###  III.2 Optical conductivity

####  III.2.1 Intravalley contribution

The leading-order diagrams for the contribution to the conductivity from the
interaction between valley-1 and valley-2 fermions are shown in Fig. 2 _a_ and
_b_. The analysis of the single-valley case follows along the same lines as in
Refs. [11, 17, 19, 21], with additional details provided in Appendix B.1.

Figure 2: Diagrams for the optical conductivity of a two-valley quantum
critical system. a) Contribution from the intravalley interaction in valley 1.
Thick solid line: Green’s function of valley 1; double wavy line: the $U_{11}$
component of the interaction, Eq. (30a); solid triangle: renormalized current
vertex. The capital letters label the $D+1$ momenta: $P=({\bm{p}},\nu_{m})$,
$K=({\bm{k}},\nu_{m}^{\prime})$, $Q=({\bm{q}},\Omega_{m})$, $W=({\bf
0},\omega_{m})$. b) Contribution from the induced intravalley interaction in
valley 2. Thin solid line: Green’s function of valley 2; zig-zag line:
$U_{22}$ component of the interaction, Eq. (30c); open triangle: renormalized
current vertex. c) Intervalley (drag) contribution. Double dashed line: the
$U_{12}$ component of the interaction, Eq. (30b).

The real part of the optical conductivity is non-zero only if the effective
interaction is dynamic. For the self-energy (SE) and Maki-Thompson (MT)
diagrams (_a1-a3_ and _b1-b3_ in Figs. 2), it implies that one can subtract
off the static part of the interaction,
$U_{ss}^{\text{st}}({\bm{q}})=U_{ss}({\bm{q}},0)$, such that the double wavy
and zigzag lines in these diagrams replaced by the corresponding dynamic
interactions,

$\displaystyle
U_{ss}^{\text{dyn}}({\bm{q}},\Omega_{m})=U_{ss}({\bm{q}},\Omega_{m})-U_{ss}({\bm{q}},0),$
(40)

expanded to linear order in $|\Omega_{m}|$. In the FL regime, the Green’s
functions in all diagrams can be approximated by their quasiparticle
forms777There is no double counting in adding self-energy corrections to the
renormalized Green’s functions in diagrams _a1_ and _a2_ , and _b1_ and _b2_.
Indeed, Eq. (41) accounts only for the real part of the self-energy, while the
double wavy line in these diagrams produces, after analytic continuation, the
imaginary part of the self-energy.

$\displaystyle
G_{s}({\bm{k}},\nu_{m})=\frac{1}{\frac{i\nu_{m}}{Z_{s}}-\varepsilon_{{\bm{k}},s}},\,s=1,2,$
(41)

where $Z_{s}$ is the renormalization factor of valley $s$, and where we have
taken into account that the theory becomes local near QCP, i.e., the self-
energy depends primarily on the frequency but not on the momentum [42].

Filled and blank triangles in Fig. 2 represent renormalized current vertices,
$\bm{\Lambda}_{{\bm{k}},s}$. For an isotropic spectrum and forward-type
scattering, such a vertex can be written as the product of the bare group
velocity and the charge vertex, while the latter is related to the $Z$-factor
via the Ward identity [43]:

$\displaystyle\bm{\Lambda}_{{\bm{k}},s}=\bm{v}_{{\bm{k}},s}\Gamma_{s}=\frac{\bm{v}_{{\bm{k}},s}}{Z_{s}}.$
(42)

With Eqs. (41) and (42) taken into account, the sum of the SE and MT diagrams
is reduced to (cf. Appendix B.1)

$\displaystyle\sigma^{\text{SE+MT}}_{ss}(\omega_{m})=-\frac{e^{2}}{2D\omega^{3}_{m}}\sum_{K,Q}({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})^{2}U_{ss}^{\text{dyn}}(Q)\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right],$
(43)

where $K=({\bm{k}},i\nu_{m})$, $Q=({\bm{q}},i\Omega_{m})$, $W=({\bf
0},i\omega_{m})$, $\sum_{K}\equiv{(2\pi)}^{D+1}\int d\nu_{m}\int d^{D}k$, and
$G_{s}(K)\equiv G_{s}({\bm{k}},\nu_{m})$. Here and thereafter, we set
$\omega_{m}>0$ without loss of generality.

In contrast to diagrams _a1-a3_ and _b1-b3_ , the interaction lines in the
Aslamazov-Larkin (AL) diagrams (_a4,a5_ and _b4,b5_ in Fig. (2)) can be
replaced by their static limit,

$\displaystyle U_{ss}^{\text{st}}({\bm{q}})=U_{ss}({\bm{q}},\Omega_{m}=0),$
(44)

while dynamics comes from the Green’s functions forming the triangles. The two
AL diagrams can be combined into the following expression (cf. Appendix B.2)

$\displaystyle\sigma^{\text{AL}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2D\omega^{3}_{m}}\sum_{K,Q}\sum_{P}({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})\cdot({\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s})\left[U^{\text{st}}_{ss}({\bm{q}})\right]^{2}G_{s}(P)G_{s}(P-Q)$
(45)
$\displaystyle\times\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right].$

Combining Eqs. (43) and (45), and taking into account that typical momenta and
energy transfers in the FL regime are in the range (34), we obtain for the
total intravalley contribution (see Appendix C)

$\displaystyle\sigma_{\text{intra}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\sum_{s}\left(\sigma_{ss}^{\mathrm{SE+MT}}+\sigma_{ss}^{\mathrm{AL}}\right)=\sum_{s}\frac{e^{2}C_{D}N_{\mathrm{F},s}\mathcal{O}_{D}}{2D\omega^{3}_{m}\varv_{\mathrm{F},s}\bar{m}_{s}^{2}}\int\frac{dqq^{D}}{(2\pi)^{D}}\int\frac{d\Omega_{m}}{2\pi}\left(|\Omega_{m}+\omega_{m}|+|\Omega_{m}-\omega_{m}|-2|\Omega_{m}|\right)U^{\mathrm{eff}}_{ss}({\bm{q}},\Omega_{m}),$
(46)

where

$\displaystyle
U^{\mathrm{eff}}_{ss}({\bm{q}},\Omega_{m})=U^{\text{dyn}}_{ss}({\bm{q}},\Omega_{m})-\left[U_{ss}^{\text{st}}({\bm{q}})\right]^{2}\Pi_{ss}^{\text{dyn}}({\bm{q}},\Omega_{m})$
(47)

is the effective intravalley interaction,

$\displaystyle\Pi_{ss}^{\text{dyn}}({\bm{q}},\Omega_{m})=C_{D}N_{\mathrm{F},s}\frac{|\Omega_{m}|}{\varv_{\mathrm{F},s}q}$
(48)

is the dynamic part of the polarization bubble, and
$\bar{m}_{s}=k_{\text{F},s}/\varv_{\text{F},s}$ is the density-of-states mass
evaluated on the FS. In the absence of the intervalley interaction, the
intravalley interaction in is reduced to the first term in Eq. (30a). In this
case, $U^{\mathrm{eff}}_{ss}({\bm{q}},\Omega_{m})$ vanishes and, to get a
finite conductivity, one needs to expand the electron velocities near the FS.
As a result, the conductivity is suppressed by a factor of
$\omega_{m}^{2}\ln|\omega_{m}|$ in 2D and $\omega_{m}^{2}$ in 3D [17, 18, 19,
20, 21]. For the case of coupled valleys, however,
$U^{\mathrm{eff}}_{ss}({\bm{q}},\Omega_{m})$ is finite because the dynamic
part of the critical interactions in Eqs. (36a)-(36c) comes from both valleys,
while $\Pi_{ss}^{\text{dyn}}({\bm{q}},\Omega_{m})$ comes only from one of the
valleys. Explicitly, we obtain

$\displaystyle U^{\mathrm{eff}}_{11}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{C_{D}}{N_{\mathrm{F},1}}\frac{q^{*4}}{(q^{2}+q_{\text{B}}^{2})^{2}}\left(\frac{1}{\varv^{*}_{\mathrm{F}}}-\frac{1}{\varv_{\mathrm{F},1}}\right)\frac{|\Omega_{m}|}{q}=\frac{C_{D}\lambda_{12}}{N_{\mathrm{F},1}}\frac{q^{*4}}{(q^{2}+q_{\text{B}}^{2})^{2}}\frac{|\Omega_{m}|}{\varv_{\mathrm{F},2}q},$
(49a) $\displaystyle U^{\mathrm{eff}}_{22}({\bm{q}},\Omega_{m})$
$\displaystyle=$
$\displaystyle\frac{C_{D}\lambda_{12}}{N_{\mathrm{F},2}}\frac{q^{*4}}{(q^{2}+q_{\text{B}}^{2})^{2}}\left(\frac{1}{\varv^{*}_{\mathrm{F}}}-\frac{1}{\varv_{\mathrm{F},2}}\right)\frac{|\Omega_{m}|}{q}=\frac{C_{D}\lambda_{12}}{N_{\mathrm{F},2}}\frac{q^{*4}}{(q^{2}+q_{\text{B}}^{2})^{2}}\frac{|\Omega_{m}|}{\varv_{\mathrm{F},1}q}.$
(49b)

Note that the two intravalley interactions are related by a permutation of the
valley indices $1\leftrightarrow 2$.

Performing straightforward integrations over $q$ and $\Omega_{m}$, and
analytically continuing to real frequencies, we obtain the intravalley part of
the conductivity as

$\displaystyle\operatorname{\mathrm{Re}}\sigma_{\text{intra}}(\omega)$
$\displaystyle=$ $\displaystyle
A_{D}e^{2}\lambda_{12}\frac{q^{*4}}{\varv_{\mathrm{F},1}\varv_{\mathrm{F},2}q_{\text{B}}^{4-D}}\left(\frac{1}{\bar{m}_{1}^{2}}+\frac{1}{\bar{m}_{2}^{2}}\right),$
(50)

where $A_{2}=1/48\pi^{2}$ and $A_{3}=1/576$.

####  III.2.2 Intervalley contribution

The intervalley contribution to the conductivity is depicted graphically in
Fig. 2 _c_. Note that the SE and MT diagrams are absent for the intervalley
case, because they involve swapping of fermions between the valleys, which is
not allowed in our model. Therefore, we have to consider only the AL diagrams.
As with other AL diagrams, the full intervalley interaction (the double-dashed
line) is replaced by its static value [cf. Eq. (36b)],
$U_{12}^{\text{st}}({\bm{q}})=u_{12}q^{*2}/(q^{2}+q_{\text{B}}^{2})$. As shown
in Appendix B.2, the sum of diagrams in panel _c_ of Fig. 2 can be written as

$\displaystyle\sigma_{\text{inter}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}\sum^{\prime}_{P}({\bm{v}}_{{\bm{k}}+{\bm{q}},1}-{\bm{v}}_{{\bm{k}},1})\cdot({\bm{v}}_{{\bm{p}},2}-{\bm{v}}_{{\bm{p}}-{\bm{q}},2})\left[U^{\text{st}}_{12}({\bm{q}})\right]^{2}G_{2}(P)G_{2}(P-Q)$
(51)
$\displaystyle\times\left[2G_{1}(K)G_{1}(K+Q)-G_{1}(K)G_{1}(K+Q+W)-G_{1}(K+Q)G_{1}(K+W)\right].$

The rest of the calculations is identical to those in Secs. III.2.1 and
Appendix C, and the final result for the intervalley part reads

$\displaystyle\operatorname{\mathrm{Re}}\sigma_{\text{inter}}(\omega)$
$\displaystyle=$
$\displaystyle-2A_{D}e^{2}\lambda_{12}\frac{q^{*4}}{\bar{m}_{1}\bar{m}_{2}\varv_{\mathrm{F},1}\varv_{\mathrm{F},2}q_{\text{B}}^{4-D}}.$
(52)

Note that $\varv_{\mathrm{F},1}$ and $\varv_{\mathrm{F},2}$ in the last
equation are the absolute values of the Fermi velocities in the corresponding
valleys, while masses $\bar{m}_{1}$ and $\bar{m}_{2}$ are positive for
electron-like valleys and negative for hole-like valleys. Therefore, the
intervalley contribution to the conductivity is negative, if both valleys are
either electron-like or hole-like, and positive, if one of the valleys is
electron-like and and another one is hole-like. This is exactly the same
effect that one encounters in Coulomb drag between two physically separate
layers: the sign of the drag conductivity depends on the sign of the relative
charge of the carriers in two layers [44]. Also note that, like the
intravalley contribution, the intervalley one contains only the density-of-
states rather than the band mass. This means that the intervalley (drag)
contribution is non-zero even for the Dirac single-particle spectrum, for
which the band mass vanishes but the density-of-state mass is finite.
Therefore, the result is not sensitive to whether the system has a particle-
hole symmetry or not.888The difference between the particle-hole symmetric and
asymmetric cases shows up only in subleading terms which contain higher powers
of frequency.

####  III.2.3 Total conductivity

Adding up the intra- and intervalley contributions, Eqs. (50) and (52)], we
obtain the total conductivity as

$\displaystyle\operatorname{\mathrm{Re}}\sigma(\omega)$ $\displaystyle=$
$\displaystyle\operatorname{\mathrm{Re}}\sigma_{\text{intra}}(\omega)+\operatorname{\mathrm{Re}}\sigma_{\text{inter}}(\omega)$
(53) $\displaystyle=$ $\displaystyle
A_{D}e^{2}\left(\frac{1}{\bar{m}_{1}}-\frac{1}{\bar{m}_{2}}\right)^{2}\lambda_{12}\frac{q^{*4}}{\varv_{\mathrm{F},1}\varv_{\mathrm{F},2}q_{\text{B}}^{4-D}}.$

Note that the conductivity vanishes for the case of identical valleys, ie.,
for $\bar{m}_{1}=\bar{m}_{2}$, in agreement with the result of Sec. II.3.

As in Refs. [19, 21], the frequency scaling of the conductivity is obtained by
replacing $q_{\text{B}}$ by $|\omega|^{1/3}$ in Eq. (53), which yields

$\displaystyle\operatorname{\mathrm{Re}}\sigma(\omega)\propto\left(\frac{1}{\bar{m}_{1}}-\frac{1}{\bar{m}_{2}}\right)^{2}\lambda_{12}|\omega|^{-(4-D)/3}.$
(54)

This recovers the scaling forms, $\omega^{-2/3}$ in 2D [33] and
$\omega^{-1/3}$ in 3D, obtained under the assumption of a current-relaxing
process of an unspecified type.

##  IV Conclusions

We have investigated the impact of Fermi surface (FS) topology and electron-
electron interaction (_ee_) interactions on the optical conductivity of a
Fermi liquid (FL). Our findings demonstrate that in a FL with a multi-valley
FS and distinct effective masses in different valleys, _ee_ scattering
facilitates a rapid current relaxation due to intervalley drag.

We derived the optical conductivity of a two-valley system, both in the FL
regime and near the Ising-nematic quantum-critical point (QCP). Our results
show that in the FL regime, intervalley drag restores the Gurzhi scaling form
of the conductivity,
$\operatorname{\mathrm{Re}}\sigma(\omega,T)=\sigma_{0}\left(1+4\pi^{2}T^{2}/\omega^{2}\right)$.
This restoration occurs because the geometric constraints on _ee_ scattering
are lifted in systems with multiply connected FSs, permitting new scattering
channels. Consequently, this effect generates a significant differential
response at the threshold of the channel opening, which emerges at the
Lifshitz transition from a single-valley to a multi-valley FS. We propose
seeking this effect in biased bilayer and rhombohedral trilayer graphenes.

Near the QCP, the intervalley contribution to the conductivity scales as
$|\omega|^{-2/3}$ in 2D and as $|\omega|^{-1/3}$ in 3D, thus providing a
specific current-relaxing process that aligns with longstanding theoretical
predictions [33].

###### Acknowledgements.

The work of Y.G. was supported by the Simons Foundation Grant No. 1249376.
A.L. acknowledges financial support by the National Science Foundation (NSF)
Grant No. DMR-2203411 and H.I. Romnes Faculty Fellowship provided by the
University of Wisconsin-Madison Office of the Vice Chancellor for Research and
Graduate Education with funding from the Wisconsin Alumni Research Foundation.
A.K. was supported by the NSF Grant No. DMR-2037654. A.V. Ch. was supported
the by U.S. Department of Energy, Office of Science, Basic Energy Sciences,
under Award No. DE-SC0014402; D. L. M. was supported by the NSF grant
DMR-2224000. A. V. Ch., Y.G., A. L., and D. L. M. acknowledge the hospitality
of the Kavli Institute for Theoretical Physics, Santa Barbara, supported by
the NSF grants PHY-1748958 and PHY-2309135.

##  Appendix A Fermi Golden Rule

(a)

(b)

(c)

(d)

Figure 3: Amplitudes for creation of two electron-hole pairs.

This appendix provides a computation of the imaginary part of the retarded
current-current correlator, $\mathcal{K}(\omega)$, to the second order of
perturbation theory in _ee_ interactions. We use the Källén–Lehmann
representation, in essence applying the FGR.

An elementary process contributing to $\mathcal{K}(\omega)$ involves the
excitation of two electron-hole pairs, one in each of valleys $s$ and
$s^{\prime}$, by a photon of frequency $\omega$. The contribution is given by:

$\delta\mathcal{K}^{\prime\prime}(\omega)=2\pi\sum_{\bm{q}}\left\lvert\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})\right\rvert^{2}\delta(\omega+\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s}+\varepsilon_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}-\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}-\varepsilon_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}})\,.$
(55)

Here, the matrix element
$\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})$ takes into
account the virtual states shown in Fig 3. 999The exchange processes are
disregarded for a long-ranged interaction. The energy denominator coming from
the propagator
$\mathcal{G}_{s}={(\varepsilon+\dfrac{\omega}{2}-\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s})}^{-1}$
simplifies to
$\varepsilon+\tfrac{\omega}{2}-(\varepsilon-\tfrac{\omega}{2})=\omega$ for a
process shown in Figs. 3(a) (and similarly for 3(d)), and $-\omega$ for
processes in Figs. 3(b),3(c), since the incoming electrons are on the mass
shell. Combining all four amplitudes, we get

$\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})=\frac{u_{ss^{\prime}}(\bm{q})}{\omega}\left(\bm{v}_{\bm{k}+\tfrac{\bm{q}}{2},s}+\bm{v}_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}-\bm{v}_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}-\bm{v}_{\bm{k}-\tfrac{\bm{q}}{2},s}\right)\,.$
(56)

The total contribution to $\operatorname{\mathrm{Im}}\mathcal{K}(\omega)$ sums
over all electron-hole pair creation and annihilation processes across both
valleys, integrating over all initial and final states to give

$\displaystyle\operatorname{\mathrm{Im}}\mathcal{K}(\omega)={}$
$\displaystyle\frac{2\pi}{4}\sum_{\stackrel{{\scriptstyle\bm{k}\bm{k^{\prime}}\bm{q}}}{{ss^{\prime}}}}\lvert\bm{\mathfrak{M}}_{ss^{\prime}}(\bm{k},\bm{k}^{\prime},\bm{q})\rvert^{2}\left\\{n_{\bm{k}-\tfrac{\bm{q}}{2},s}n_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}\left(1-n_{\bm{k}+\tfrac{\bm{q}}{2},s}\right)\left(1-n_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}\right)-\left(1-n_{\bm{k}-\tfrac{\bm{q}}{2},s}\right)\left(1-n_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}\right)n_{\bm{k}+\tfrac{\bm{q}}{2},s}n_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}}\right\\}$
$\displaystyle{}\times\delta(\omega+\varepsilon_{\bm{k}-\tfrac{\bm{q}}{2},s}+\varepsilon_{\bm{k}^{\prime}+\tfrac{\bm{q}}{2},s^{\prime}}-\varepsilon_{\bm{k}+\tfrac{\bm{q}}{2},s}-\varepsilon_{\bm{k}^{\prime}-\tfrac{\bm{q}}{2},s^{\prime}})\,,$
(57)

with equilibrium distribution function $n_{\bm{k},s}$ accounting for the
available states for electron transitions, and $1/4$ accounting for double
counting. Finally, the dissipative component of the conductivity,
$\operatorname{\mathrm{Re}}\sigma(\omega)$, is determined via the Kubo formula
to give Eq. (II.1) of the main text.

##  Appendix B Combining the conductivity diagrams

In this Appendix, we demonstrate how the diagrams for the optical conductivity
in Fig. 2 can be grouped together in such a way that the constraints imposed
by momentum conservation become explicit.

###  B.1 Self-energy and Maki-Thompson diagrams

We begin with the self-energy (SE) and Maki-Thompson (MT) diagrams in the
intravalley channel, _a1_ -_a3_ and _b1_ -_b3_ in Fig. 2. Introducing the
fermionic self-energy, $\Sigma_{s}(K)=-\sum_{Q}G_{s}(K+Q)U_{ss}(Q)$, we write
the sum of two SE diagrams as

$\displaystyle\sigma_{ss}^{\mathrm{SE}}(\omega_{m})=\frac{e^{2}}{DZ_{s}^{2}\omega_{m}}\sum_{K}\varv_{s,{\bm{k}}}^{2}\left[G^{2}_{s}(K)\Sigma_{s}(K)G_{s}(K+W)+G_{s}(K)G^{2}_{s}(K+W)\Sigma_{s}(K+Q)\right]\Sigma_{s}(K),$
(58)

where we took into account Eq. (42) for the current vertex and, as in the main
text, $K=(i\nu_{m},{\bm{k}})$ and $W=(i\omega_{m},{\bf 0})$. Using the
identity

$\displaystyle
G_{s}(K)G_{s}(K+W)=\frac{Z_{s}}{i\omega_{m}}\left[G_{s}(K)-G_{s}(K+Q)\right],$
(59)

Eq. (58) can be written as

$\displaystyle\sigma_{ss}^{\mathrm{SE}}(\omega_{m})=-\frac{1}{D\omega^{3}_{m}}\sum_{K}\varv_{s,{\bm{k}}}^{2}\left[G_{s}(K)-G_{s}(K+W)\right]\left[\Sigma_{s}(K+W)-\Sigma_{s}(K)\right].$
(60)

Now we recall that the dissipative part of the conductivity comes only from
the dynamic part of the interaction, as defined by Eq. (40). Therefore,
$\Sigma_{s}(K)\to-\sum_{K}G_{s}(K+Q)U_{ss}^{\mathrm{dyn}}(Q)$.

Repeating the same steps for the MT diagrams, we obtain for the sum of the SE
and MT contributions

$\displaystyle\sigma^{\text{SE+MT}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}{\bm{v}}_{{\bm{k}},s}\cdot({\bm{v}}_{{\bm{k}},s}-{\bm{v}}_{{\bm{k}}+{\bm{q}},s})U_{ss}^{\mathrm{dyn}}(Q)$
(61)
$\displaystyle\times\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right].$

The expression above can be re-written in a more symmetric form. Relabeling
$K+Q\to K$ and then $Q\to-Q$, while keeping in mind that
$U_{ss}^{\text{dyn}}(Q)$ is an even function of $Q$, we obtain an equivalent
form of Eq. (61):

$\displaystyle\sigma^{\text{SE+MT}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}{\bm{v}}_{{\bm{k}}+{\bm{q}},s}\cdot({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})U_{ss}^{\mathrm{dyn}}(Q)$
(62)
$\displaystyle\times\left[2G_{s}(K+Q)G_{s}(K)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right].$

Taking a half-sum of Eqs. (61) and (62), we arrive at

$\displaystyle\sigma^{\text{SE+MT}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{2D\omega^{3}_{m}}\sum_{K,Q}({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})^{2}U_{ss}^{\mathrm{dyn}}(Q)$
(63)
$\displaystyle\times\left[2G_{s}(K+Q)G_{s}(K)-G_{s}(K+Q)G_{s}(K+W)-G_{s}(K+Q)G_{s}(K+W)\right],$

which is Eq. (43) of the main text. The advantage of the last equation is that
the “transport factor”,
$({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})^{2}$, which suppresses
the contribution from small-$q$ scattering, is now explicitly quadratic in $q$
for $q\to 0$.

###  B.2 Aslamazov-Larkin diagrams

We now turn to the Aslamazov-Larkin (AL) diagrams, considering simultaneously
the intravelly (diagrams _a4_ , _a5_ , _b4_ , and _b5_ in Fig. 2) and
intervalley parts (diagrams _c1-c4_ in Fig. 2).

####  B.2.1 Particle-hole channel

The particle-hole (_ph_) diagrams _a4_ , _b4_ , and _c1_ can be written as

$\displaystyle\sigma^{\text{ph}}_{ss^{\prime}}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{D\omega_{m}Z_{s}Z_{s^{\prime}}}\sum_{K,P,Q}{\bm{v}}_{{\bm{k}},s}\cdot{\bm{v}}_{{\bm{p}}-{\bm{q}},s^{\prime}}\left[U^{\text{st}}_{ss^{\prime}}({\bm{q}})\right]^{2}G_{s}(K)G_{s}(K+W)G_{s^{\prime}}(P-Q)G_{s^{\prime}}(P-Q+W)G_{s}(K+Q)G_{s^{\prime}}(P),$
(64)

where $s=s^{\prime}=1,2$ in diagrams _a4_ and _b4_ , and $s=1$, $s^{\prime}=2$
in diagram _c1_. Diagram _c2_ for $\sigma^{\mathrm{ph}}_{21}$ is a mirror
image of diagram _c1_ , so that
$\sigma^{\mathrm{ph}}_{21}(\omega)=\sigma^{\mathrm{ph}}_{12}(\omega)$.

Applying the identity (59) to the two first pairs of the Green’s functions in
Eq. (64) and opening the brackets, we obtain

$\displaystyle\sigma^{\text{ph}}_{ss^{\prime}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{D\omega_{m}^{3}}\sum_{K,P,Q}{\bm{v}}_{{\bm{k}},s}\cdot{\bm{v}}_{{\bm{p}}-{\bm{q}},s^{\prime}}\left[U^{\text{st}}_{ss^{\prime}}({\bm{q}})\right]^{2}\left[G_{s}(K)-G_{s}(K+W)\right]\left[G_{s^{\prime}}(P-Q)-G_{s^{\prime}}(P-Q+W)\right]G_{s^{\prime}}(K+Q)G_{s^{\prime}}(P)$
(65) $\displaystyle=$
$\displaystyle\frac{e^{2}}{D\omega_{m}^{3}}\sum_{K,P,Q}{\bm{v}}_{{\bm{k}},s}\cdot{\bm{v}}_{{\bm{p}}-{\bm{q}},s^{\prime}}\left[U^{\text{st}}_{ss^{\prime}}({\bm{q}})\right]^{2}\left[G_{s}(K)G_{s}(K+Q)G_{s^{\prime}}(P-Q)G_{s^{\prime}}(P)+G_{s}(K+W)G_{s}(K+Q)G_{s^{\prime}}(P-Q+W)G_{s^{\prime}}(P)\right.$
$\displaystyle\left.-G_{s}(K)G_{s}(K+Q)G_{s^{\prime}}(P-Q+W)G_{s^{\prime}}(P)-G_{s}(K+W)G_{s}(K+Q)G_{s^{\prime}}(P-Q)G_{s^{\prime}}(P)\right].$

In the second term of the last equation we relabel the momenta (under the sum
over $K$ and $Q$) as

$\displaystyle
G_{s}(K+W)G_{s}(K+Q)G_{s^{\prime}}(P-Q+W)G_{s^{\prime}}(P)\underbrace{\to}_{K+W\leftrightarrow
K}G_{s}(K)G_{s}(K+Q-W)G_{s^{\prime}}(P-Q+W)G_{s^{\prime}}(P)$
$\displaystyle\underbrace{\to}_{Q-W\leftrightarrow
Q}G_{s}(K)G_{s}(K+Q)G_{s^{\prime}}(P-Q)G_{s^{\prime}}(P),$ (66)

upon which it coincides with the first term. (Note that a replacement
$Q-W\leftrightarrow Q$ affects neither the velocity nor static interaction,
because $W$ has only the frequency component.) Finally, relabeling
$Q-W\leftrightarrow Q$ in the third term, we obtain

$\displaystyle\sigma^{\text{ph}}_{ss^{\prime}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{D\omega_{m}^{3}}\sum_{K,P,Q}({\bm{v}}_{{\bm{k}},s}\cdot{\bm{v}}_{{\bm{p}}-{\bm{q}},s^{\prime}})\left[U^{\text{st}}_{ss^{\prime}}({\bm{q}})\right]^{2}\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+W)G_{s}(K+Q)\right]$
(67) $\displaystyle\times G_{s^{\prime}}(P-Q)G_{s^{\prime}}(P).$

####  B.2.2 Particle-particle channel

Next, we turn to the particle-particle (_pp_) diagrams _a5_ , _b5_ , _c3_ ,
and _c4_ in Fig. 2. Algebraically,

$\displaystyle\sigma^{\text{pp}}_{ss^{\prime}}(\omega_{m})=-\frac{e^{2}}{D\omega_{m}Z_{s}Z_{s^{\prime}}}\sum_{K,P,Q}({\bm{v}}_{s,{\bm{k}}}\cdot{\bm{v}}_{s^{\prime},{\bm{p}}})\left[U^{\text{st}}_{ss^{\prime}}({\bm{q}})\right]^{2}G_{s}(K)G_{s}(K+W)G_{s^{\prime}}(P)G_{s^{\prime}}(P+W)G_{s}(K+Q+W)G_{s^{\prime}}(P-Q),$
(68)

where $s=s^{\prime}=1,2$ for diagrams _a5_ and _b5_ , and $s=1$,
$s^{\prime}=2$ for diagrams _c3_ and _c4_. As for the particle-hole case,
diagram _c4_ is a mirror image of _c3_ , such that
$\sigma^{\text{pp}}_{21}(\omega_{m})=\sigma^{\text{pp}}_{12}(\omega_{m})$.
Applying Eq. (59) to the first two pairs of Green’s functions, we obtain

$\displaystyle\sigma^{\text{pp}}_{ss^{\prime}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{D\omega_{m}^{3}}\sum_{K,P,Q}({\bm{v}}_{s,{\bm{k}}}\cdot{\bm{v}}_{s^{\prime},{\bm{p}}})\left[U^{\text{st}}_{ss^{\prime}}({\bm{q}})\right]^{2}\left[G_{s}(K)G_{s}(K+Q+W)G_{s^{\prime}}(P)G_{s^{\prime}}(P-Q)\right.$
(69)
$\displaystyle\left.+G_{s}(K+W)G_{s}(K+Q+W)G_{s^{\prime}}(P+W)G_{s^{\prime}}(P-Q)-G_{s}(K)G_{s}(K+Q+W)G_{s^{\prime}}(P+W)G_{s^{\prime}}(P-Q)\right.$
$\displaystyle\left.-G_{s}(K+W)G_{s}(K+Q+W)G_{s^{\prime}}(P)G_{s^{\prime}}(P-Q)\right].$

In the second and third terms in the last equation, we relabel the momenta
(under the sum over $P$ and $Q$) as

$\displaystyle 2^{\text{nd}}:\;$ $\displaystyle
G_{s}(K+W)G_{s}(K+Q+W)G_{s^{\prime}}(P+W)G_{s^{\prime}}(P-Q)\underbrace{\to}_{P+W\leftrightarrow
P}G_{s}(K+W)G_{s}(K+Q+W)G_{s^{\prime}}(P)G_{s^{\prime}}(P-Q-W)$
$\displaystyle\underbrace{\to}_{Q+W\leftrightarrow
Q}G_{s}(K+W)G_{s}(K+Q)G_{s^{\prime}}(P)G_{s^{\prime}}(P-Q),$ $\displaystyle
3^{\text{rd}}:\;$ $\displaystyle
G_{s}(K)G_{s}(K+Q+W)G_{s^{\prime}}(P+W)G_{s^{\prime}}(P-Q)\underbrace{\to}_{P+W\leftrightarrow
P}G_{s}(K)G_{s}(K+Q+W)G_{s^{\prime}}(P)\ G_{s^{\prime}}(P-Q-W)$ (70)
$\displaystyle\underbrace{\to}_{Q+W\leftrightarrow
Q}G_{s}(K)G_{s}(K+Q)G_{s^{\prime}}(P)G_{s^{\prime}}(P-Q).$

Finally, relabeling $K+W\leftrightarrow K$ in the fourth term, we obtain the
same combination of the Green’s functions as in the final expression for the
particle-hole channel, Eq. (67), but with an opposite sign:

$\displaystyle\sigma^{\text{pp}}_{ss^{\prime}}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{D\omega_{m}^{3}}\sum_{K,P,Q}({\bm{v}}_{s,{\bm{k}}}\cdot{\bm{v}}_{s^{\prime},{\bm{p}}})\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+W)G_{s}(K+Q)\right]$
(71) $\displaystyle\times G_{s^{\prime}}(P-Q)G_{s^{\prime}}(P).$

####  B.2.3 Combined contributions of the particle-hole and particle-particle
channels

The total AL contribution to the conductivity due to intravalley interaction
in valleys 1 and 2 is given by the sum of the _ph_ and _pp_ parts, which are
diagonal in valley indices. Adding up equations Eqs. (67) and (71) with
$s=s^{\prime}$, we find for the intravalley contribution

$\displaystyle\sigma^{\text{AL}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle\sigma^{\text{ph}}_{ss}(\omega_{m})+\sigma^{\text{pp}}_{ss}(\omega_{m})=-\frac{e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}\sum^{\prime}_{P}{\bm{v}}_{{\bm{k}},s}\cdot({\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s})\left[U^{\text{st}}_{ss}({\bm{q}})\right]^{2}G_{s}(P)G_{s}(P-Q)$
(72)
$\displaystyle\times\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right],$

where the prime over $\sum_{P}$ indicates that only the dynamic part of the
result is to be retained.

The contribution to the conductivity due to intervalley interaction is given
by sum of diagrams in panel _c_ of Fig. 2:

$\displaystyle\sigma_{\text{inter}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\sigma^{\text{ph}}_{12}(\omega_{m})+\sigma^{\text{pp}}_{12}(\omega_{m})+\sigma^{\text{ph}}_{21}(\omega_{m})+\sigma^{\text{pp}}_{21}(\omega_{m})=2\left[\sigma^{\text{ph}}_{12}(\omega_{m})+\sigma^{\text{pp}}_{12}(\omega_{m})\right]$
(73) $\displaystyle=$
$\displaystyle-\frac{2e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}\sum^{\prime}_{P}{\bm{v}}_{{\bm{k}},1}\cdot({\bm{v}}_{{\bm{p}},2}-{\bm{v}}_{{\bm{p}}-{\bm{q}},2})\left[U^{\text{st}}_{12}({\bm{q}})\right]^{2}G_{2}(P)G_{2}(P-Q)$
$\displaystyle\times\left[2G_{1}(K)G_{1}(K+Q)-G_{1}(K)G_{1}(K+Q+W)-G_{1}(K+Q)G_{1}(K+W)\right].$

####  B.2.4 Symmetrized form of the Aslamazov-Larkin contribution

Like the SE+MT contribution, the AL contribution can also be re-written in a
more symmetric form. To obtain such a form for the intravalley AL
contribution, we relabel $K+Q\to K$, $P-Q\to P$, and $Q\to-Q$ in Eq. (72) with
the result

$\displaystyle\sigma^{\text{AL}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}\sum^{\prime}_{P}{\bm{v}}_{{\bm{k}}+{\bm{q}},s}\cdot({\bm{v}}_{{\bm{p}}-{\bm{q}},s}-{\bm{v}}_{{\bm{p}},s})\left[U^{\text{st}}_{ss}({\bm{q}})\right]^{2}G_{s}(P)G_{s}(P-Q)$
(74)
$\displaystyle\times\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right].$

Taking a half sum of Eqs. (72) and (74), we arrive at the symmetrized form

$\displaystyle\sigma^{\text{AL}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2D\omega^{3}_{m}}\sum_{K,Q}\sum^{\prime}_{P}({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})\cdot({\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s})\left[U^{\text{st}}_{ss}({\bm{q}})\right]^{2}G_{s}(P)G_{s}(P-Q)$
(75)
$\displaystyle\times\left[2G_{s}(K)G_{s}(K+Q)-G_{s}(K)G_{s}(K+Q+W)-G_{s}(K+Q)G_{s}(K+W)\right].$

As in the SE+MT case, the transport factor in the last equation is manifestly
quadratic in $q^{2}$, which is Eq. (45) of the main text.

Likewise, the symmetrized form of the intervalley contribution in Eq. (73) can
be reduced to

$\displaystyle\sigma_{\text{inter}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{D\omega^{3}_{m}}\sum_{K,Q}\sum^{\prime}_{P}({\bm{v}}_{{\bm{k}}+{\bm{q}},1}-{\bm{v}}_{{\bm{k}},1})\cdot({\bm{v}}_{{\bm{p}},2}-{\bm{v}}_{{\bm{p}}-{\bm{q}},2})\left[U^{\text{st}}_{12}({\bm{q}})\right]^{2}G_{2}(P)G_{2}(P-Q)$
(76)
$\displaystyle\times\left[2G_{1}(K)G_{1}(K+Q)-G_{1}(K)G_{1}(K+Q+W)-G_{1}(K+Q)G_{1}(K+W)\right],$

which coincides with Eq. (51) of the main text.

##  Appendix C Intravalley contribution to the conductivity of a nearly-
critical two-valley system: Computational details

In this Appendix, we present a detailed calculation of the conductivity of a
nearly-critical two-valley system due to intravalley interaction, given by the
sum of Eqs. (43) and (45) of the main text.

At the first step, we integrate over the frequency components of $K$ and $P$,
arriving at

$\displaystyle\sigma^{\text{SE+MT}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}N_{\mathrm{F},s}^{2}}{2D\omega^{3}_{m}}\int\frac{dqq^{D-1}}{(2\pi)^{D}}\int\frac{d\Omega_{m}}{2\pi}\int
d\mathcal{O}_{{\bm{q}},D}\int\frac{d\mathcal{O}_{{\bm{k}},D}}{\mathcal{O}_{D}}\int
d\varepsilon_{{\bm{k}},s}({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})^{2}U_{ss}^{\text{dyn}}(Q)\left[n_{\text{F}}(\varepsilon_{{\bm{k}},s})-n_{\text{F}}(\varepsilon_{{\bm{k}}+{\bm{q}},s})\right]$
(77a)
$\displaystyle\times\mathcal{G}_{s}({\bm{k}},{\bm{q}},\Omega_{m},\Omega_{m}+\omega_{m},\Omega_{m}-\omega_{m}),$
$\displaystyle\sigma^{\text{AL}}_{ss}(\omega_{m})$ $\displaystyle=$
$\displaystyle\frac{e^{2}N^{2}_{\mathrm{F},s}}{2D\omega^{3}_{m}}\int\frac{dqq^{D-1}}{(2\pi)^{D}}\int\frac{d\Omega_{m}}{2\pi}\int
d\mathcal{O}_{{\bm{q}},D}\int\frac{d\mathcal{O}_{{\bm{k}},D}}{\mathcal{O}_{D}}\int
d\varepsilon_{{\bm{k}},s}\int\frac{d\mathcal{O}_{{\bm{p}},D}}{\mathcal{O}_{D}}\int
d\varepsilon_{{\bm{p}},s}({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})\cdot({\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s})\left[U^{\text{st}}_{ss}({\bm{q}})\right]^{2}$
(77b)
$\displaystyle\times\left[n_{\text{F}}(\varepsilon_{{\bm{k}},s})-n_{\text{F}}(\varepsilon_{{\bm{k}}+{\bm{q}},s})\right]\left[n_{\text{F}}(\varepsilon_{{\bm{p}}-{\bm{q}},s})-n_{\text{F}}(\varepsilon_{{\bm{p}},s})\right]\mathcal{G}_{s}({\bm{k}},{\bm{q}},\Omega_{m},\Omega_{m}+\omega_{m},\Omega_{m}-\omega_{m})\mathcal{G}_{s}({\bm{p}}-{\bm{q}},{\bm{q}},\Omega_{m},\Omega_{m},0),$

where

$\displaystyle\mathcal{G}_{s}({\bf
m},{\bm{q}},\omega_{1},\omega_{2},\omega_{3})=Z_{s}\left(\frac{2}{\frac{i\omega_{1}}{Z_{s}}-\varepsilon_{{\bf
m}+{\bm{q}},s}+\varepsilon_{{\bf
m},s}}-\frac{1}{\frac{i\omega_{2}}{Z_{s}}-\varepsilon_{{\bf
m}+{\bm{q}},s}+\varepsilon_{{\bf
m},s}}-\frac{1}{\frac{i\omega_{3}}{Z_{s}}-\varepsilon_{{\bf
m}+{\bm{q}},s}+\varepsilon_{{\bf m},s}}\right),$ (78)

$N_{\mathrm{F},s}$ is the density of the states of the $s^{\mathrm{th}}$
valley at the corresponding Fermi energy, $n_{F}(\varepsilon)$ is the Fermi
function,
$d\mathcal{O}_{{\bm{k}},3}=\sin\phi_{\bm{k}}d\phi_{\bm{k}}d\varphi_{\bm{k}}$
with $\phi_{\bm{k}}\in(0,\pi)$ and $\varphi_{\bm{k}}\in(0,2\pi)$ being the
polar and azimuthal angles of a spherical system in 3D, respectively,
$\mathcal{O}_{3}=4\pi$, $d\mathcal{O}_{{\bm{k}},2}=d\phi_{\bm{k}}$ with
$\phi_{\bm{k}}\in(-\pi,\pi)$ being the azimuthal angle of a polar system in
2D, and $\mathcal{O}_{2}=2\pi$. Now we take into account that typical momenta
transfers are small: $q\sim q_{\text{B}}\ll k_{\mathrm{F},s}$. Therefore, the
dispersions and the Fermi functions can be expanded to order $\mathcal{O}(q)$:

$\displaystyle\varepsilon_{{\bf k}+{\bm{q}},s}-\varepsilon_{{\bf
k},s}\approx\varv_{\text{F},s}q\cos\phi_{\bf k},\,\varepsilon_{{\bf
p},s}-\varepsilon_{{\bf p}-{\bm{q}},s}\approx\varv_{\text{F},s}q\cos\phi_{\bf
p},$ (79)

and

$\displaystyle n_{\text{F}}(\varepsilon_{{\bf
k},s})-n_{\text{F}}(\varepsilon_{{\bf k}+{\bm{q}},s})$ $\displaystyle\approx$
$\displaystyle\delta(\varepsilon_{{\bf k},s})\varv_{\text{F},s}q\cos\phi_{\bf
k},$ $\displaystyle n_{\text{F}}(\varepsilon_{{\bf
p}-{\bm{q}},s})-n_{\text{F}}(\varepsilon_{{\bf p},s})$ $\displaystyle\approx$
$\displaystyle\delta(\varepsilon_{{\bf p},s})\varv_{\text{F},s}q\cos\phi_{\bf
p},$ (80)

where all the angles are measured from the direction of ${\bm{q}}$. Due to the
last equation, the integrations over $\varepsilon_{{\bm{k}},s}$ and
$\varepsilon_{{\bm{p}},s}$ project the integrands onto the FS.

Next, we turn to the “transport factors” in Eqs. (77a) and (77b). For an
isotropic but otherwise arbitrary single-particle dispersion, considered in
this paper, the group velocity in the $s^{\mathrm{th}}$ valley can be written
as

$\displaystyle{\bm{v}}_{{\bm{k}},i}=\bm{\nabla}\varepsilon_{{\bm{k}},s}=\frac{{\bm{k}}}{k}\varepsilon_{s}^{\prime}(k),$
(81)

where $\varepsilon_{s}(k)\equiv\varepsilon_{{\bm{k}},s}$ and
$\varepsilon^{\prime}_{s}(k)=d\varepsilon_{s}(k)/dk$. To linear order in $q$,
we have

$\displaystyle{\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s}$
$\displaystyle=$
$\displaystyle\left({\bm{q}}\cdot\bm{\nabla}\right){\bm{v}}_{{\bm{k}},s}=\frac{{\bm{q}}}{k}\varepsilon^{\prime}_{s}(k)+(\hat{\bm{k}}\cdot{\bm{q}})\hat{\bm{k}}\left(\varepsilon^{\prime\prime}_{s}(k)-\frac{\varepsilon^{\prime}_{s}(k)}{k}\right),$
(82a) $\displaystyle{\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s}$
$\displaystyle=$
$\displaystyle\frac{{\bm{q}}}{p}\varepsilon^{\prime}_{s}(p)+(\hat{\bm{p}}\cdot{\bm{q}})\hat{\bm{p}}\left(\varepsilon^{\prime\prime}_{s}(p)-\frac{\varepsilon^{\prime}_{s}(p)}{p}\right),$
(82b)

where $\hat{\bm{k}}={\bm{k}}/k$ and $\hat{\bm{p}}={\bm{p}}/p$. Therefore, the
transport factors, evaluated on the Fermi surface(s), are given by

$\displaystyle({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})^{2}\big{|}_{k=k_{\mathrm{F},s}}$
$\displaystyle=$ $\displaystyle
q^{2}\left(\frac{\sin^{2}\phi_{\bm{k}}}{\bar{m}^{2}_{s}}+\frac{\cos^{2}\phi_{\bm{k}}}{m^{*2}_{s}}\right),$
(83a)
$\displaystyle({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})\cdot({\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s})\big{|}_{k=p=k_{\mathrm{F},s}}$
$\displaystyle=$ $\displaystyle
q^{2}\left\\{\frac{1}{\bar{m}_{s}^{2}}+(\cos^{2}\phi_{\bm{k}}+\cos^{2}\phi_{\bm{p}})\frac{1}{\bar{m}_{s}}\left(\frac{1}{m_{s}^{*}}-\frac{1}{\bar{m}_{s}}\right)\right.$
(83b)
$\displaystyle\left.+\cos\phi_{\bm{k}}\cos\phi_{\bm{p}}\cos\vartheta_{{\bm{k}}{\bm{p}}}\left(\frac{1}{m_{s}^{*}}-\frac{1}{\bar{m}_{s}}\right)^{2}\right\\},$

where $\vartheta_{{\bm{k}}{\bm{p}}}=\angle({\bm{k}},{\bm{p}})$, and

$\displaystyle\frac{1}{\bar{m}_{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\text{F},s}}\varepsilon^{\prime}_{i}(k)|_{k=k_{\text{F},s}}=\frac{\varv_{\text{F},s}}{k_{\text{F},s}}$
(84a) $\displaystyle\frac{1}{m^{*}_{s}}$ $\displaystyle=$
$\displaystyle\varepsilon^{\prime\prime}_{s}(k)|_{k=k_{\text{F},s}}$ (84b)

are the (inverse) density-of-states and band masses of the $s^{\text{th}}$
valley, respectively.

In the FL regime, typical momentum and energy transfers are such that
$|\Omega_{m}|\sim\omega_{m}\ll\varv_{\text{F},s}q$. Then, as it follows from
Eqs. (78) and (79), the relevant values of $\cos\phi_{\bm{k}}$ and
$\cos\phi_{\bm{p}}$ are small, i.e.,
$|\cos\phi_{{\bm{k}},{\bm{p}}}|\sim\omega_{m}/\varv_{\mathrm{F},1}q\ll 1$,
which means that $\phi_{{\bm{k}},{\bm{p}}}\approx\pm\pi/2$ in 2D and
$\phi_{{\bm{k}},{\bm{p}}}\approx\pi/2$ in 3D. In this limit, the transport
factors in Eqs. (83a) and (83b) are reduced to to a simpler form

$\displaystyle({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})^{2}\big{|}_{k=k_{\mathrm{F},1}}$
$\displaystyle=$
$\displaystyle({\bm{v}}_{{\bm{k}}+{\bm{q}},s}-{\bm{v}}_{{\bm{k}},s})\cdot({\bm{v}}_{{\bm{p}},s}-{\bm{v}}_{{\bm{p}}-{\bm{q}},s})\big{|}_{k=k_{\mathrm{F},1},p=k_{\mathrm{F},1}}=\frac{q^{2}}{\bar{m}_{s}^{2}}.$
(85)

The rest of the integrands in Eqs. (77a) and (77b), $\cos\phi_{\bm{k}}$ and
$\cos\phi_{\bm{p}}$ are linearized near $\phi_{{\bm{k}},{\bm{p}}}=\pm\pi/2$ in
2D and near $\phi_{{\bm{k}},{\bm{p}}}=\pi/2$ in 3D, respectively, and the
angular integrals are solved to linear order in
$\omega_{m}/\varv_{\mathrm{F},s}q$. Since $U_{ss}^{\mathrm{dyn}}$ and
$\Pi^{\mathrm{dyn}}_{11}$ depend only on the magnitude of ${\bm{q}}$, the
integration over $\mathcal{O}_{{\bm{q}},D}$ simply gives a factor of
$\mathcal{O}_{D}$. Adding up the SE+MT and AL contributions, we obtain

$\displaystyle\sigma_{\text{intra}}(\omega_{m})$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{e^{2}C_{D}\mathcal{O}_{D}N_{\mathrm{F},s}}{2D\omega^{3}_{m}\varv_{\mathrm{F},s}\bar{m}_{s}^{2}}\int\frac{dqq^{D}}{(2\pi)^{D}}\int\frac{d\Omega_{m}}{2\pi}\left(|\Omega_{m}+\omega_{m}|+|\Omega_{m}-\omega_{m}|-2|\Omega_{m}|\right)\left\\{U^{\text{dyn}}_{ss}({\bm{q}},\Omega_{m})-\left[U_{ss}^{\text{st}}({\bm{q}})\right]^{2}\Pi_{ss}^{\text{dyn}}({\bm{q}},\Omega_{m})\right\\},$

where $C_{2}=1$ and $C_{3}=\pi/2$, which coincides with Eq. (46) of the main
text.

##  Appendix D “High-energy” approach to quantum criticality

In the previous section, we treated all intra- and intervalley interactions on
the same footing, within RPA, while quantum criticality was imposed by tuning
the interaction in valley-1 to the critical value. We will refer to this
approach as to a “low-energy” one. Another widely accepted way to construct an
effective theory of quantum criticality is to assume that critical bosons are
formed out of “high-energy” fermions with energies comparable to the
ultraviolet scale of the model. Bosons are assumed to be described by the
Ornstein-Zernike susceptibility

$\displaystyle\chi_{0}({\bm{q}})=\frac{\mu}{q^{2}+q_{\text{B}}^{2}},$ (87)

which incorporates the static part of $\Pi_{11}({\bm{q}})$. In general,
$\chi_{0}({\bm{q}})$ also contains $\Omega^{2}$ term coming from the internal
dynamics of bosons, but we assume that it is less important than Landau
damping by low-energy fermions. We will refer to this approach as to a “high-
energy” one.

The goal of this section is to show that the high-energy approach has to be
treated with extra caution when applied to a two-valley system. One reason for
caution is the renormalization of $q_{\text{B}}$ by intra-valley interaction
$u_{12}$. Another, more fundamental reason, is the need to keep the static
$\Pi_{11}({\bm{q}})$ in the formulas.

Indeed, a formal extension of the high-energy approach to the two-valley case
yields the action of the following form:

$\displaystyle
S=S_{\text{f}0}+S_{\text{b}0}+S_{\text{fb}}+S_{\text{int},12}+S_{\text{int},22}$
(88)

where

$\displaystyle S_{\text{f}0}=\sum_{s=1,2}\int
d\tau\sum_{\bm{k}}\bar{c}_{{\bm{k}},s}(\tau)\left(\partial_{\tau}+\varepsilon_{{\bm{k}},s}\right)c_{{\bm{k}},s}(\tau)$
(89)

is the action for free fermions with dispersions $\varepsilon_{{\bm{k}},s}$
($s=1,2$),

$\displaystyle S_{\text{b}0}=\int
d\tau\sum_{\bm{q}}\frac{\varphi_{\bm{q}}(\tau)\varphi_{-{\bm{q}}}(\tau)}{\chi_{0}({\bm{q}})}$
(90)

is the the bosonic part,

$\displaystyle S_{\text{fb}}=u\int
d\tau\sum_{{\bm{k}},{\bm{q}}}F_{{\bm{k}}}c^{\dagger}_{{\bm{k}}+\frac{{\bm{q}}}{2},1}(\tau)c_{{\bm{k}}-\frac{{\bm{q}}}{2},1}(\tau)\varphi_{-{\bm{q}}}(\tau)$
(91)

describes the coupling of valley-1 fermions to critical bosons, and

$\displaystyle S_{\text{int},12}$ $\displaystyle=$ $\displaystyle u_{12}\int
d\tau\sum_{{\bm{k}},{\bm{p}},{\bm{q}}}\bar{c}_{{\bm{k}}+\frac{{\bm{q}}}{2},1}(\tau)\bar{c}_{{\bm{p}}-\frac{{\bm{q}}}{2},2}(\tau)c_{{\bm{p}}+\frac{{\bm{q}}}{2},2}(\tau)c_{{\bm{k}}-\frac{{\bm{q}}}{2},1}(\tau)$
(92a) $\displaystyle S_{\text{int},22}$ $\displaystyle=$
$\displaystyle\frac{u_{22}}{2}\int
d\tau\sum_{{\bm{k}},{\bm{p}},{\bm{q}}}\bar{c}_{{\bm{k}}+\frac{{\bm{q}}}{2},2}(\tau)\bar{c}_{{\bm{p}}-\frac{{\bm{q}}}{2},2}(\tau)c_{{\bm{p}}+\frac{{\bm{q}}}{2},2}(\tau)c_{{\bm{k}}-\frac{{\bm{q}}}{2},2}(\tau)$
(92b)

describe non-critical interactions between valleys 1 and valley 2, and in
valley 2, respectively.

Integrating out bosons via the Hubbard-Stratonovich transformation, we obtain

$\displaystyle S_{\text{int},11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d\tau\sum_{{\bm{k}},{\bm{p}},{\bm{q}}}u_{11}({\bm{q}})\bar{c}_{{\bm{k}}+\frac{{\bm{q}}}{2},1}(\tau)\bar{c}_{{\bm{p}}-\frac{{\bm{q}}}{2},1}(\tau)c_{{\bm{k}}-\frac{{\bm{q}}}{2},1}(\tau)c_{{\bm{p}}+\frac{{\bm{q}}}{2},1}(\tau),$
(93)

where

$\displaystyle
u_{11}({\bm{q}})=-u^{2}\chi_{0}({\bm{q}})=-\frac{w}{q^{2}+q_{\text{B}}^{2}}$
(94)

is the $q$-dependent effective interaction with valley 1 with $w=u^{2}\mu$. As
we said earlier, the static part of $\Pi_{11}({\bm{q}})$ is incorporated into
$u_{11}({\bm{q}})$ as a portion of $q_{\text{B}}$. Other parts of the action
are not affected by the Hubbard-Stratonovich transformation. The full action
can now be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{\text{f}0}+\frac{1}{2}\int
d\tau\sum_{{\bm{k}},{\bm{p}},{\bm{q}}}\sum_{ss^{\prime}=1,2}u_{ss^{\prime}}({\bm{q}})\bar{c}_{{\bm{k}}+\frac{{\bm{q}}}{2},s}(\tau)\bar{c}_{{\bm{p}}-\frac{{\bm{q}}}{2},s^{\prime}}(\tau)c_{{\bm{p}}+\frac{{\bm{q}}}{2},s^{\prime}}(\tau)c_{{\bm{k}}-\frac{{\bm{q}}}{2},s}(\tau)$
(95)

where

$\displaystyle\hat{u}({\bm{q}})=\begin{pmatrix}u_{11}({\bm{q}})&u_{12}\\\
u_{12}&u_{22}\\\ \end{pmatrix},$ (96)

is a $2\times 2$ matrix of “bare” interactions. In this formulation,
interaction between low-energy fermions introduces Landau damping, but there
are no interaction terms that would contain static $\Pi_{11}({\bm{q}})$. As
the consequence, if we extract the effective interactions from the the action,
we would obtain

$\displaystyle\tilde{U}_{11}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\tilde{U}_{11,a}({\bm{q}},\Omega_{m})+\tilde{U}_{11,b}({\bm{q}},\Omega_{m}),$
(97a) $\displaystyle\tilde{U}_{11,a}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{u_{11}({\bm{q}})}{1-u_{11}({\bm{q}})\Pi^{\text{dyn}}_{11}({\bm{q}},\Omega_{m})}=-\frac{w}{q^{2}+q_{\text{B}}^{2}+\gamma|\Omega_{m}|/q},$
(97b) $\displaystyle\tilde{U}_{11,b}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{u_{12}^{2}\Pi_{22}({\bm{q}},\Omega_{m})}{\left[1-u_{11}({\bm{q}})\Pi^{\text{dyn}}_{11}({\bm{q}},\Omega_{m})\right]^{2}}=-\frac{N_{\mathrm{F},2}u_{12}^{2}(q^{2}+q_{\text{B}}^{2})^{2}}{\left(q^{2}+q_{\text{B}}^{2}+\gamma|\Omega_{m}|/q\right)^{2}},$
(97c) $\displaystyle\tilde{U}_{12}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{u_{12}}{1-u_{11}({\bm{q}})\Pi^{\text{dyn}}_{11}({\bm{q}},\Omega_{m})}=u_{12}\frac{q^{2}+q_{\text{B}}^{2}}{q^{2}+q_{\text{B}}^{2}+\gamma|\Omega_{m}|/q},$
(97d) $\displaystyle\tilde{U}_{22}({\bm{q}},\Omega_{m})$ $\displaystyle=$
$\displaystyle\frac{u_{12}^{2}\Pi^{\text{dyn}}_{11}({\bm{q}},\Omega_{m})}{1-u_{11}({\bm{q}})\Pi_{11}^{\text{dyn}}({\bm{q}},\Omega_{m})}=\frac{u_{12}^{2}}{w}\frac{(q^{2}+q_{\text{B}}^{2})\gamma|\Omega_{m}|/q}{q^{2}+q_{\text{B}}^{2}+\gamma|\Omega_{m}|/q}$
(97e)

where $\gamma=C_{D}wN_{\mathrm{F},1}/\varv_{\mathrm{F},1}$ and where we
neglected the difference between $R_{1}$ and $R$ in Eqs. (31a) and
(LABEL:R12).

Comparing with Eqs. (30a), (30b), and (30c) we see that
$\tilde{U}_{11,a}({\bm{q}},\Omega_{m})$ is the same as the first term in the
last line in (30a), but other terms are different. The difference is two-fold.
First, all denominators are powers of
$1-u_{11}({\bm{q}})\Pi^{\text{dyn}}_{11}({\bm{q}},\Omega_{m})$, which is the
same as $R_{1}$ in (31a). The denominators in (30a), (30b), and ( 30c) contain
$R$, which differs from $R_{1}$ by
$\Pi_{11}({\bm{q}},\Omega_{m})\Pi_{22}({\bm{q}},\Omega_{m})u_{12}^{2}$, see
Eq. (31a). Second, the terms (97e) all contain small $u_{11}({\bm{q}})$ in the
numerator, while the corresponding terms in (30a), (30b), and (30c) contain
the bare $u_{11}$. The two terms differ by a static $\Pi_{11}({\bm{q}})$:
$u^{-1}_{11}({\bm{q}})=u^{-1}_{11}-\Pi_{11}({\bm{q}})$. This static piece is
missing if we formally extend the high-energy approach from a one-valley case,
where it is valid, to the two-valley case.

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