#  Kohn-Luttinger Superconductivity and Inter-Valley Coherence
in Rhombohedral Trilayer Graphene

Yi-Zhuang You  Department of Physics, University of California, San Diego, CA
92093, USA  Ashvin Vishwanath  Department of Physics, Harvard University,
Cambridge, MA 02138, USA

###### Abstract

Motivated by recent experiments on ABC-stacked rhombohedral trilayer graphene
(RTG) which observed spin-valley symmetry-breaking and superconductivity, we
study instabilities of the RTG metallic state to symmetry breaking orders. We
find that interactions select the inter-valley coherent order (IVC) as the
preferred ordering channel over a wide range, whose theoretically determined
phase boundaries agree well with experiments on both the hole and electron
doped sides. The Fermi surfaces near van Hove singularities admit partial
nesting between valleys, which promotes both inter-valley superconductivity
and IVC fluctuations. We investigate the interplay between these fluctuations
and the Hunds (intervalley spin) interaction using a renormalization group
approach. For antiferromagnetic Hund’s coupling, intervalley pairing appears
in the spin-singlet channel with enhanced $T_{c}$, that scales with the
dimensionless coupling $g$ as $T_{c}\sim\exp(-1/\sqrt{g})$ , compared to the
standard $\exp(-1/g)$ scaling. In its simplest form, this scenario assumes a
sign change in the Hund’s coupling on increasing hole doping. On the other
hand, the calculation incorporates breaking of the independent spin rotations
between valleys from the start, and strongly selects spin singlet over spin
triplet pairing, and naturally occurs in proximity to the IVC, consistent with
observations.

Introduction. — The recent discovery of spin-valley-ordered metals[1] and
superconductivity[2] in the ABC-stacked rhombohedral trilayer graphene (RTG)
provides a new platform to investigate strongly correlated phases of matter in
graphene-based systems[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The
RTG band structure features van Hove singularities with the divergent density
of states (DOS) near charge neutrality[17], which can be further enhanced by a
perpendicular electric displacement field. Around the van Hove singularities,
multiple symmetry-breaking metallic phases are discovered[1] with reduced spin
and valley degrees of freedom, as identified by the doubling or quadrupling of
quantum oscillation frequencies. At low temperature, superconducting (SC)
phases are further observed[2] tightly adjacent to the spin-valley symmetry
breaking transitions. In particular, the most prominent SC phase (denoted as
SC1 in Ref. 2) appears to be a spin-singlet SC closely proximate to a spin-
unpolarized symmetry-breaking metallic phase. Phonon-mediated pairing
mechanism[18] was soon proposed as a possible scenario. However, it remains
unclear why the SC phase follows the boundary of the symmetry-breaking phase
so closely and why the enhanced resistivity from electron-phonon scattering is
not observed above $T_{c}$ [2].

In this work, we investigate an alternative scenario where the SC originates
from the Kohn-Luttinger mechanism [19] and its adjacent symmetry-breaking
phase is an inter-valley coherence (IVC) [20, 21, 22] phase. Both SC and IVC
orders benefit from the inter-valley nesting presented near the van Hove
singularity, which naturally explains their tight proximity. Our theory
predicts an inter-valley spin-singlet $s$-wave pairing for the SC1 phase with
a critical temperature strongly enhanced by the van Hove singularity. Fig.
1(b) summarizes the schematic phase diagram based on our analysis.

Figure 1: (a) Lattice structure (top view) of rhombohedral trilayer graphene.
$A_{l}$ and $B_{l}$ label the sublattices of the $l$th graphene layer. (b)
Schematic phase diagram as a function of density and displacement field,
obtained here using different techniques. The IVC phase boundary is set by an
RPA instability. Additional spin-valley phase boundaries arising from Stoner
instability are not captured. Singlet superconductivity emerges in proximity
to the IVC order with antiferromagnetic Hunds coupling $J_{H}$. The assume
sign change of $J_{H}$ is schematically shown.

Band Structure Modeling. — The RTG consists of three layers of honeycomb
lattices in ABC-stacking as shown in Fig. 1(a). Its electronic states near
charge neutrality mainly reside on $A_{1}$ and $B_{3}$ sites, whose low-energy
band structure can be described by an effective two-band model[17, 23, 24]

$H_{0}=\sum_{{\bm{k}},\xi}c_{{\bm{k}},\xi}^{\dagger}h_{\xi}({\bm{k}})c_{{\bm{k}},\xi},$
(1)

where
$c_{{\bm{k}},\xi}=(c_{{\bm{k}},\xi,A_{1}},c_{{\bm{k}},\xi,B_{3}})^{\intercal}$
denotes the electron annihilation operator. $\xi=+$($-$) labels $K$
($K^{\prime}$) valley, and ${\bm{k}}$ labels the momentum deviation from the
corresponding valley center. The band Hamiltonian takes the form of
$h_{\xi}({\bm{k}})=(\epsilon_{\bm{k}}^{\text{s}}-\mu)\sigma^{0}+(\xi\alpha_{\bm{k}}^{\text{ch}}+\epsilon_{\bm{k}}^{\text{tr}})\sigma^{1}+\beta_{\bm{k}}^{\text{ch}}\sigma^{2}+\epsilon_{\bm{k}}^{\text{gap}}\sigma^{3}$,
with
$\alpha_{\bm{k}}^{\text{ch}}+\mathrm{i}\beta_{\bm{k}}^{\text{ch}}=\frac{v_{0}^{3}}{\gamma_{1}^{2}}(k_{x}+\mathrm{i}k_{y})^{3}$,
$\epsilon_{\bm{k}}^{\text{s}}=(\delta+\frac{u_{a}}{3})-(\frac{2v_{0}v_{4}}{\gamma_{1}}+u_{a}\frac{v_{0}^{2}}{\gamma_{1}^{2}}){\bm{k}}^{2}$,
$\epsilon_{\bm{k}}^{\text{tr}}=\frac{\gamma_{2}}{2}-\frac{2v_{0}v_{3}}{\gamma_{1}}{\bm{k}}^{2}$,
and
$\epsilon_{\bm{k}}^{\text{gap}}=u_{d}(1-\frac{v_{0}^{2}}{\gamma_{1}^{2}}{\bm{k}}^{2})$.
We adopt the parameters proposed in Ref. 1, namely $\gamma_{0}=3.1\text{eV}$,
$\gamma_{1}=380\text{meV}$, $\gamma_{2}=-15\text{meV}$,
$\gamma_{3}=-290\text{meV}$, $\gamma_{4}=-141\text{meV}$,
$\delta=-10.5\text{meV}$, $u_{a}=-6.9\text{meV}$, and
$v_{i}=\sqrt{3}a\gamma_{i}/2$ (for $i=0,3,4$ with $a=0.246\text{nm}$ being the
lattice constant). Their physical meanings are well documented in Ref. 17. In
particular, the parameters $u_{d}$ and $\mu$ are experimentally tunable by a
dual-gate device[1, 2], where $u_{d}=(u_{1}-u_{3})/2$ is the potential
difference between the outer layers (which is approximately proportional to
the applied displacement field) and $\mu=-(u_{1}+u_{2}+u_{3})/3$ is the
(overall) chemical potential (assuming $u_{l}$ to be the electronic potential
in the $l$th layer).

Figure 2: (a) Density of state (DOS) v.s. the electron density $n_{e}$, under
a typical displacement field $u_{d}=30$meV. Insets show the $K$ valley Fermi
surfaces at the corresponding doping level (those at van Hove singularities
are highlighted in red). (b-c) Ideal scenarios of inter-valley nesting at the
van Hove singularity on the (b) hole-doped side and the (c) electron-doped
side. The Fermi sea area exactly doubles when the IVC order develops under
perfect nesting. (d-e) Perfect inter-valley pairing protected by the time-
reversal symmetry on the (d) hole-doped side and the (e) electron-doped side.

A prominent feature of the RTG band structure is the presence of van Hove
singularities in both the conduction and valence bands, where the DOS diverges
due to Lifshitz transitions of the Fermi surface topology, as demonstrated in
Fig. 2(a). On the hole-doped side (in the valence band), the van Hove
singularity features an interwoven circular and triangular Fermi surface,
whose intersection points are where the larger annular Fermi surface pinches
off into smaller Fermi pockets. On the electron-doped side (in the conduction
band), the van Hove singularity features a three-leaf clover-shaped Fermi
surface when the three smaller pockets are about to merge. These particular
shapes of Fermi surfaces at van Hove singularities are well-nested between $K$
and $K^{\prime}$ valleys, as illustrated in Fig. 2(b,c). Under repulsive
interactions, the inter-valley nesting leads to a strong instability towards
the inter-valley coherence (IVC) order, described by the order parameter
$c_{{\bm{k}},K}^{\dagger}c_{{\bm{k}},K^{\prime}}$, which is energetically
favored as it gaps out the nested part of the Fermi surface and reduces the
DOS at the Fermi level. Furthermore, the time-reversal symmetry ensures that
band dispersions around the two valleys are related by
$\epsilon_{K^{\prime}}({\bm{k}})=\epsilon_{K}(-{\bm{k}})$, which admits
perfect inter-valley pairing of electrons, as illustrated in Fig. 2(d,e). If
the interaction is renormalized to attractive, the system will likely develop
inter-valley superconductivity in the $s$-wave channel which can fully gap the
Fermi surface.

Interaction and Leading Instability. — To analyze these leading instabilities
in the RTG system, we introduce the following local interaction of electrons

$H_{\text{int}}=\int\mathrm{d}^{2}{\bm{x}}\;\frac{U}{2}(n_{K}^{2}+n_{K^{\prime}}^{2})+Vn_{K}n_{K^{\prime}}+H_{\rm
Hunds},$ (2)

where $n_{\xi}=c_{\xi}^{\dagger}c_{\xi}$ (for $\xi=K,K^{\prime}$) denotes the
charge operator of the $\xi$ valley. The interaction Hamiltonian
$H_{\text{int}}$ in Eq. (2) takes the most general form that respects the
time-reversal, the charge $\mathrm{U}(1)_{\mathrm{c}}$, valley
$\mathrm{U}(1)_{\mathrm{v}}$ (approximate), and spin
$\mathrm{SU}(2)_{\mathrm{s}}$ symmetries of the band model $H_{0}$, where $U$
describes the intra-valley density interaction, $V$ describes the inter-valley
density interaction, and $H_{\rm Hunds}$ describes the Hund’s coupling. At the
lattice scale (at the bare level), it is expected that $U,V>0$ are repulsive
and $H_{\rm Hunds}$ is small compared to $U,V$ terms. Dropping the Hunds
coupling, the symmetry gets enlarged to
$\mathrm{U}(1)_{\mathrm{c}}\times\mathrm{U}(1)_{\mathrm{v}}\times\mathrm{SO}(4)$,
where
$\mathrm{SO}(4)\simeq\mathrm{SU}(2)_{\mathrm{s}}^{K}\times\mathrm{SU}(2)_{\mathrm{s}}^{K^{\prime}}$
consists of independent spin rotation symmetries in each valley[25]. Further
taking $U=V$ will enlarge the symmetry of the interaction to
$\mathrm{U}(1)_{\mathrm{c}}\times\mathrm{SU}(4)$, where valley and spin
degrees of freedoms are degenerate[26]. Of course, the kinetic terms break
this SU(4) down to
$\mathrm{U}(1)_{\mathrm{c}}\times\mathrm{U}(1)_{\mathrm{v}}\times\mathrm{SO}(4)$.

Table 1: Complete list of local fermion bilinear operators $\Phi$, classified by symmetry representations of $\mathrm{U}(1)_{\mathrm{c}}\times\mathrm{U}(1)_{\mathrm{v}}\times\mathrm{SO}(4)$ following Ref. 25. $g_{0}^{\Phi}$ is the bare coupling in each channel (calculated at $H_{\text{Hunds}}=0$ with $\mathrm{SO}(4)$ symmetry). operator $\Phi$ | $\mathrm{U}(1)_{\mathrm{c}}$ | $\mathrm{U}(1)_{\mathrm{v}}$ | $\mathrm{SO}(4)$ | $g_{0}^{\Phi}$
---|---|---|---|---
$n_{c}=n_{K}+n_{K^{\prime}}$ | 0 | 0 | $\mathbf{1}$ | $\frac{1}{4}(U+V)$
$n_{v}=n_{K}-n_{K^{\prime}}$ | 0 | 0 | $\mathbf{1}^{\prime}$ | $\frac{1}{4}(U-V)$
${\bm{S}}_{\xi}=c_{\xi}^{\dagger}{\bm{\sigma}}c_{\xi}$ | 0 | 0 | $\mathbf{6}$ | $-\frac{1}{12}U$
$I^{\mu}=c_{K}^{\dagger}\sigma^{\mu}c_{K^{\prime}}$ | 0 | 2 | $\mathbf{4}\oplus\mathbf{4}^{\prime}$ | $-\frac{1}{8}V$
$\Delta^{\mu}=c_{K}^{\intercal}\mathrm{i}\sigma^{2}\sigma^{\mu}c_{K^{\prime}}$ | 2 | 0 | $\mathbf{4}\oplus\mathbf{4}^{\prime}$ | $\frac{1}{8}V$
$\Delta_{\xi}=c_{\xi}^{\intercal}\mathrm{i}\sigma^{2}c_{\xi}$ | 2 | 2 | $2(\mathbf{1}\oplus\mathbf{1}^{\prime})$ | $\frac{1}{8}U$

For simplicity, we take the interaction to be repulsive and $\mathrm{SU}(4)$
symmetric by setting $U=V>0$ and $H_{\rm Hunds}=0$, and apply the random phase
approximation (RPA)[27, 28, 29] approach to analyze the Fermi surface
instabilities towards all possible local fermion bilinear orders $\Phi$, as
enumerated in Tab. 1 (strictly speaking, the total charge density $n_{c}$ can
not order spontaneously, as it is controlled by the chemical potential). For
each fermion bilinear operator
$\Phi_{{\bm{q}}}=\frac{1}{2}\sum_{{\bm{k}}}\chi_{-{\bm{k}}+{\bm{q}}}^{\intercal}\Phi\chi_{{\bm{k}}}$
generally expressed in the Majorana basis $\chi_{\bm{k}}$, we evaluate its
bare static (zero frequency) susceptibility
$\chi_{0}^{\Phi}({\bm{q}})\equiv\langle\Phi_{{\bm{q}}}^{\dagger}\Phi_{{\bm{q}}}\rangle_{0}$
on the thermal equilibrium state of the band Hamiltonian $H_{0}$ at a low
temperature $T=0.29$K ($25\upmu$eV). Then we rewrite the interaction
$H_{\text{int}}=g_{0}^{\Phi}\sum_{{\bm{q}}}\Phi_{{\bm{q}}}^{\dagger}\Phi_{{\bm{q}}}+\cdots$
in the same channel to extract the bare coupling $g_{0}^{\Phi}$ (see the last
column of Tab. 1). The RPA corrected coupling is then given by
$g_{\text{RPA}}^{\Phi}({\bm{q}})=g_{0}^{\Phi}(1+g_{0}^{\Phi}\chi_{0}^{\Phi}({\bm{q}}))^{-1}$.
A strong negative (attractive) coupling $g_{\text{RPA}}^{\Phi}$ indicates a
strong ordering tendency of the corresponding order parameter $\Phi$. The peak
value
$\hat{g}_{\text{RPA}}^{\Phi}\equiv\mathop{\mathrm{max}}_{\bm{q}}g_{\text{RPA}}^{\Phi}({\bm{q}})$
is taken and plotted in Fig. 3 as a function of the bare interaction $U$ in
various channels, at the van Hove singularity on the (a) hole-doped and (b)
electron-doped sides. In either cases, the most attractive coupling appears in
the IVC channel, which is associated with the operator
$I^{\mu}_{\bm{q}}=\sum_{{\bm{k}}}c_{{\bm{k}}+{\bm{q}},K}^{\dagger}\sigma^{\mu}c_{{\bm{k}},K^{\prime}}$.
Comparing Fig. 3(a) and (b), the electron-doped side has a stronger IVC
instability, which is consistent with the stronger van Hove singularity on the
electron-doped side, as shown in Fig. 2(b). We would like to comment that the
RPA analysis only captures the nesting/pairing instability. The spin or valley
polarization (${\bm{S}}_{\xi}$ or $n_{v}$ ordering) under Stoner instability
is outside the scope of RPA calculation. How the IVC order may compete with
the spin-valley polarization will be left for future study.

Figure 3: The peak value of RPA corrected coupling
$\hat{g}_{\text{RPA}}^{\Phi}$ in different ordering channels v.s. the bare
interaction strength $U=V$ under the displacement field $u_{d}=30$meV at the
van Hove singularity on (a) the hole-doped side and (b) the electron-doped
side. The IVC channel is always the leading instability. The insets shows the
wavevector dependence of the bare susceptibility
$\chi_{0}^{I^{\mu}}({\bm{q}})$ in the IVC channel.

Inter-Valley Coherence. — The momentum distributions of the bare IVC
susceptibility $\chi_{0}^{I^{\mu}}({\bm{q}})$ are shown in the insets of Fig.
3, which peaks at ${\bm{q}}={\bm{0}}$ under optimal nesting near van Hove
singularities. Therefore, when the bare interaction is strong enough,
$I^{\mu}_{{\bm{q}}}$ will condense at zero momentum (modulo the inter-valley
momentum), leading to a uniform IVC order at a large scale. Nevertheless, the
IVC order does carry the inter-valley momentum difference, which should appear
as a charge/spin density wave (CDW/SDW) with Kekulé pattern on the lattice
scale[30, 31]. In the absence of the $\mathrm{SO}(4)$ symmetry-breaking
interaction $H_{\rm Hunds}$, the four components of the IVC order $I^{\mu}$
($\mu=0,1,2,3$) are degenerate as an $\mathrm{SO}(4)$ vector, where $I^{0}$ is
the spin-singlet (CDW-like) IVC and ${\bm{I}}=(I^{1},I^{2},I^{3})$ is the
spin-triplet (SDW-like) IVC. This degeneracy can be split by the Hund’s
interaction either as the spin coupling
$H_{\text{Hunds}}=-J_{H}{\bm{S}}_{K}\cdot{\bm{S}}_{K^{\prime}}$ or as the IVC
coupling $H_{\text{Hunds}}=-\tilde{J}_{H}{\bm{I}}^{\dagger}\cdot{\bm{I}}$
(note that the two types of couplings are related by
${\bm{S}}_{K}\cdot{\bm{S}}_{K^{\prime}}=2{\bm{I}}^{\dagger}\cdot{\bm{I}}+3n_{K}n_{K^{\prime}}$
in the local interaction limit when the couplings are momentum independent).
For example, an antiferromagnetic coupling ($J_{H}<0$ or $\tilde{J}_{H}<0$)
will favor the spin-singlet IVC. The spin-singlet IVC order could be directly
spotted in the scanning tunneling microscopy (STM) image[32].

Figure 4: The maximal bare static IVC susceptibility
$\hat{\chi}_{0}^{I^{\mu}}$ as a function of the electron density $n_{e}$ and
the displacement field $u_{d}$, calculated at temperature $T=1.8$K
($155\upmu$eV). The mean-field phase boundary between the IVC phase and the
Fermi liquid (FL) phase is determined by assuming the inter-valley density
interaction to be $V=42$eV (at lattice scale). Inside the hole-doped IVC
phase, the dashed curve separates the IVC order of zero momentum from that of
finite momentum.

Within the RPA approach, the ordering transition happens when the effective
coupling $g_{\text{RPA}}^{\Phi}({\bm{q}})$ diverges at some momentum
${\bm{q}}$ where $1+g_{0}^{\Phi}\chi_{0}^{\Phi}({\bm{q}})=0$. This leads to
the mean-field equation for the IVC order

$\hat{\chi}_{0}^{I^{\mu}}\equiv\max_{\bm{q}}\chi_{0}^{I^{\mu}}({\bm{q}})=-\frac{1}{g_{0}^{I^{\mu}}}=\frac{8}{V}.$
(3)

As shown in Fig. 4, the maximal bare static susceptibility
$\hat{\chi}_{0}^{I^{\mu}}$ varies with both the displacement field $u_{d}$ and
the chemical potential $\mu$ (where $\mu$ can be translated to the electron
density $n_{e}$ assuming a rigid band). By solving the mean-field equation Eq.
(3), the IVC phase boundary can be traced out in the $n_{e}$-$u_{d}$ plane,
see Fig. 4. The phase diagram looks most similar to the experiment[1] by
taking $V\approx 42$eV 111Note the analogous Stoner parameter used in Ref [1]
was in the same ball park $\sim 30$eV. Near the van Hove singularities
($n_{e}\sim\pm 0.5\times 10^{12}\text{cm}^{-2}$), uniform
(${\bm{q}}={\bm{0}}$) IVC order is favored due to good nesting. Doping away
from the van Hove singularity (on the hole-doped side), the IVC order can
shift to finite momentum ${\bm{q}}$. The finite-${\bm{q}}$ IVC order will
either break the three-fold lattice rotation symmetry (if the IVC order
parameter condenses at a single ${\bm{q}}$ point [kwan2021kekule]), or break
the (larger-scale) translation symmetry (if it condenses to symmetry-related
${\bm{q}}$ points coherently [25]).

As shown in Fig. 5, the IVC ordering splits the van Hove singularities in the
original band structure, and significantly reduces the Fermi surface DOS.
However, it does not fully gap out the Fermi-level degrees of freedom.
Instead, it merges the two sets of Fermi surfaces from separate valleys to
create a single set of Fermi surfaces shared between valleys, as illustrated
in Fig. 2(c,d). Thus the IVC order effectively locks the valley degrees of
freedom together and doubles the Fermi sea area, which provides a possible
explanation for the doubling of quantum oscillation frequency (from
$f_{\nu}=1/4$ to $1/2$) observed in experiment[1]. As the system remains
metallic, it is further subject to other Fermi surface instability. For
example, the hexagonal inner Fermi surface (highlighted in red in Fig. 5) near
the hole-under-doped van Hove singularity is well-nested between parallel
sides (as indicated by arrows), which may lead to a secondary SDW order on top
of the IVC order, such that the quantum oscillation frequency could be further
doubled (to $f_{\nu}=1$). While near the hole-over-doped van Hove singularity,
the circular Fermi surface (highlighted in green in Fig. 5) exhibits a strong
pairing instability, which could potentially give rise to superconductivity.

Figure 5: Density of state (DOS) and Fermi surface topology under a uniform
(${\bm{q}}={\bm{0}}$) IVC order that opens $\sim 0.4\text{meV}$ partial gap
near the original van Hove singularities on (a) the hole-doped side and (b)
the electron-doped side.

Superconductivity. — It is recently discovered in experiment [2] that
superconducting (SC) phases emerge at low temperature within the Fermi liquid
(FL) phase, and are closely adjacent to the boundary of the spin-valley
symmetry breaking phase (which we identified as the IVC phase). However,
signatures of enhanced scattering from the underlying pairing-glue modes is
not observed above the SC transition temperature [2], which points to the
Kohn-Luttinger mechanism[19, 34, 35, 36, 37] for superconductivity based on
the intrinsic instability of the FL. To explore this possibility, we perform
the patch renormalization group (RG) analysis [38, 39, 40, 41, 42, Betouras]
that tracks the RG flow of interactions at one-loop level. We will focus on
the van Hove singularity in a single band, described by
$H_{0}=\sum_{{\bm{k}},\xi}c_{{\bm{k}},\xi}^{\dagger}\epsilon_{\xi}({\bm{k}})c_{{\bm{k}},\xi}$,
where the band dispersion $\epsilon_{\xi}({\bm{k}})$ can be obtained by
diagonalizing the two-band model in Eq. (1) and pick one band of interest. It
will be convenient to decompose the dispersion
$\epsilon_{\xi}({\bm{k}})=\xi\pi_{{\bm{k}}}+\mu_{{\bm{k}}}$ in terms of the
time-reversal even $\mu_{{\bm{k}}}=\mu_{-{\bm{k}}}$ and odd
$\pi_{{\bm{k}}}=-\pi_{-{\bm{k}}}$ components, and redefine
$\epsilon_{\bm{k}}\equiv\pi_{\bm{k}}+\mu_{\bm{k}}$.

We consider the Hund’s interaction in the form of inter-valley spin coupling
$H_{\text{Hunds}}=-J_{H}{\bm{S}}_{K}\cdot{\bm{S}}_{K^{\prime}}$ projected onto
states near the Fermi surface. Note, $J_{H}<0$ corresponds to
antiferromagnetic Hund’s interaction. By treating the two valleys as two
patches in the patch RG framework, we can employ the general approach
developed in Ref. 38 to renormalize the interactions in Eq. (2), and obtain
the following RG equations (see Appx. A.1 for derivation)

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}U&=-\tfrac{1}{2}d_{{\bm{0}}}^{\text{pp}}U^{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}V&=-\tfrac{1}{2}(1-d_{{\bm{Q}}}^{\text{ph}})(V^{2}+3J_{H}^{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}J_{H}&=-J_{H}((1+d_{{\bm{Q}}}^{\text{ph}})J_{H}+(1-d_{{\bm{Q}}}^{\text{ph}})V),\\\
\end{split}$ (4)

where the RG “time” $y=\Pi_{{\bm{Q}}}^{\text{pp}}(E)$ and the nesting
parameters
$d_{{\bm{0}}}^{\text{pp}}(y)=\partial_{y}\Pi_{{\bm{0}}}^{\text{pp}}(E)$,
$d_{{\bm{Q}}}^{\text{ph}}(y)=\partial_{y}\Pi_{{\bm{Q}}}^{\text{ph}}(E)$ are
related to the particle-particle (pp) or particle-hole (ph) ladder diagram
$\Pi$ down to the energy scale $E$ at the momentum transfer of either
${\bm{0}}$ (intra-valley) or ${\bm{Q}}$ (inter-valley). More explicitly, in
the zero-temperature limit,
$\Pi_{{\bm{Q}}}^{\text{pp}}(E)=\sum_{|\epsilon_{\bm{k}}|>E}\frac{1}{|\epsilon_{\bm{k}}|}$,
$\Pi_{{\bm{0}}}^{\text{pp}}(E)=\sum_{|\mu_{\bm{k}}|>E}\frac{\Theta(|\mu_{\bm{k}}|-|\pi_{\bm{k}}|)}{|\mu_{\bm{k}}|}$,
$\Pi_{{\bm{Q}}}^{\text{ph}}(E)=\sum_{|\pi_{\bm{k}}|>E}\frac{\Theta(|\pi_{\bm{k}}|-|\mu_{\bm{k}}|)}{|\pi_{\bm{k}}|}$.
At the van Hove singularity (assuming the density of state diverges as
$2\nu_{0}\ln(\Lambda/\epsilon)$ with $\Lambda$ being the UV cuttoff), the RG
time will increase double logarithmically $y=\nu_{0}\ln^{2}(\Lambda/E)$ with
the decreasing energy scale $E$, while the nesting parameters vary within the
range $0\leq d_{{\bm{0}}}^{\text{pp}},d_{{\bm{Q}}}^{\text{ph}}\leq 1$ and
typically decay with $y$. Similar RG equations can be derived, if the Hund’s
interaction $H_{\text{Hunds}}=-\tilde{J}_{H}{\bm{I}}^{\dagger}\cdot{\bm{I}}$
takes the form of IVC coupling (see Appx. A.2).

Figure 6: (a) RG flow diagram of $(V,J_{H})$ with a typical choice of the
nesting parameter $d_{\bm{Q}}^{\text{ph}}=0.5$. Fixed rays/points of RG flows
are classified by symmetries. They separately control different
superconducting (SC) and Fermi liquid (FL) phases. (b) The critical
$|J_{H}/V|$ ratio for the spin-singlet SC and FL transition as a function of
$d_{\bm{Q}}^{\text{ph}}$. (c) Critical exponents for singlet
($\gamma_{\Delta^{0}}$) and triplet-paring ($\gamma_{{\bm{\Delta}}}$)
susceptibility at the SC transition, as well as exponents for IVC orders
($\gamma_{I^{0}}$, $\gamma_{{\bm{I}}}$).

For repulsive interaction, the intra-valley density interaction $U$ is always
marginally irrelevant and decouples from the flow of $V$ and $J_{H}$, so we
only need to focus on the RG flow in the $V$-$J_{H}$ plane. A typical flow
diagram is shown in Fig. 6(a). We first look at the $J_{H}=0$ axis, where the
interaction is $\mathrm{SO}(4)$ symmetric. Based on the RG equation Eq. (4),
the inter-valley density interaction $V$ always decrease under RG, which will
either flow to the Gaussian (free) fixed point at $V=0$ (if $V>0$ initially),
or flow away towards $V\to-\infty$ (if $V<0$ initially) along the
$\mathrm{SO}(4)$ fixed ray in Fig. 6(a). A strong and negative $V$ will drive
an instability towards the inter-valley SC pairing
$\Delta^{\mu}=c_{K}^{\intercal}\mathrm{i}\sigma^{2}\sigma^{\mu}c_{K^{\prime}}$
($\mu=0,1,2,3$) presumably in the $s$-wave channel (as $\Delta^{\mu}$ gains
energy the most from a negative $V$ interaction due to the
$\frac{V}{8}|\Delta^{\mu}|^{2}$ effective potential listed in Tab. 1). Under
the $\mathrm{SO}(4)$ symmetry, the spin-singlet $\Delta^{0}$ and spin-triplet
${\bm{\Delta}}=(\Delta^{1},\Delta^{2},\Delta^{3})$ pairings are degenerate.
However, the degeneracy can be split by the Hund’s interaction
$H_{\text{Hunds}}$, which could either be ferromagnetic ($J_{H}>0$) or
antiferromagnetic ($J_{H}<0$) depending on microscopic details[43, 44]. In any
case, the $\mathrm{SO}(4)$ fixed ray is unstable under the perturbation of
$J_{H}$, and flows to either the $\mathrm{SU}(2)_{\text{s}}^{\text{triplet}}$
or $\mathrm{SU}(2)_{\text{s}}^{\text{singlet}}$ fixed rays in Fig. 6(a), which
separately control the spin-triplet or spin-singlet inter-valley SC phases.

Although the $\mathrm{SO}(4)$-breaking interaction $J_{H}$ might be small
compare to $U$ and $V$, it plays an important role as a dangerously irrelevant
perturbation to help an initially repulsive ($V>0$) interaction to evade the
attraction of the Gaussian fixed point and to change the sign to attractive
($V<0$) that enables the $s$-wave SC order. However, the bare $(J_{H}/V)$
ratio (on the $V>0$ side) must exceed a certain critical value (in terms of
magnitude) to achieve this effect. Let us focus on the transition between
spin-singlet SC phase and the FL phase, as marked by the two-away arrow in
Fig. 6(a), the corresponding critical ratio is given by
$(J_{H}/V)_{c}=d_{V}/d_{J}^{-}$ where $d_{V}=1-d_{\bm{Q}}^{\text{ph}}$ and
$d_{J}^{\pm}=\pm
2(1-d_{\bm{Q}}^{\text{ph}}+(d_{\bm{Q}}^{\text{ph}})^{2})^{1/2}-1-d_{\bm{Q}}^{\text{ph}}$,
which can be reasonably small if the inter-valley nesting is almost perfect
(i.e. $d_{\bm{Q}}^{\text{ph}}\to 1$), see Fig. 6(b). Notice that a good
nesting is also the condition for IVC instability on the mean-field level,
which explains why the SC phase (e.g. SC1 in Ref. 2) comes so close to the
boundary of the IVC phase in the experiment, as both phases want to take
advantage from the inter-valley nesting. However, on the spin-triplet SC side,
the critical ratio is given by $(J_{H}/V)_{c}=d_{V}/d_{J}^{+}$, which is
bounded by $(J_{H}/V)_{c}\geq 1$ for all range of $d_{\bm{Q}}^{\text{ph}}$.
Therefore, if we assume $J_{H}$ to be small compare to $V$, the spin-triplet
SC will be unlikely to develop inside the FL phase (unless adjacent to other
FM spin-ordered phases).

Focusing on the spin-singlet SC phase for $(J_{H}/V)<(J_{H}/V)_{c}$, the RG
flow will diverge at some critical scale $y_{c}=1/V_{\text{eff}}$ along the
$\mathrm{SU}(2)_{\text{s}}^{\text{singlet}}$ fixed ray, where
$V_{\text{eff}}=\frac{1}{2}(d_{J}^{-}J_{H}-d_{V}V)>0$. This sets a critical
temperature

$T_{c}\sim\Lambda\exp(-1/\sqrt{\nu_{0}V_{\text{eff}}}).$ (5)

The $\sqrt{V_{\text{eff}}}$ dependence was highlighted in several existing
literature[45, 46, 41]. It results in a $T_{c}$ that is strongly enhanced
compared to the standard Bardeen-Cooper-Schrieffer (BCS) result
$T_{c}\sim\exp(-1/\nu_{0}V_{\text{eff}})$, which could put the proposed Kohn-
Luttinger mechanism to work in the experimentally accessible temperature
range.

To determine the nature of the ordered phase below $T_{c}$, we examine how the
susceptibility
$\chi^{\Phi}\equiv\langle\Phi^{\dagger}\Phi\rangle\sim(y_{c}-y)^{-\gamma_{\Phi}}$
diverges near the critical scale $y_{c}$ by computing the critical exponent
$\gamma_{\Phi}$ for various order parameters $\Phi$ (as listed in Tab. 1).
Following Ref. 38, 41 (see Appx. A.3 for details), along the
$\mathrm{SU}(2)_{\text{s}}^{\text{singlet}}$ fixed ray, we find
$\gamma_{n_{c}}=\gamma_{n_{v}}=\gamma_{{\bm{S}}_{\xi}}=\gamma_{\Delta_{\xi}}=0$
with the remaining non-trivial exponents
$\gamma_{I^{\mu}},\gamma_{\Delta^{\mu}}$ plotted in Fig. 6(c). The spin-
singlet inter-valley pairing $\Delta^{0}$ has indeed the largest exponent
$\gamma_{\Delta^{0}}$ (hence the fastest divergence) as the leading
instability, confirming the proposed SC phase below $T_{c}$. We also note that
the CDW-type IVC order $I^{0}$ is secondary and becomes degenerate with
$\Delta^{0}$ at perfect nesting. The RG analysis assumes infinitesimal
interactions. When the bare interaction is small but not close to the fixed
ray, sub-leading instability (such as the IVC order) may first occur at the
mean-field level before the RG can reverse the sign of $V$ and trigger the SC
instability, which is compatible with the RPA result.

Another remarkable phenomenon is that the small $\mathrm{SO}(4)$ symmetry
breaking interaction $J_{H}$ can be amplified by the RG flow. As seen in Fig.
6(c), both the spin-triplet SC and spin-triplet IVC orders (${\bm{\Delta}}$
and ${\bm{I}}$) are suppressed under RG (with negative exponents), which is
consistent with the experimental observation that no spin-triplet SC remains
when the spin-singlet SC is suppressed by an external in-plane magnetic field
(i.e. Pauli limited superconductivity[47, 48]).

Discussion. — Before closing, let us highlight some issues raised by
experiments. In the vicinity of SC1 we have assumed antiferromagnetic spin
coupling $J_{H}<0$, [2] . The same sign of $J_{H}$ is observed in other
graphene structures, e.g. monolayers and bilayers [49, 50]. However, the spin-
split Fermi surfaces at smaller hole doping points to $J_{H}>0$, implying that
it changes sign with doping, as illustrate in Fig. 1(b). Clarifying this
evolution will require a microscopic calculation of $J_{H}$, an important
future task. Across the $J_{H}=0$ line, the IVC phase can be split into the
spin-singlet ($J_{H}<0$) and spin-triplet ($J_{H}>0$) sub-phases, see Fig. 1 A
different mechanism for IVC fluctuation superconductivity was discussed in
[25], even when SO(4) symmetry is preserved. There, topological
superconductivity was obtained and the details of that mechanism are rather
different from the one we explored here. We propose that the spin unpolarized
PIP phase could correspond to an IVC phase. However, our RPA approach does not
capture the Stoner instability of spin or valley polarized states, which is
another competitive valley-spin symmetry breaking order near the van Hove
singularity. We will leave the competition between IVC and Stoner
ferromagnetism for future investigation.

Next, we note that the observation of a Pauli limiting field in SC1 is
significant, particularly if we assume that the in-plane magnetic field
couples entirely through the Zeeman effect. As a side remark, although it is
natural to ascribe the observed Pauli limiting behavior to the Zeeman
coupling, we note that potentially, an in-plane field could also lead to an
orbital suppression of superconductivity, due to finite layer separation.
Future experiments probing in-plane field anisotropy, expected for the orbital
but not for the Zeeman coupling, should help isolate the relevant coupling. At
present, the absence of any in-plane field effect on the putative triplet
superconductor SC2, and the observation of the Pauli limit in SC1, suggests
pure Zeeman coupling.) Under this assumption, the Pauli limiting behavior
strongly suggests that anisotropy terms like $J_{H}$ play a significant role
in superconductivity. If instead pairing was primarily determined by
$\mathrm{SO}(4)$ symmetric interactions, then one would expect both singlet
and triplet superconductors to be nearly degenerate. The $\mathrm{SO}(4)$
breaking anisotropies would lead only to small splittings in $T_{c}$ implying
that on destroying the spin-singlet superconductor with Zeeman field, a spin-
triplet superconductor, with slightly reduced $T_{c}$ would be established
beyond the Pauli limit. Indeed, such behavior is observed in a moiré material
of mirror-symmetric twisted trilayer graphene[51], but is not observed in the
RTG SC1 phase. This indicates a potentially different origin for
superconductivity here compared to moiré graphene. Our pairing mechanism
emphasizes the role of $J_{H}$ from the start, in contrast to those that rely
predominantly on $\mathrm{SO}(4)$ symmetric interactions, either electronic or
those based on acoustic phonons. The Hund’s coupling $J_{H}$ plays an
indispensable role to allow the inter-valley density interaction $V$ to change
sign under RG, which enables the $s$-wave SC to emerge even under a repulsive
bare interaction. A weak coupling $J_{H}$ also flows strong under RG, which
explains the absence of spin-triplet SC phase above the Pauli limit, even if
the $\mathrm{SO}(4)$ symmetry is only weakly broken at the bare level.

Finally, it is interesting to compare and contrast magic-angle graphene
(bilayer and mirror-symmetric trilayer) and RTG. On one hand, differences
include the fact that the transition temperatures in the former are larger
than the latter by over an order of magnitude despite comparable electronic
densities, and show signatures of strong coupling superconductivity [6, 7, 9,
16, 15] along with strange metal normal state behavior [52, 53, 54]. The width
in doping supporting superconductivity is also dramatically different in the
two cases. However, we note that the proximity of isospin order and
superconductivity is a common feature in both. Comparing these two remarkable
superconductors will be a useful future exercise.

Summary. — We combine the mean-field/RPA approach and the patch RG analysis to
investigate the IVC and inter-valley SC orders in RTG systems. The particular
Fermi surface topology at the van Hove singularity enables a good inter-valley
nesting, which promotes both the IVC and SC ordering under local electronic
interactions. The IVC picture nicely reproduces the general shape of the phase
diagram and explains the doubling of quantum oscillation frequency. It also
leaves room for cascades of secondary ordered phases when the system is
further doped away from the van Hove singularity. The diverging density of
state near the van Hove singularity also strongly renormalizes the interaction
in the FL phase, which, particularly in the presence of an inter-valley
antiferromagnetic $J_{H}<0$ coupling, causes the inter-valley density
interaction to become attractive at low-energy, favoring the inter-valley
spin-singlet $s$-wave SC with an enhanced critical temperature $T_{c}$
compared to the standard BCS formula. The RG flow can also amplify the effect
of the inter-valley Hunds interaction, leading to a significant splitting of
spin-singlet and spin-triplet SC states, even if the $\mathrm{SO}(4)$ breaking
anisotropy is weak at the lattice scale.

Notes Added: While completing this manuscript, related works Refs. [55, 56,
57, 58, 59] appeared, which propose alternative pairing mechanisms and sign
changing or topological superconductivity. In particular Ref. [56] identified
an IVC ground state through a microscopic calculation.

Acknowledgements: We thank Fa Wang, Mike Zaletel, Taige Wang, and especially
Andrea Young for stimulating discussions. YZY was supported by a startup fund
provided by UCSD and a UC Hellmann Fellowship. AV was supported by a Simons
Investigator award.

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##  Appendix A RG Analysis

###  A.1 Derivation of RG Equation

We start with the local interaction of electrons (here we take a Hund’s
coupling in the form of spin coupling and carry out the derivation, we will
discuss the other type of Hund’s coupling in the form of IVC coupling later)

$H_{\text{int}}=\int\mathrm{d}^{2}{\bm{x}}\;\frac{U}{2}(n_{K}^{2}+n_{K^{\prime}}^{2})+Vn_{K}n_{K^{\prime}}-J_{H}{\bm{S}}_{K}\cdot{\bm{S}}_{K^{\prime}},$
(6)

where $n_{\xi}=c_{\xi}^{\dagger}c_{\xi}$ and
${\bm{S}}_{\xi}=c_{\xi}^{\dagger}{\bm{\sigma}}c_{\xi}$ (for
$\xi=K,K^{\prime}$) are respectively the charge and spin density operators
near the $\xi$ valley, and
$c_{\xi}=(c_{\xi\uparrow},c_{\xi\downarrow})^{\intercal}$ is the electron
annihilation operator. To derive the RG equation more systematically, it will
be convenient to rewrite the interaction Eq. (6) in the Majorana fermion basis
(by rewriting the electron operator as $c_{\xi\sigma}=(\chi_{\xi\sigma
1}+\mathrm{i}\chi_{\xi\sigma 2})/\sqrt{2}$),

$H_{\text{int}}=\int\mathrm{d}^{2}{\bm{x}}\;\sum_{a,b,c,d}V_{abcd}\chi_{a}\chi_{b}\chi_{c}\chi_{d},$
(7)

where $a,b,c,d=1,\cdots,8$ runs over eight Majorana components
(valley$\times$spin$\times$particle-hole), and $V_{abcd}$ is a totally
antisymmetric tensor. In particular, its representative components are given
by

$\begin{split}&V_{1234}=V_{5678}=-U,\\\ &V_{1256}=V_{3478}=-V+J_{H},\\\
&V_{1278}=V_{3456}=-V-J_{H},\\\
&V_{1357}=V_{1458}=V_{2367}=V_{2468}=V_{1368}=V_{1476}=V_{2385}=V_{2457}=J_{H}.\end{split}$
(8)

The advantage of using the Majorana basis is that the interaction vertex $V$
receives one-loop correction via a single unified ladder diagram (which
unifies all types of ladder/bubble/vertex-correction diagrams in the complex
fermion basis). By perturbative expansion, the corrected interaction vertex at
the energy scale $E$ is given by

$V_{abcd}(E)=\raisebox{-15.0pt}{\includegraphics[width=20.0pt]{diag_V}}+\sum_{k\in\mathcal{K}_{E}}\raisebox{-20.0pt}{\includegraphics[width=60.0pt]{diag_bubble}}+\cdots=V_{abcd}-V_{abc^{\prime}d^{\prime}}\Pi_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}(E)V_{a^{\prime}b^{\prime}cd}+\cdots,$
(9)

where $\mathcal{K}_{E}$ denotes the frequency-momentum space beyond the energy
scale $E$. Here $\Pi(E)$ denotes the ladder kernel at the energy scale $E$ (by
integrating out virtual processes beyond the scale $E$),

$\Pi_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}(E)=\sum_{k\in\mathcal{K}_{E}}G_{0}(k)_{c^{\prime}b^{\prime}}G_{0}(-k)_{d^{\prime}a^{\prime}},$
(10)

where $G_{0}(k)$ denotes the bare fermion propagator in the Majorana basis.
Upon the reduction of the energy scale $E\to E-\mathrm{d}E$, the RG equation
will be given by

$\mathrm{d}V_{abcd}=-V_{abc^{\prime}d^{\prime}}\mathrm{d}\Pi_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}V_{a^{\prime}b^{\prime}cd}.$
(11)

To evaluate ladder kernel $\Pi(E)$, we take a single-band model in the
momentum space (we assume that the interaction has been projected to the band
of interest)

$H_{0}=\sum_{{\bm{k}},\xi}c_{\xi,{\bm{k}}}^{\dagger}\epsilon_{\xi,{\bm{k}}}c_{\xi,{\bm{k}}},$
(12)

where $\xi$ labels the $K$ and $K^{\prime}$ valleys, and
$\epsilon_{\xi,{\bm{k}}}$ be the band dispersion near the $\xi$ valley. By the
time-reversal symmetry, the band dispersion $\epsilon_{K,{\bm{k}}}$ near the
$K$ valley must be related to that $\epsilon_{K^{\prime},{\bm{k}}}$ near the
$K^{\prime}$ valley by
$\epsilon_{K^{\prime},{\bm{k}}}=\epsilon_{K,-{\bm{k}}}$. Let us define
$\epsilon_{K,{\bm{k}}}=\epsilon_{{\bm{k}}}=\pi_{{\bm{k}}}+\mu_{{\bm{k}}}$, in
terms of the odd $\pi_{-{\bm{k}}}=-\pi_{{\bm{k}}}$ and even
$\mu_{-{\bm{k}}}=\mu_{{\bm{k}}}$ parity parts. Then the dispersion around the
$K^{\prime}$ valley can be expressed as
$\epsilon_{K^{\prime},{\bm{k}}}=\epsilon_{K,-{\bm{k}}}=\pi_{-{\bm{k}}}+\mu_{-{\bm{k}}}=-\pi_{{\bm{k}}}+\mu_{{\bm{k}}}$.
Therefore the dispersion around both valleys can be unified in one formula
$\epsilon_{\xi,{\bm{k}}}=\xi\pi_{{\bm{k}}}+\mu_{{\bm{k}}}$ (with $\xi=\pm$ for
$K$ and $K^{\prime}$ valleys). This enables us to rewrite Eq. (12) in the
Majorana basis

$H_{0}=\sum_{{\bm{k}},\xi=\pm}c_{\xi,{\bm{k}}}^{\dagger}(\xi\pi_{{\bm{k}}}+\mu_{{\bm{k}}})c_{\xi,{\bm{k}}}=\sum_{{\bm{k}}}c_{{\bm{k}}}^{\dagger}(\pi_{{\bm{k}}}\sigma^{30}+\mu_{{\bm{k}}}\sigma^{00})c_{{\bm{k}}}=\frac{1}{2}\sum_{{\bm{k}}}\chi_{-{\bm{k}}}^{\intercal}(\pi_{{\bm{k}}}\sigma^{300}+\mu_{{\bm{k}}}\sigma^{002})\chi_{{\bm{k}}},$
(13)

therefore the band electron action reads

$S_{0}=\frac{1}{2}\sum_{k}\chi_{-k}^{\intercal}(-\mathrm{i}\omega\sigma^{000}+\pi_{{\bm{k}}}\sigma^{300}+\mu_{{\bm{k}}}\sigma^{002})\chi_{k},$
(14)

where $k=(\mathrm{i}\omega,{\bm{k}})$ is the frequency-momentum vector. We can
write down the bare propagator

$G_{0}(k)\equiv-\langle\chi_{k}\chi_{-k}^{\intercal}\rangle=-(-\mathrm{i}\omega\sigma^{000}+\pi_{{\bm{k}}}\sigma^{300}+\mu_{{\bm{k}}}\sigma^{002})^{-1}=\sum_{{\bm{s}}}\frac{P^{\bm{s}}}{\mathrm{i}\omega-
s_{1}\pi_{{\bm{k}}}-s_{2}\mu_{{\bm{k}}}},$ (15)

where the sign vector ${\bm{s}}=(s_{1},s_{2})$ (with $s_{1},s_{2}=\pm$) labels
different Majorana bands, and $P^{\bm{s}}$ is projection operator too the
Majorana band

$P^{\bm{s}}=\frac{\sigma^{0}+s_{1}\sigma^{3}}{2}\otimes\sigma^{0}\otimes\frac{\sigma^{0}+s_{2}\sigma^{2}}{2}.$
(16)

Substitute Eq. (15) into Eq. (10), the ladder kernel $\Pi$ can be evaluated as

$\begin{split}\Pi_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}(E)&=\sum_{\mathrm{i}\omega}\sum_{k\in\mathcal{K}_{E}}\sum_{{\bm{s}},{\bm{s}}^{\prime}}\frac{P^{{\bm{s}}}_{c^{\prime}b^{\prime}}}{\mathrm{i}\omega-
s_{1}\pi_{{\bm{k}}}-s_{2}\mu_{{\bm{k}}}}\frac{P^{{\bm{s}}^{\prime}}_{d^{\prime}a^{\prime}}}{-\mathrm{i}\omega-s^{\prime}_{1}\pi_{-{\bm{k}}}-s^{\prime}_{2}\mu_{-{\bm{k}}}}\\\
&=\sum_{k\in\mathcal{K}_{E}}\sum_{{\bm{s}},{\bm{s}}^{\prime}}-\frac{n_{F}(s_{1}\pi_{\bm{k}}+s_{2}\mu_{\bm{k}})-n_{F}(s^{\prime}_{1}\pi_{\bm{k}}-s^{\prime}_{2}\mu_{\bm{k}})}{(s_{1}-s^{\prime}_{1})\pi_{\bm{k}}+(s_{2}+s^{\prime}_{2})\mu_{\bm{k}}}P^{{\bm{s}}}_{c^{\prime}b^{\prime}}P^{{\bm{s}}^{\prime}}_{d^{\prime}a^{\prime}}\end{split}$
(17)

To further simplify, we con introduce the relative sign vector
${\bm{\eta}}=(\eta_{1},\eta_{2})$ such that
${\bm{s}}^{\prime}={\bm{\eta}}\odot{\bm{s}}$ (i.e.
$s^{\prime}_{i}=\eta_{i}s_{i}$ for $i=1,2$), then the ladder kernel can be
classified into four different cases by ${\bm{\eta}}=++,+-,-+,--$

$\Pi(E)=\sum_{{\bm{\eta}}}\Pi^{{\bm{\eta}}}(E)Q^{{\bm{\eta}}},$ (18)

where $Q^{{\bm{\eta}}}$ is the two-particle projection operator,

$Q^{{\bm{\eta}}}=\sum_{{\bm{s}}}P^{{\bm{s}}}\otimes
P^{{\bm{\eta}}\odot{\bm{s}}}$ (19)

and the ladder diagram $\Pi^{{\bm{\eta}}}(E)$ is given by

$\Pi^{{\bm{\eta}}}(E)=\sum_{k\in\mathcal{K}_{E}}-\frac{n_{F}(\pi_{\bm{k}}+\mu_{\bm{k}})-n_{F}(\eta_{1}\pi_{\bm{k}}-\eta_{2}\mu_{\bm{k}})}{(1-\eta_{1})\pi_{\bm{k}}+(1+\eta_{2})\mu_{\bm{k}}},$
(20)

which can be enumerated in different ${\bm{\eta}}$-channels as follows:

  * •

intra-valley particle-particle fluctuation (${\bm{\eta}}=++$)

$\Pi_{{\bm{0}}}^{\text{pp}}(E)\equiv\Pi^{++}(E)=\sum_{k\in\mathcal{K}_{E}}-\frac{n_{F}(\pi_{\bm{k}}+\mu_{\bm{k}})-n_{F}(\pi_{\bm{k}}-\mu_{\bm{k}})}{2\mu_{\bm{k}}}\stackrel{{\scriptstyle
T\to
0}}{{=}}\sum_{|\mu_{\bm{k}}|>E}\frac{\Theta(|\mu_{\bm{k}}|-|\pi_{\bm{k}}|)}{|\mu_{\bm{k}}|},$
(21)

  * •

intra-valley particle-hole fluctuation (${\bm{\eta}}=+-$)

$\Pi_{{\bm{0}}}^{\text{ph}}(E)\equiv\Pi^{+-}(E)=0,$ (22)

  * •

inter-valley particle-particle fluctuation (${\bm{\eta}}=-+$)

$\Pi_{{\bm{Q}}}^{\text{pp}}(E)\equiv\Pi^{-+}(E)=\sum_{k\in\mathcal{K}_{E}}-\frac{n_{F}(\epsilon_{\bm{k}})-n_{F}(-\epsilon_{\bm{k}})}{2\epsilon_{\bm{k}}}\stackrel{{\scriptstyle
T\to 0}}{{=}}\sum_{|\epsilon_{\bm{k}}|>E}\frac{1}{|\epsilon_{\bm{k}}|},$ (23)

  * •

inter-valley particle-hole fluctuation (${\bm{\eta}}=--$)

$\Pi_{{\bm{Q}}}^{\text{ph}}(E)\equiv\Pi^{--}(E)=\sum_{k\in\mathcal{K}_{E}}-\frac{n_{F}(\pi_{\bm{k}}+\mu_{\bm{k}})-n_{F}(-\pi_{\bm{k}}+\mu_{\bm{k}})}{2\pi_{\bm{k}}}\stackrel{{\scriptstyle
T\to
0}}{{=}}\sum_{|\pi_{\bm{k}}|>E}\frac{\Theta(|\pi_{\bm{k}}|-|\mu_{\bm{k}}|)}{|\pi_{\bm{k}}|}.$
(24)

Suppose the chemical potential is tuned to a van Hove singularity, and the
density of state diverges logarithmically $2\nu_{0}\ln(\Lambda/\epsilon)$ as
$\epsilon\to 0$. Then among all four $\Pi^{\bm{\eta}}(E)$, only
$\Pi_{{\bm{Q}}}^{\text{pp}}(E)=\Pi^{-+}(E)\sim\nu_{0}\ln^{2}(\Lambda/E)$ will
diverge double-logarithmically as $E\to 0$. Therefore instead of using the log
energy cutoff $(-\ln E)$ as the RG scale, we would better use
$y=\Pi_{{\bm{Q}}}^{\text{pp}}(E)$ as the RG scale. Reducing the energy scale
$E$ corresponds to increasing the scale $y$. Introduce the following nesting
parameters to track how the other ladder diagram scales with $y$,

$d_{{\bm{0}}}^{\text{pp}}=\partial_{y}\Pi_{{\bm{0}}}^{\text{pp}}=\frac{\mathrm{d}\Pi_{{\bm{0}}}^{\text{pp}}}{\mathrm{d}\Pi_{{\bm{Q}}}^{\text{pp}}},d_{{\bm{Q}}}^{\text{ph}}=\partial_{y}\Pi_{{\bm{Q}}}^{\text{ph}}=\frac{\mathrm{d}\Pi_{{\bm{Q}}}^{\text{ph}}}{\mathrm{d}\Pi_{{\bm{Q}}}^{\text{pp}}},$
(25)

then we have

$\mathrm{d}\Pi^{++}=d_{{\bm{0}}}^{\text{pp}}\mathrm{d}y,\mathrm{d}\Pi^{+-}=0,\mathrm{d}\Pi^{-+}=\mathrm{d}y,\mathrm{d}\Pi^{--}=d_{{\bm{Q}}}^{\text{ph}}\mathrm{d}y.$
(26)

Because $\Pi_{{\bm{0}}}^{\text{pp}}(E)$ and $\Pi_{{\bm{Q}}}^{\text{ph}}(E)$ do
not diverge as fast as $\Pi_{{\bm{Q}}}^{\text{pp}}(E)$ as $E\to 0$, we expect
the nesting parameters to fall in the regime of
$d_{{\bm{0}}}^{\text{pp}},d_{{\bm{Q}}}^{\text{ph}}\in[0,1]$.

With the above parametrization of $\mathrm{d}\Pi^{\bm{\eta}}$, we substitute
Eq. (18) into Eq. (11), and obtain

$\mathrm{d}V_{abcd}=-\sum_{{\bm{\eta}}}V_{abc^{\prime}d^{\prime}}Q^{\bm{\eta}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}V_{a^{\prime}b^{\prime}cd}\;\mathrm{d}\Pi^{\bm{\eta}}=-V_{abc^{\prime}d^{\prime}}(d_{{\bm{0}}}^{\text{pp}}Q^{\bm{++}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}+Q^{\bm{-+}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}+d_{{\bm{Q}}}^{\text{ph}}Q^{\bm{--}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}})V_{a^{\prime}b^{\prime}cd}\;\mathrm{d}y.$
(27)

Plug in the interaction vertex $V$ in Eq. (LABEL:eq:V) and the two-particle
projection operator $Q^{{\bm{eta}}}$ in Eq. (19) into Eq. (27), we arrive at
the RG equation

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}U&=-\tfrac{1}{2}d_{{\bm{0}}}^{\text{pp}}U^{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}V&=-\tfrac{1}{2}(1-d_{{\bm{Q}}}^{\text{ph}})(V^{2}+3J_{H}^{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}J_{H}&=-J_{H}((1+d_{{\bm{Q}}}^{\text{ph}})J_{H}+(1-d_{{\bm{Q}}}^{\text{ph}})V).\end{split}$
(28)

We can show the RG equation is consistent with similar results in existing
literature. In Ref. 38, the one-loop RG of the following interaction was
studied

$H_{\text{int}}=\sum_{\alpha\neq\beta}(g_{1}c_{\alpha}^{\dagger}c_{\beta}^{\dagger}c_{\alpha}c_{\beta}+g_{2}c_{\alpha}^{\dagger}c_{\beta}^{\dagger}c_{\beta}c_{\alpha}+g_{3}c_{\alpha}^{\dagger}c_{\alpha}^{\dagger}c_{\beta}c_{\beta}+)+\sum_{\alpha}g_{4}c_{\alpha}^{\dagger}c_{\alpha}^{\dagger}c_{\alpha}c_{\alpha},$
(29)

and the following RG equation was obtained

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y^{\prime}}g_{1}&=-2d_{1}g_{1}(g_{1}-g_{2})+2d_{2}g_{1}g_{4}-2d_{3}g_{1}g_{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y^{\prime}}g_{2}&=d_{1}(g_{2}^{2}+g_{3}^{2})+2d_{2}(g_{1}-g_{2})g_{4}-d_{3}(g_{1}^{2}+g_{2}^{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y^{\prime}}g_{3}&=-2g_{3}g_{4}-2d_{1}g_{3}(g_{1}-2g_{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y^{\prime}}g_{4}&=-(g_{3}^{2}+g_{4}^{2})+d_{2}(g_{1}^{2}+2g_{1}g_{2}-g_{2}^{2}+g_{4}^{2}),\\\
\end{split}$ (30)

where the RG time is $y^{\prime}=\Pi_{{\bm{0}}}^{\text{pp}}$ and the nesting
parameters are defined as

$d_{1}=\frac{\mathrm{d}\Pi_{{\bm{Q}}}^{\text{ph}}}{\mathrm{d}\Pi_{{\bm{0}}}^{\text{pp}}},d_{2}=\frac{\mathrm{d}\Pi_{{\bm{0}}}^{\text{ph}}}{\mathrm{d}\Pi_{{\bm{0}}}^{\text{pp}}},d_{3}=\frac{\mathrm{d}\Pi_{{\bm{Q}}}^{\text{pp}}}{\mathrm{d}\Pi_{{\bm{0}}}^{\text{pp}}}.$
(31)

We first need to change the RG time from
$y^{\prime}=\Pi_{{\bm{0}}}^{\text{pp}}$ to $y=\Pi_{{\bm{Q}}}^{\text{pp}}$ and
rewrite the RG equation in terms of our nesting parameters in Eq. (25) (which
amounts to replacing
$\frac{\mathrm{d}}{\mathrm{d}y^{\prime}}=(1/d_{\bm{0}}^{\text{pp}})\frac{\mathrm{d}}{\mathrm{d}y}$,
$d_{1}=d_{\bm{Q}}^{\text{ph}}/d_{\bm{0}}^{\text{pp}}$, $d_{2}=0$,
$d_{3}=1/d_{\bm{0}}^{\text{pp}}$),

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}g_{1}&=-2d_{{\bm{Q}}}^{\text{ph}}g_{1}(g_{1}-g_{2})-2g_{1}g_{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}g_{2}&=d_{{\bm{Q}}}^{\text{ph}}(g_{2}^{2}+g_{3}^{2})-(g_{1}^{2}+g_{2}^{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}g_{3}&=-2d_{\bm{0}}^{\text{pp}}g_{3}g_{4}-2d_{{\bm{Q}}}^{\text{ph}}g_{3}(g_{1}-2g_{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}g_{4}&=-d_{\bm{0}}^{\text{pp}}(g_{3}^{2}+g_{4}^{2}).\\\
\end{split}$ (32)

Then by comparing with Eq. (6), we identify the couplings in Eq. (29) as

$g_{1}=J_{H},g_{2}=\frac{1}{2}(V+J_{H}),g_{3}=0,g_{4}=\frac{1}{2}U,$ (33)

then the RG equation becomes

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}J_{H}&=-d_{{\bm{Q}}}^{\text{ph}}J_{H}(J_{H}-V)-J_{H}(J_{H}+V),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}\tfrac{1}{2}(V+J_{H})&=d_{{\bm{Q}}}^{\text{ph}}\tfrac{1}{4}(V+J_{H})^{2}-(J_{H}^{2}+\tfrac{1}{4}(V+J_{H})^{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}(\tfrac{1}{2}U)&=-d_{\bm{0}}^{\text{pp}}(\tfrac{1}{4}U^{2}).\\\
\end{split}$ (34)

After some simple algebra, it is straightforward to show that the RG equation
is consistent with our derivation.

###  A.2 Other Form of Hund’s Interaction

Another form [25] of Hund’s interaction in terms of IVC operators
${\bm{I}}=c_{K}^{\dagger}{\bm{\sigma}}c_{K^{\prime}}$, (Ref. 56 points out
that this has a better justified microscopic origin in the present problem).
In terms of the IVC Hund’s coupling, the total interaction will be
reparametrized as

$H_{\text{int}}=\int\mathrm{d}^{2}{\bm{x}}\;\frac{U}{2}(n_{K}^{2}+n_{K^{\prime}}^{2})+\tilde{V}n_{K}n_{K^{\prime}}-\tilde{J}_{H}{\bm{I}}^{\dagger}\cdot{\bm{I}}.$
(35)

We have assume the limit that the interaction is very local, such that there
is no momentum dependence of the coupling coefficients. In this limit, we can
use the following operator identity

${\bm{I}}^{\dagger}\cdot{\bm{I}}=-\frac{3}{2}n_{K}n_{K^{\prime}}+\frac{1}{2}{\bm{S}}_{K}\cdot{\bm{S}}_{K^{\prime}}$
(36)

to connect Eq. (35) and Eq. (6), and establish the following linear relations
between couplings $(V,J_{H})$ and $(\tilde{V},\tilde{J}_{H})$,

$\left[\begin{matrix}V\\\
J_{H}\end{matrix}\right]=\left[\begin{matrix}1&3/2\\\
0&1/2\end{matrix}\right]\left[\begin{matrix}\tilde{V}\\\
\tilde{J}_{H}\end{matrix}\right].$ (37)

Therefore the RG equations

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}U&=-\tfrac{1}{2}d_{{\bm{0}}}^{\text{pp}}U^{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}V&=-\tfrac{1}{2}(1-d_{{\bm{Q}}}^{\text{ph}})(V^{2}+3J_{H}^{2}),\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}J_{H}&=-J_{H}((1+d_{{\bm{Q}}}^{\text{ph}})J_{H}+(1-d_{{\bm{Q}}}^{\text{ph}})V),\\\
\end{split}$ (38)

can also be written as

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}U&=-\tfrac{1}{2}d_{{\bm{0}}}^{\text{pp}}U^{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}\tilde{V}&=-\tfrac{1}{2}(1-d_{{\bm{Q}}}^{\text{ph}})\tilde{V}^{2}+\tfrac{3}{2}\tilde{J}_{H}^{2},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}\tilde{J}_{H}&=-\tilde{J}_{H}((2-d_{{\bm{Q}}}^{\text{ph}})\tilde{J}_{H}+(1-d_{{\bm{Q}}}^{\text{ph}})\tilde{V}),\\\
\end{split}$ (39)

The RG equation for $U$ is not affected by the reparameterization, and is
decoupled. The RG flow in the $(V,J_{H})$ or $(\tilde{V},\tilde{J}_{H})$
planes are compared in Fig. 7.

Figure 7: (a) RG flow in $(V,J_{H})$ with spin Hund’s interaction following
Eq. (38). (b) RG flow in $(\tilde{V},\tilde{J}_{H})$ with IVC Hund’s
interaction following Eq. (39). Both RG flow diagrams are calculated for a
nesting parameter $d_{\bm{Q}}^{ph}=0.5$. Note the red (blue) regions
correspond to spin singlet (triplet) superconductor, while the grey is the
metal (Fermi liquid).

###  A.3 RG Fixed Rays and Oder Parameter Scaling

We will focus on the $(V,J_{H})$ basis, and discuss the RG fixed rays of the
flow equation Eq. (38). By solving the RG equation, the asymptotic behavior
near $y\to y_{c}$ is expected to be

$V=\frac{\gamma_{V}}{y_{c}-y},J_{H}=\frac{\gamma_{J_{H}}}{y_{c}-y}.$ (40)

Substitute Eq. (40) to Eq. (38), one can determine the coefficients
$\gamma_{V}$ and $\gamma_{J_{H}}$ by the following algebraic equations

$\begin{split}\gamma_{V}&=\tfrac{1}{2}(1-d_{\bm{Q}}^{\text{ph}})(\gamma_{V}^{2}+3\gamma_{J_{H}}^{2}),\\\
\gamma_{J_{H}}&=-\gamma_{J_{H}}((1+d_{\bm{Q}}^{\text{ph}})\gamma_{J_{H}}+(1-d_{\bm{Q}}^{\text{ph}})\gamma_{V}).\end{split}$
(41)

The solutions are:

  * •

Gaussian fixed point

$\gamma_{V}=\gamma_{J}=0;$ (42)

  * •

$\mathrm{SO}(4)$ fixed ray

$\gamma_{V}=-\frac{2}{1-d_{\bm{Q}}^{\text{ph}}},\gamma_{J_{H}}=0;$ (43)

  * •

$\mathrm{SU}(2)_{\text{s}}^{\text{singlet}}$ fixed ray

$\gamma_{V}=-\frac{1}{1-d_{\bm{Q}}^{\text{ph}}}\left(1-\frac{1+d_{\bm{Q}}^{\text{ph}}}{2\sqrt{1-d_{\bm{Q}}^{\text{ph}}+(d_{\bm{Q}}^{\text{ph}})^{2}}}\right),\gamma_{J_{H}}=-\frac{1}{2\sqrt{1-d_{\bm{Q}}^{\text{ph}}+(d_{\bm{Q}}^{\text{ph}})^{2}}};$
(44)

  * •

$\mathrm{SU}(2)_{\text{s}}^{\text{triplet}}$ fixed ray

$\gamma_{V}=-\frac{1}{1-d_{\bm{Q}}^{\text{ph}}}\left(1+\frac{1+d_{\bm{Q}}^{\text{ph}}}{2\sqrt{1-d_{\bm{Q}}^{\text{ph}}+(d_{\bm{Q}}^{\text{ph}})^{2}}}\right),\gamma_{J_{H}}=\frac{1}{2\sqrt{1-d_{\bm{Q}}^{\text{ph}}+(d_{\bm{Q}}^{\text{ph}})^{2}}}.$
(45)

To study the scaling of fermion bilinear order parameters along the RG flow,
one can consider deforming the Hamiltonian with infinitesimal fermion bilinear
terms $H=H_{0}+\delta H$,

$\delta H=\int\mathrm{d}^{2}x\sum_{a,b}X_{ab}\chi_{a}\chi_{b},$ (46)

where $X$ is an antisymmetric imaginary matrix representing the fermion
bilinear order. The correction of $X$ is given by
$X_{ab}(E)=X_{ab}-V_{abc^{\prime}d^{\prime}}\Pi_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}(E)X_{a^{\prime}b^{\prime}}+\cdots$,
which leads to the RG equation

$\mathrm{d}X_{ab}=-V_{abc^{\prime}d^{\prime}}\mathrm{d}\Pi_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}X_{a^{\prime}b^{\prime}}=-V_{abc^{\prime}d^{\prime}}(d_{{\bm{0}}}^{\text{pp}}Q^{\bm{++}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}+Q^{\bm{-+}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}+d_{{\bm{Q}}}^{\text{ph}}Q^{\bm{--}}_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}})X_{a^{\prime}b^{\prime}}.$
(47)

Along the eigen direction (labeled by $\Phi$) of the RG flow, we have

$\tfrac{\mathrm{d}}{\mathrm{d}y}X_{\Phi}=V_{\Phi}X_{\Phi},$ (48)

where $V_{\Phi}$ denotes the eigen value (which depends on $V_{abcd}$
linearly). Near the critical point $y\to y_{c}$, $V_{\Phi}$ diverges as
$\gamma_{\Phi}/(y_{c}-y)$, then the RG equation implies

$\frac{\mathrm{d}}{\mathrm{d}y}X_{\Phi}=\frac{\gamma_{\Phi}}{y_{c}-y}X_{\Phi},$
(49)

whose solution is

$X_{\Phi}(y)\propto(y_{c}-y)^{-\gamma_{\Phi}}.$ (50)

Then the susceptibility diverges near $y_{c}$ as

$\chi_{\Phi}(y)=\frac{X_{\alpha}(y)}{X_{\Phi}(0)}\sim(y_{c}-y)^{-\gamma_{\Phi}}.$
(51)

Therefore, a large positive exponent $\gamma_{\Phi}$ indicates a stronger
instability in the eigen channel of the fermion bilinear ordering. In the
following, we will apply this general framework to different bilinear orders.

  * •

Inter-Valley superconductivity
$\Delta^{\mu}=c_{K}^{\intercal}\mathrm{i}\sigma^{2}\sigma^{\mu}c_{K^{\prime}}$.
The RG equation reads

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}\Delta^{0}&=-2(3J_{H}+V)\Delta^{0},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}{\bm{\Delta}}&=2(J_{H}-V){\bm{\Delta}},\\\
\end{split}$ (52)

therefore $\gamma_{\Delta^{0}}=-2(3\gamma_{J_{H}}+\gamma_{V})$ and
$\gamma_{{\bm{\Delta}}}=2(\gamma_{J_{H}}-\gamma_{V})$.

  * •

Inter-Valley coherence $I^{\mu}=c_{K}^{\dagger}\sigma^{\mu}c_{K^{\prime}}$.
The RG equation reads

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}I^{0}&=-2d_{\bm{Q}}^{\text{ph}}(3J_{H}-V)I^{0},\\\
\tfrac{\mathrm{d}}{\mathrm{d}y}{\bm{I}}&=2d_{\bm{Q}}^{\text{ph}}(J_{H}+V){\bm{I}},\\\
\end{split}$ (53)

therefore
$\gamma_{\Delta^{0}}=-2d_{\bm{Q}}^{\text{ph}}(3\gamma_{J_{H}}-\gamma_{V})$ and
$\gamma_{{\bm{\Delta}}}=2d_{\bm{Q}}^{\text{ph}}(\gamma_{J_{H}}+\gamma_{V})$.

  * •

Intra-Valley superconductivity
$\Delta_{\xi}=c_{\xi}^{\intercal}\mathrm{i}\sigma^{2}c_{\xi}$. The RG equation
reads

$\begin{split}\tfrac{\mathrm{d}}{\mathrm{d}y}\Delta_{\xi}&=-2d_{\bm{0}}^{\text{pp}}U\Delta_{\xi},\end{split}$
(54)

therefore $\gamma_{\Delta_{\xi}}=-2d_{\bm{0}}^{\text{pp}}\gamma_{U}$.

  * •

Valley charge density $n_{v}=n_{K}-n_{K^{\prime}}$. The RG equation is trivial
$\tfrac{\mathrm{d}}{\mathrm{d}y}n_{v}=0$, so $\gamma_{n_{v}}=0$.

  * •

Valley charge density ${\bm{S}}_{\xi}=c_{\xi}^{\dagger}{\bm{\sigma}}c_{\xi}$.
The RG equation is trivial $\tfrac{\mathrm{d}}{\mathrm{d}y}{\bm{S}}_{\xi}=0$,
so $\gamma_{{\bm{S}}_{\xi}}=0$.