# Origin of Superconductivity in Rhombohedral Trilayer Graphene: Quasiparticle
Pairing within the Inter-Valley Coherent Phase

Chun Wang Chau  Department of Physics, Hong Kong University of Science and
Technology, Clear Water Bay, Hong Kong, China  Shuai A. Chen<EMAIL_ADDRESS>Department of Physics, Hong Kong University of Science and Technology, Clear
Water Bay, Hong Kong, China  K. T. Law<EMAIL_ADDRESS>Department of Physics,
Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong,
China

###### Abstract

Superconductivity (SC) is observed in rhombohedral trilayer graphene (RTG)
with an extremely low charge carrier densities, between the normal metal state
and a probable inter-valley coherent (IVC) state. The measured coherence
length in RTG is roughly two orders of magnitude shorter than the value
predicted by a conventional BCS theory. To resolve these discrepancies, we
propose that the RTG SC phase arises from the pairing of quasiparticles within
the IVC phase. We illustrate the SC behavior using gapped Dirac cones with the
chemical potential close to the conduction band’s bottom and establish the
Ginzburg-Landau theory. Our findings indicate that the mean-field transition
temperature, $T_{\mathrm{MF}}$, is primarily determined by the density of
states of quasiparticles within the IVC phase, and is more suppressed in
comparison with the BCS relation. Interestingly, we discover that the
coherence length behaves according to $\xi\sim 1/\sqrt{T_{\mathrm{MF}}}$. When
the chemical potential lies within the band gap, the conventional coherence
length becomes small, while the quantum metric contribution can diverge.
Applying our approach to a microscopic model for RTG, our predictions align
well with experimental data and effectively capture key measurable features of
quantities such as the mean-field transition temperature $T_{\mathrm{MF}}$ and
the coherence length $\xi$. Furthermore, we reveal that the quantum metric
contribution, stemming from the multi-band properties of the IVC
quasiparticles, plays a significant role in determining the coherence length
around the transition regime from superconductivity to IVC. Lastly,
experimental predictions are discussed on bebaviors of the upper critical
field and coherence length at low temperatures.

###### Contents

  1. I Introduction
  2. II Background
    1. II.1 Discrepancies in Superconductivity of RTG
    2. II.2 Introduction to Intervalley coherence state
  3. III Toy model: Superconductivity at band bottom of gapped Dirac fermions
    1. III.1 Methodology
    2. III.2 Application to Gapped Dirac Fermion at Band Bottom
  4. IV Superconductivity from pairing of IVC quasiparticle
    1. IV.1 Prediction from toy model
    2. IV.2 Numerical calculation on IVC-SC state on the microscopic model
  5. V Conclusion
  6. A Details of 6-band Hamiltonian for RTG
  7. B Derivation of free energy
  8. C Quantum metric of massive Dirac cones
  9. D Ginzburg-Landau Theory of Massive Dirac Cone
    1. D.1 Case 1: $m\gg T_{\mathrm{MF}}$
      1. D.1.1 Beyond band bottom
    2. D.2 Case 2: $T_{\mathrm{MF}}\gg m$
  10. E Discussion on pairing and time-reversal symmetry in IVC

##  I Introduction

Graphene heterostructures, in particular twisted bilayer graphene, have been
observed experimentally to host various strongly correlated phases [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11]. However, many of these heterostructures are
difficult to realize in experiments due to structural instability [12, 13].
Recently, breakthrough [14, 15, 16] has been made in a more stable graphene-
based system, namely ABC-stacked rhombohedral trilayer graphene [17, 18]
(RTG), and has opened up exciting opportunities for studying strongly
correlated phases of matter [19, 20, 21, 22]. By adjusting the electron
density and applying electric fields perpendicular to the material [23],
researchers can customize and investigate these states in RTG. One
particularly intriguing development is the discovery of superconductivity in
RTG, which manifests in two distinct phases referred to as SC1 and SC2 in
[14]. SC1 occurs within the flavor-symmetric phase and exhibits a maximum
critical temperature of $T_{c1}\sim 100\mathrm{mK}$ and superconducting
coherence length $\xi\sim 200\mathrm{nm}$. Several theoretical explanations
have been proposed to elucidate the mechanism behind SC1 in RTG. These
proposals encompass diverse pairing mechanisms of electrons in a normal metal
phase, such as electron-phonon coupling [24, 25], Kohn-Luttinger-like
mechanism [26, 27, 28, 29], direct coupling mediated by Coulomb interaction
[30, 31, 32], and pairing facilitated by the proximity to a correlated metal
[33, 34, 35]. However, the influence of the neighboring symmetry-breaking
phase, which is likely an intervalley coherent state, on the properties of the
superconducting states remains unclear. This lack of understanding is
especially relevant when it comes to experimental measurements such as the
transition temperature and coherence length.

In this paper, we propose that superconductivity SC1 observed in RTG does not
arise from the pairing between bare electrons in the normal metal state, but
instead is due to direct pairing between quasiparticles of the intervalley
coherence state (IVC). We first argue that there is a clear discrepancy from
the experimental result [14] both in the mean-field temperature and upper
critical field, when we assumed direct pairing of electrons. Instead, we
proposed that when in close vicinity with the phase transition from IVC to a
normal metal state, where the chemical potential cuts the band top of the
lower energy IVC quasiparticle band, would increase in density of state thus
favoring direct pairing between IVC quasiparticles, inducing s-wave
superconductivity where the IVC order parameter remains intact. We then
analytically study the unique properties of superconductivity near the band
gap using a toy model that captures the key feature of the IVC state, namely
nearly massless Dirac fermions located at $K$ and $K^{\prime}$ valleys. By
parameterizing the toy model accordingly with the RTG, we approximate the
coherence length that is similar in value to what is observed in experiments,
and explained a narrower window of superconductivity as opposed to the
Bardeen-Cooper-Schrieffer (BCS) theory [36] on a highly dispersive band. In
closing of the paper, we perform numerical calculations explicitly on the IVC
states of RTG, calculating the mean field temperature and upper critical
field, which matches well and capture key features of the experimental result.
We also commented on the possible implication of quantum metric, in inducing
the transition between the superconducting state and the IVC state.

In Sec. II.1, we examine the observed superconductivity SC1 in the experiment
and present evidence that it cannot be attributed to pairing between bare
electrons. In Sec. II.2, we delve into the IVC state and explore how SC1 can
arise from the direct pairing between IVC quasiparticles. In Section III, we
develop a simplified model that captures the essential characteristics of the
IVC quasiparticle bands. We discuss how superconductivity beyond the BCS limit
occurs when the chemical potential is close to the band gap. In Sec. IV.1,
utilizing the toy model, we provide justification for our assertion that SC1
originates from the IVC states and is observed when the chemical potential is
in proximity to the top of the lower energy IVC quasiparticle band. In Sec.
IV.2, we perform numerical calculations on the microscopic model to determine
the mean-field temperature and coherence length for superconductivity induced
by the pairing of IVC quasiparticles. These results are then compared with the
experimental findings. Additionally, we discuss how the quantum metric may
play a significant role in facilitating the transition from the
superconducting state to the IVC state.

##  II Background

###  II.1 Discrepancies in Superconductivity of RTG

Figure 1:  (a) Schematic illustration of the lattice structure of RTG. Note
that only $A_{1}$ and $B_{3}$ do not have direct interlayer hopping, and hence
are of lower energy in comparison with other sites. As such at the low charge
carrier density, they act as active sites of the system. (b) The density of
states at the valence band of electrons in a normal metal phase. There is a
jump at $n\approx-2.9\times 10^{12}\mathrm{cm}^{-2}$, indicating a transition
to an annulus Fermi surface from a complete one. Also, there is a Van Hove
singularity when $n\approx-0.5\times 10^{12}\mathrm{cm}^{-2}$, corresponding
to Lifshitz transition to disconnected pockets. (c) Contour plot of the band
structure at K valley. When the chemical potential varies, the Fermi surface
topology is changed from disconnected pockets to annulus, and then to a
completed surface, giving rise to the density of states in (b). (d) The phase
diagram from the experiment in Ref. 14. The normal metal state is left
uncolored. The SC1 occurs in between the normal metal state and the IVC state
in a very narrow window of charge carrier densities. In obtaining (b) and (c),
we set the potential difference between the outer layers of RTG as
$u_{d}=34.5$meV, and the horizontal dotted dashed line in (d) corresponds to
$u_{d}=34.5$meV.

For RTG, where the lattice structure is illustrated in Fig. 1(a),
superconducting state SC1 is observed within the window of charge carrier
density $n\approx-1.93\times 10^{12}\mathrm{cm}^{-2}$ to $-1.8\times
10^{12}\mathrm{cm}^{-2}$ when $u_{d}$, which is the potential difference
between the outer layers of RTG [17], is approximately $34.5$meV in experiment
[14]. Despite the increase in density of state as illustrated in Fig. 1(b) due
to the trigonal warping [18] of the band structure as shown in Fig. 1(c), when
we further decrease the carrier density, competing states, for example, the
spin polarized state [15], are more energy favorable, and thus a
superconducting state is not observed, until SC2 [14], which is deep within
the half-metal regime. A schematic for the phase diagram is illustrated in
Fig. 1(d). Focusing on SC1, at $n\approx-1.85\times 10^{12}\mathrm{cm}^{-2}$,
corresponding to chemical potential of $\mu\approx-43.4\mathrm{meV}$, the
temperature is maximum, at $T_{\mathrm{MF}}\approx 120\mathrm{mK}$. By fitting
the maximum transition temperature at the specific charge carrier density, we
can estimate an attractive interaction strength of approximately $U\approx
4\mathrm{eV}$ using the self-consistency equation in a mean-field theory, as
discussed in detail in Sec. III.1. Indeed, when considering electron pairing
in the normal metal state and examining a wide range of charge carrier
densities, a noticeable discrepancy emerges between the predicted trend of the
transition temperature according to the mean-field theory and the experimental
observations illustrated in Fig. 2(a). This discrepancy highlights the
inadequacy of the naive mean-field theory and the consideration of pairing in
the normal metal state in capturing the intricacies of the superconducting
phase.

Significant contradiction from the experimental result can be observed when we
attempt to calculate the superconducting coherence length. For the charge
carrier density $n\approx-1.85\times 10^{12}\mathrm{cm}^{-2}$, where the
transition temperature is maximum, under BCS relation [36], the coherence
length can be approximated by $\xi_{0}\sim 0.18\hbar v_{F}/k_{B}T_{c}$, with
$\hbar v_{F}\approx 6.6\times 10^{-10}$eVnm for graphene heterostructure.
Using $T_{c}\approx 120$mK, we can approximate the coherence length as $\sim
10^{4}$nm, which is roughly two orders of magnitude larger than the coherence
length ($150-250$nm) extracted from the experiment using the perpendicular
upper critical field $H_{c2}={\phi_{0}}/{2\pi\xi^{2}}$ where
$\phi_{0}={h}/{2e}$ is the superconducting quantum flux. If we additionally
account for the low charge carrier density, from numerical calculations, at
base temperature ($\approx 50$mK) of the experiment, the coherence length lies
between $1650-1800$nm, which is still an order of magnitude larger than the
experimental result. A plot of $H_{c2}$ from numerical calculation is shown in
Fig. 2(b).

These discrepancies from experiments are clear indications that the
superconductivity SC1 observed in RTG may not originate from the normal metal
state, namely pairing of bare electrons. Instead, SC1 might emerge from the
neighboring competing state, which is referred to as the partially isospin
polarized (PIP) state [14, 15] and most likely corresponds to the intervalley
coherence state (IVC) [26, 34]. In this paper, we propose that SC1, is
characterized by both the IVC order parameter and the superconducting order
parameter simultaneous. In other words, the Cooper pair in SC1, instead of
originating from pairing between electrons, is from the direct pairing of
quasiparticles of the IVC state, thus referred to as IVC-SC in the remainder
of this paper. In addition, we propose that the occurrence of IVC-SC in close
proximity to the phase boundary between the IVC state and the normal metal
state is not merely coincidental but rather a key characteristic. In our
model, the IVC state, characterized by two quasiparticle bands, transitions
into the normal metal state when the chemical potential starts to intersect at
the top of the lower energy band. Hence, at the phase boundary between the IVC
state and the normal metal state, the enhanced density of states would benefit
the superconductivity.

###  II.2 Introduction to Intervalley coherence state

As discussed in Fig. 1(b), the density of states of electrons diverges at the
charge carrier density $n\approx-0.5\times 10^{12}\mathrm{cm}^{-2}$. At this
doping level, the K and K′ valley Fermi surfaces are almost perfectly nested
together as illustrated in Fig. 2(c). Assuming a repulsive interaction between
K and K′ valley, the intervalley nesting would energetically favor the IVC
state, for which a gap is opened at the nested portion of the Fermi surfaces.
To further study the IVC state, we can define the mean-field Hamiltonian:

$H_{\mathrm{MF}}=\sum_{\tau,\tau^{\prime},\mathbf{q}}\psi_{\tau,\mathbf{q}}^{\dagger}h_{\tau,\tau^{\prime}}(\mathbf{q})\psi_{\tau^{\prime},\mathbf{q}},$
(1)

with the Hamiltonian matrix as

$h_{\tau,\tau^{\prime}}(\mathbf{q})=\epsilon_{s}(\mathbf{q})I_{2}+\bm{\Delta}(\mathbf{q})\cdot\bm{\sigma}.$
(2)

The $\bm{\Delta}(\mathbf{q})$ is related to the IVC order parameter,

$\\!\\!\\!\\!\bm{\Delta}(\mathbf{q})\\!=(|\bm{\Delta}_{\mathrm{IVC}}(\mathbf{q})|\cos(\phi_{\mathbf{q}}),|\bm{\Delta}_{\mathrm{IVC}}(\mathbf{q})|\sin(\phi_{\mathbf{q}}),\epsilon_{a}(\mathbf{q})).$
(3)

where $I_{2}$ is the $2\times 2$ identity matrix and
$\bm{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are Pauli matrices. In Eq.
(3),
$\epsilon_{s}(\mathbf{q})=-\mu+\frac{1}{2}[\epsilon_{K}(\mathbf{q})+\epsilon_{K^{\prime}}(\mathbf{q})]$
and
$\epsilon_{a}(\mathbf{q})=\frac{1}{2}[\epsilon_{K}(\mathbf{q})-\epsilon_{K^{\prime}}(\mathbf{q})]$
are the symmetric and antisymmetric parts of the normal metal state
dispersion, respectively, and
$\Delta_{\mathrm{IVC}}(\mathbf{q})=|\bm{\Delta}_{\mathrm{IVC}}(\mathbf{q})|\exp(i\phi_{\mathbf{q}})$
is the order parameter for the IVC state. Note that the IVC order parameter
has a phase, that can be well matched with the phase of the intervalley form
factor [34]. The dispersion spectrum of the IVC state is given by:

$\epsilon_{\pm}(\mathbf{q})=\epsilon_{s}(\mathbf{q})\pm|\bm{\Delta}(\mathbf{q})|.$
(4)

Figure 2:  Predictions from the theory by assuming SC1 from normal metal
phase. (a) The mean-field transition temperature (red curve) by assuming
superconductivity from a normal metal phase. The blue region is extracted for
SC1 from the experiment in Ref. 14. The predictions deviate from the
experimental measurements in particular at the phase boundary $n=-1.8\times
10^{12}\mathrm{cm}^{-2}$. (b) Out-of-plane upper critical field $H_{c2}$ by
assuming superconductivity from a normal metal phase. The two curves
correspond to $T=50$mK (Blue) and $T=100$mK (Red). The predictions are of two
orders of magnitude smaller than the experimental results $\sim 10$mT in Ref.
14. Within the window of superconductivity, the upper critical field is
predicted to increase monotonically, as contrast to the experimental
observations where $H_{c2}$ reach a maximum at $n\approx-1.82\times
10^{12}\mathrm{cm}^{-2}$ and decreases to $0$ at $n\approx-1.8\times
10^{12}\mathrm{cm}^{-2}$. (c) Fermi surfaces corresponding to the Lifshitz
transition from the annulus to isolated pockets, when $u_{d}=34.5$meV at
$n=-0.5\times 10^{12}\mathrm{cm}^{-2}$. The red and blue regions correspond to
the valence band at K and K′ valleys, respectively. The Fermi surfaces exhibit
nearly perfect nesting, thus having strong instability under repulsive
interaction. (d) Dispersion spectrum of the quasiparticles in Eq. (4) in the
IVC state at different momenta $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$. The dashed
lines enclose the chemical potential of a very narrow regime where a
superconducting state SC1 is observed in experiments and the inset is to zoom
in to show the regime.

The dispersion spectrum for IVC states is shown in Fig. 2(d). With IVC states
defined in the valley basis, correspond to two quasiparticle bands, with Bloch
states:

$\displaystyle\psi_{+,\mathbf{q}}$
$\displaystyle=\frac{1}{\sqrt{2|\bm{\Delta}(\mathbf{q})|}}\begin{pmatrix}e^{-i\phi_{\mathbf{q}}}\sqrt{|\bm{\Delta}(\mathbf{q})|+\epsilon_{a}(\mathbf{q})}\\\
\sqrt{|\bm{\Delta}(\mathbf{q})|-\epsilon_{a}(\mathbf{q})}\end{pmatrix},$ (5)
$\displaystyle\psi_{-,\mathbf{q}}$
$\displaystyle=\frac{1}{\sqrt{2|\bm{\Delta}(\mathbf{q})|}}\begin{pmatrix}-e^{-i\phi_{\mathbf{q}}}\sqrt{|\bm{\Delta}(\mathbf{q})|-\epsilon_{a}(\mathbf{q})}\\\
\sqrt{|\bm{\Delta}(\mathbf{q})|+\epsilon_{a}(\mathbf{q})}\end{pmatrix}.$ (6)

Note that time reversal symmetry remains intact after the formation of IVC
state [37], which is crucial for the existence of superconductivity in our
proposal by assuming an s-wave pairing. We should also retrieve an effective
attractive interaction, to enable the formation of Cooper pairs from IVC
states. By adopting the argument in Ref. 26, involving the introduction of a
weaker anti-ferromagnetic Hund’s coupling compared to the repulsive
intervalley interaction, it becomes possible to achieve an effective
attractive intervalley interaction. This ultimately results in the emergence
of a spin-singlet superconducting phase. As shown in Fig. 2(d), the density of
states of the quasiparticles peak at the bottom of the conduction band (the
blue line), which allows us to introduce a toy model of massive Dirac cones
with the chemical potential close to the band bottom in Sec. III.

##  III Toy model: Superconductivity at band bottom of gapped Dirac fermions

###  III.1 Methodology

To further study the IVC-SC state, we construct a simplified model that
captures the key features of the IVC quasiparticles’ band structure, to
analytically study superconductivity when the chemical potential is in close
vicinity with the band gap. In our toy model, similar to RTG, we have two
Dirac cones, of Fermi velocity $v$, centered at the $K$ and $K^{\prime}$
valley respectively. By acting with an external potential, for example a
displacement field, we open up a mass gap $m$ between the originally touching
Dirac bands. The system can be described effectively in the continuum limit by
massive Dirac fermion Hamiltonian at close proximity to each valley:

$\displaystyle h_{\tau}(\mathbf{q})$
$\displaystyle=v\mathbf{q}\cdot(\tau\sigma_{x}\hat{\mathbf{e}}_{x}+\sigma_{y}\hat{\mathbf{e}}_{y})+m\sigma_{z},$
(7)

where $\tau=+/-$ correspond to $K$ and $K^{\prime}$ valley respectively,
$\mathbf{q}$ is the momentum difference to the center of the valley, and
$\sigma_{i}$ are Pauli matrices that act on the sub-lattice basis. In
particular, we will focus on the quasiparticle band of dispersion spectrum
$E(q)=\sqrt{v^{2}q^{2}+m^{2}}$, and consider only on-site effective attractive
interaction between quasiparticles:

$H_{\mathrm{int}}=-U\int d^{2}\mathbf{r}\
a_{+}^{\dagger}(\mathbf{r})a_{-}^{\dagger}(\mathbf{r})a_{-}(\mathbf{r})a_{+}(\mathbf{r}),$
(8)

where $U$ is attractive interaction strength, and $a_{\tau}(\mathbf{r})$ is
the quasiparticle annihilation operator of the valleys. Via the Hubbard-
Stratonovich transformation [38], we can then introduce a bosonic field
$\Delta(\mathbf{r})=a_{-}(\mathbf{r})a_{+}(\mathbf{r})$ that corresponds to
Cooper pairs. To simplify our discussions, we will focus on the contributions
of the conduction bands of the toy model, which can be realized via a
projection by involving the Bloch waves $u_{c,\tau}(\mathbf{q})$ of the
conduction bands [39, 40],

$a_{\tau}(\mathbf{r})\rightarrow\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}e^{i\mathbf{q}\cdot\mathbf{r}}u_{c,\tau}^{*}(\mathbf{q})c_{\tau}(\mathbf{q}),$
(9)

where $c_{\tau}(\mathbf{q})$ is the quasiparticle annihilation operator of
momentum $\mathbf{q}$ for the conduction band. Using the path integral
formalism, we can evaluate the free energy of the system, which can be
decomposed into two components: the mean-field solution and the fluctuations.
This decomposition is achieved by expanding the free energy around its
extremum. Further details of the calculation can be found in the supplementary
material [37]. Focusing on the case around the transition region where the
mean-field value of the bosonic field $\Delta_{0}$ vanishes, minimizing the
mean-field solution of the free energy provides us with a self-consistency
condition for the mean-field temperature, at which transition to
superconducting state initiates:

$1=U\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{1}{2\varepsilon(\mathbf{q})}\tanh\frac{\beta_{\mathrm{MF}}\varepsilon(\mathbf{q})}{2},$
(10)

where $T_{\mathrm{MF}}=1/\beta_{\mathrm{MF}}$ is the mean-field temperature,
and for convenience we have defined
$\varepsilon(\mathbf{q})=\epsilon_{0}(\mathbf{q})-\mu$. The fluctuation of the
free energy, in particular the quadratic term $F_{2}$, can be related to the
fluctuation of the order parameter $\delta\Delta(\mathbf{k})$. After
integrating out the fermionic fields $F_{2}$ takes the form:

$F_{2}=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}|\delta\Delta(\mathbf{k})|^{2}\left[U-U^{2}\chi(\mathbf{k})\right],$
(11)

where $\chi(\mathbf{k})$ is the four-point coherence function,

$\displaystyle\chi(\mathbf{k})$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}|\Gamma(\mathbf{q},\mathbf{k})|^{2}$
$\displaystyle\phantom{=}\times\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right]+\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}{2\left[\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}.$
(12)

And we have defined the form factor
$|\Gamma(\mathbf{q},\mathbf{k})|^{2}=\left|u_{c,+}^{T}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)u_{c,-}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right|^{2}$,
which was introduced upon performing projection (9) from a multiband system
onto a single relevant band. Given time reversal invariant of the Bloch states
$u^{*}_{c,-}(-\mathbf{q})=u_{c,+}(\mathbf{q})\equiv u(\mathbf{q})$, we can
expand the form factor around $\mathbf{k}=0$ as [39]:

$|\Gamma(\mathbf{q},\mathbf{k})|^{2}=1-\sum_{i,j}k_{i}k_{j}g_{ij}(\mathbf{q}),$
(13)

where
$g_{ij}(\mathbf{q})=\frac{1}{2}\mathrm{Tr}\left[\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})\right]$
is the quantum metric, and
$P(\mathbf{q})=\left|u(\mathbf{q})\right\rangle\left\langle
u(\mathbf{q})\right|$ is the projection matrix for the Bloch state. Similarly,
we can expand the four-point coherence function $\chi(\mathbf{k})$ around
$\mathbf{k}=0$, with the assumption that the conduction band is rotational
invariant as:

$\chi(\mathbf{k})=\chi_{0}-\left(\chi_{2,\mathrm{con}}+\chi_{2,\mathrm{qm}}\right)k^{2}+\mathcal{O}(k^{4}),$
(14)

where $\chi_{2,\mathrm{con}}$ is the contribution due to dispersion of the
band, and $\chi_{2,\mathrm{qm}}$ is due to the form factor, thus the quantum
metric. Recall the definition of $F_{2}$ (11), we can construct an effective
Lagrangian that governs the fluctuation of the order parameter
$\delta\Delta(\mathbf{k})$ from as
$F_{2}=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\mathcal{L}[\delta\Delta]$:

$\displaystyle\mathcal{L}[\delta\Delta]$
$\displaystyle=\frac{1}{2m^{*}}|\nabla\delta\Delta|^{2}+\alpha|\delta\Delta|^{2}+\mathcal{O}(|\delta\Delta|^{4}),$
(15)

with the coeffients

$\displaystyle\frac{1}{2m^{*}}$
$\displaystyle=U^{2}\left(\chi_{2,\mathrm{con}}+\chi_{2,\mathrm{qm}}\right),$
$\displaystyle\alpha$ $\displaystyle=U-U^{2}\chi_{0}.$

From the effective Lagrangian, we can determine the general form of the
superconducting coherence length, as motivated by the Ginzburg-Landau theory:

$\displaystyle\xi$
$\displaystyle=\sqrt{\frac{U^{2}\left(\chi_{2,\mathrm{con}}+\chi_{2,\mathrm{qm}}\right)}{|U-U^{2}\chi_{0}|}}=\sqrt{\xi_{\mathrm{con}}^{2}+\xi_{\mathrm{qm}}^{2}},$
(16)

where the overall coherence length is governed by both the conventional
contribution and the quantum metric [39, 40]. The conventional contribution
$\xi_{\mathrm{con}}$ depends on the band dispersion with the similar form as a
conventional s-wave superconductor and it vanishes in the flat-band limit. The
quantum metric part $\xi_{\mathrm{qm}}$ exclusively appears as a multi-band
effect, and it is determined by the quantum metric $g_{ij}$. The coherence
length $\xi$ will be scaled as
$\xi=(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}})^{-1/2}\xi_{\mathrm{GL}}$
since the GL theory is developed around $T\rightarrow T_{\mathrm{MF}}$, while
we extract the GL coherence length $\xi_{\mathrm{GL}}$ as in the standard way.
In the following subsection, we will study the case that is relevant to the
IVC-SC state, for which the mass gap $m\gg T_{\mathrm{MF}}$. Details of the
analytical calculation and a brief discussion of the $m\ll T_{\mathrm{MF}}$
regime are included in the supplementary material [37].

###  III.2 Application to Gapped Dirac Fermion at Band Bottom

In this section, we will focus on chemical potential $\mu<\epsilon_{D}$. We
note that for $\mu>\epsilon_{D}$, the system can be well described by the BCS
theory, thus is highly conventional, and will not be discussed in this
section. We first consider the regime where the chemical potential satisfies
$\beta_{\mathrm{MF}}(\mu-m)\gg 1$, thus away from the mass gap. The mean-field
temperature can be solved self-consistently by Eq. (10):

$T_{\mathrm{MF}}\propto\sqrt{\rho\epsilon_{D}}\exp\left(-\frac{1}{\rho
U^{\prime}}\right),$ (17)

where $\rho$ is the density of states of the quasiparticles at the fermi
energy. We have also defined
$\frac{1}{U^{\prime}}=\frac{1}{U}-\frac{\epsilon_{D}\mathcal{A}}{4\pi v^{2}}$,
with the correction term ${\epsilon_{D}\mathcal{A}}/{4\pi v^{2}}\ll 1$. Note
that as $\mu\rightarrow\epsilon_{D}$, the BCS relationship for mean-field
temperature $T_{\mathrm{MF}}\propto\epsilon_{D}\exp\left(-1/\rho U\right)$ can
be recovered. We can also derive the conventional coherence length
$\xi_{\mathrm{con}}$, which is proportional to $v/T_{\mathrm{MF}}$, similar to
the BCS limit:

$\xi_{\mathrm{con}}\approx\frac{v}{T_{\mathrm{MF}}}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{\frac{7\zeta(3)}{32\pi^{2}}},$
(18)

where $\zeta$ is the Riemann zeta function and the
$\xi_{\mathrm{GL,con}}\propto\frac{v}{T_{\mathrm{MF}}}$. As for the quantum
metric contribution, it can be approximated by:

$\xi_{\mathrm{qm}}\approx\frac{v}{4\mu}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{\ln\left(\frac{\mu^{3}}{mT_{\mathrm{MF}}^{2}}\right)},$
(19)

where the factor $\sqrt{\ln({\mu^{3}}/{mT_{\mathrm{MF}}^{2}})}$ is in order of
unity. As such, the conventional contribution dominates, by a factor of
$\sim\beta_{\mathrm{MF}}\mu\gg 1$. Thus the quantum metric effect is
negligible and the system can be treated as conventional when away from the
band bottom.

We now consider the case where the chemical potential satisfies
$\beta_{\mathrm{MF}}(\mu-m)\ll 1$. The mean-field temperature is solved self-
consistently by Eq. (10):

$T_{\mathrm{MF}}\propto\epsilon_{D}\exp\left(-\frac{2}{\rho
U^{\prime}}\right).$ (20)

Note that there is an extra factor of two in the exponential term in
comparison with the BCS limit due to the integral range effectively halved.
This hints at superconductivity being more suppressed near the band bottom,
thus the range of doping where superconductivity can be observed, is expected
to be narrower when compared with the prediction from BCS. The conventional
coherence length in this regime is given by:

$\xi_{\mathrm{con}}\approx\frac{1}{4}\frac{v}{\sqrt{\mu
T_{\mathrm{MF}}}}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2},$
(21)

which gives rise to the Ginzburg-Landau coherence length
$\xi_{\mathrm{GL,con}}\approx\frac{1}{4}\frac{v}{\sqrt{\mu T_{\mathrm{MF}}}}$.
This relation deviates from a BCS relation $\xi_{\mathrm{BCS}}\approx
v_{F}/T_{\mathrm{MF}}$. As for the contribution of quantum metric, when the
chemical potential is at the band bottom (i.e. $\mu=m$), we have:

$\xi_{\mathrm{qm}}\approx\frac{v}{4m}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{\gamma^{2}-1+2\ln\left(\frac{\beta_{\mathrm{MF}}^{2}m^{2}}{\pi}\right)},$
(22)

where $\gamma$ is the Euler’s constant and we have the quantum metric
Ginzburg-Landau coherence length
$\xi_{\mathrm{GL,qm}}\approx\frac{v}{2m}\ln(\beta_{\mathrm{MF}}m)$. As such we
can conclude that the quantum metric contribution is smaller than, but can be
comparable with the conventional coherence length, by a factor of $\sim
2\sqrt{\ln(\beta_{\mathrm{MF}}m)/\beta_{\mathrm{MF}}m}<1$ at the conduction
band bottom.

Another intriguing scenario arises when the chemical potential is in close
proximity to the bottom of the band, but lies within the band gap. At zero
temperature, one would not typically anticipate a superconducting phase since
the ground state is an insulator. However, a SC phase can occur at the finite
temperature due to a metallic phase induced by temperature fluctuations. Of
course, the corresponding mean-field transition temperature gets suppressed.
As the transitional temperature is small such that
$\beta_{\mathrm{MF}}|\mu-m|\gg 1$, the conventional coherence length is
proportional to:

$\xi_{\mathrm{GL,con}}\propto\frac{v}{T_{\mathrm{MF}}}e^{-\beta_{\mathrm{MF}}|\mu-m|/2},$
(23)

which goes to zero when we approach the phase boundary given a fixed chemical
potential $\mu$. On the contrary, the quantum metric coherence length
diverges, and is proportional to:

$\xi_{\mathrm{GL,qm}}\propto\frac{v}{T_{\mathrm{MF}}}\sqrt{\frac{\ln(\beta_{\mathrm{MF}}m)}{\beta_{\mathrm{MF}}m}}.$
(24)

Obviously, $\xi_{\mathrm{GL,qm}}$ diverges when $T_{\mathrm{MF}}$ approaches
zero. As such, without considering the effect of the quantum metric, we expect
the Ginzburg-Landau coherence length to remain a small value as we approach
the phase boundary. On the contrary, if the effect of quantum metric is
included, we would instead predict a divergent Ginzburg-Landau coherence
length. As we will see, this case has potential implications towards the
interesting results in experimental measurements in RTG [14].

Figure 3:  Predictions from the theory by assuming SC1 originates from pairing
of quasiparticles in IVC phase: (a) mean-field transition temperature, (b)
upper critical field $H_{c2}$ at $T=50$mK, and (c) the Ginzburg-Landau
coherence lengths at $n\approx-1.82\times 10^{12}\mathrm{cm}^{-2}$. In (a),
the SC1 occurs in the blue region which is extracted from the experiment in
Ref. 14 with the transition temperature approximated by the boundary of the
blue region. The mean-field temperature captures the key features of the
experimental data, especially the disappearance of SC1 at $n\approx-1.8\times
10^{12}\mathrm{cm}^{-2}$. In (b), the green curve corresponds to $H_{c2}$ only
accounting for the conventional contribution, while the red curve additionally
accounts for quantum metric. The black dashed line indicates the chemical
potential exactly at the Dirac point of the quasiparticle band in IVC. The
blue region is extracted from [14]. Our theoretical predictions capture both
the maximal in $H_{c2}$ in close proximity with the band top, and the sharp
disappearance of superconductivity at $n\approx-1.8\times
10^{12}\mathrm{cm}^{-2}$. In (c), the green and blue curves correspond to the
quantum metric and conventional contributions respectively, and the red curve
is the overall Ginzburg-Landau coherence length $\xi_{\mathrm{GL}}$. The
conventional contribution $\xi_{\mathrm{GL,con}}$ plays a dominating role, and
the quantum metric $\xi_{\mathrm{GL,qm}}$ has a correctional effect in most
regimes. However, near the phase transition from IVC-SC to IVC state of which
the chemical potential lies between the band gap, the quantum metric
contribution $\xi_{\mathrm{GL,qm}}$ increases significantly, surpassing the
conventional contribution $\xi_{\mathrm{GL,con}}$ due to the vanishingly low
density of states of the IVC quasiparticles at the fermi energy. It leads to a
divergence of coherence length $\xi_{\mathrm{GL}}$ at the phase boundary, as
contrast to a finite value for the conventional contribution at
$n\approx-1.8\times 10^{12}\mathrm{cm}^{-2}$. Here the calculations are based
on the microscopic model for RTG in SM [38].

##  IV Superconductivity from pairing of IVC quasiparticle

###  IV.1 Prediction from toy model

As discussed in the end of section II.1, we proposed that superconductivity in
RTG originates from the pairing of quasiparticles in IVC state instead of
pairing between bare electrons. In particular, IVC-SC phase is only observed
in the vicinity of the phase boundary in between the IVC and the normal metal
state, at which the chemical potential cuts the band top of the lower energy
IVC band. As quasiparticles from the lower energy IVC band are responsible to
the formation of Cooper pairs, we can approximate the coherence length of IVC-
SC using Eq. (21). Given the band gap is $m\approx 34.5$meV, with mean-field
temperature $T_{\mathrm{MF}}=120$mK and graphene Fermi velocity $\hbar
v=6.6\times 10^{-1}$eVnm, the coherence length at base temperature $\approx
50$mK can be approximated as $126$nm, which is similar to the value
($150$-$250$nm) reported in the experiment [14]. Additionally, in the
experiment, it has been commented that the window of charge carrier density at
which superconductivity can be observed to be narrower than the prediction
from BCS theory. This can also be encapsulated by Eq. (20), which hints that
superconductivity near the band gap is more suppressed than the BCS theory,
resulting in a narrower range of doping at which superconductivity can be
observed. These pieces of evidence strongly support the notion that SC1
emerges at the band top, which can be achieved through the opening of a gap by
other order parameters, such as the IVC order parameters. This justify that
the IVC-SC state we have proposed is indeed a strong candidate for SC1
observed in RTG.

We also comment on possible implication when the chemical potential is within
the gap. Near the phase boundary, when the mean-field temperature approaches
zero, our toy model predicts the conventional contribution remains finite
while the quantum metric contribution diverges. This can be observed in our
numerical calculation on RTG in Fig. 3(c).

###  IV.2 Numerical calculation on IVC-SC state on the microscopic model

In the following, we will calculate numerically the property of the
superconducting state originating from the IVC state, with the assumption that
$|\bm{\Delta}_{\mathrm{IVC}}(\mathbf{q})|$ is independent of $\mathbf{q}$. We
use a six-band microscopic model whose details are introduced in the
supplementary material [37]. In general, one can solve for
$|\bm{\Delta}_{\mathrm{IVC}}|$ using self-consistency equation, similar to the
case of superconductivity. However, in our calculation, we will assume that at
the superconducting phase, $|\bm{\Delta}_{\mathrm{IVC}}|$ remains unchanged,
and superconductivity occurs when in close vicinity with the band top of
$\epsilon_{-}(\mathbf{q})$, thus Cooper pair is formed between
$\psi_{-,\mathbf{q}}$ states (6). Using the fact that at charge carrier
density $n\approx-1.8\times 10^{12}\mathrm{cm}^{-2}$ there is a sharp
disappearance of superconductivity, we can deduce that
$|\bm{\Delta}_{\mathrm{IVC}}|\approx 7$meV.

With the $|\bm{\Delta}_{\mathrm{IVC}}|$ determined, we can calculate the
effective attractive interaction strength, using Eq. (10), with the dispersion
set to $\epsilon_{-}(\mathbf{q})$. Numerically, the mean-field temperature is
the highest when the chemical potential is around $22\mu\mathrm{eV}$ below the
top of the band. Setting the maximal mean-field temperature as $125$mK, the
effective attractive interaction strength $U\approx 1.2$eV. Similar to the
normal metal state calculation, with the interaction strength determined,
using Eq. (10), we can determine the mean-field temperature, as illustrated in
Fig. 3(a). The mean-field temperature calculated from the IVC state, in
general, agrees with the experimental result, both reaching a maximum of at
$n\approx-1.85\times 10^{12}\mathrm{cm}^{-2}$, decrease down to $\approx 70$mK
at $n\approx-1.93\times 10^{12}\mathrm{cm}^{-2}$, and is no longer
superconducting for $n>-1.8\times 10^{12}\mathrm{cm}^{-2}$. We note that for
$n>-1.82\times 10^{12}\mathrm{cm}^{-2}$, where the chemical potential no
longer cut the IVC band that is responsible for the formation of Cooper pair,
superconductivity can still emerge from finite temperature effect. We also
note the mismatch in temperature could be due to ignorance of other bands,
which has been neglected when calculating mean-field temperature.

We can calculate the coherence length using Eqs. (14-16), thus checking if the
corresponding upper critical field $H_{c2}$ matches with the experimental
result. A plot of $H_{c2}$ at base temperature $50$mK, calculated from the IVC
state dispersion is shown in Fig. 3(b). We note that both the numerical
calculations and the experimental measurement of $H_{c2}$ have a maximal at
density $n\approx-1.82\times 10^{12}\mathrm{cm}^{-2}$. Our calculation can
also well capture the drop in $H_{c2}$ beyond the band top, with the little
discrepancy possibly originating from the mismatch in mean-field temperature
calculated numerically from the transition temperature near
$n\approx-1.82\times 10^{12}\mathrm{cm}^{-2}$. We also comment that the
quantum metric could have a correctional effect to $H_{c2}$, of approximately
$5\%$ at $n\approx-1.82\times 10^{12}\mathrm{cm}^{-2}$. In fact, within the
band gap, the variation in chemical potential is insignificant as illustrated
in Fig 2(d), and the drop in $H_{c2}$ can mostly be accounted to the decrease
in mean-field temperature. In particular, the experimental temperature
$T=50$mK approaches the mean-field transition temperature at
$n\approx-1.8\times 10^{12}\mathrm{cm}^{-2}$, resulting in $H_{c2}=0$.
However, the quantum metric effect can set in if one considers the zero-
temperature coherence length, that is, the Ginzburg-Landau coherence length
$\xi_{\mathrm{GL}}$, which is calculated in Fig. 3(c). In most regions of
charge carrier densities, the conventional contribution
$\xi_{\mathrm{GL,con}}$ (blue line in Fig. 3) dominates and will approach a
finite value. By comparison, the quantum metric contribution
$\xi_{\mathrm{GL,qm}}$ will increase and finally diverge when the charge
carrier density gets close to the phase boundary between the IVC-SC state and
IVC state. This originates from the vanishingly low charge carrier density
contributed by the relevant band at the Fermi energy when the chemical
potential lies between the band gap and is very close to the band bottom.
Therefore, we can make a prediction of a divergent coherence length $\xi$ thus
a vanishing upper critical field $H_{c2}$ when the lower temperature can be
realized for experimental measurements around $n=-1.8\times 10^{12}$cm12.

We should also remark that in our model the IVC-SC state has effectively
extended the IVC order parameter into the normal metal state, where IVC state
was originally not observed [14]. As such, contrary to the bare electron
picture where IVC and SC orders are competing orders, in the SC-IVC state,
where there exists effective attractive interaction, thus Cooper pair
formation directly in between the IVC states, the SC order combining with the
IVC order becomes more energy favorable than the normal metal state.

##  V Conclusion

We begin with an introduction to RTG, where superconductivity is observed
within a narrow window of the low charge carrier densities. Via the numerical
calculation of the mean-field temperature and upper critical field $H_{c2}$,
we have deduced that the superconductivity of RTG is unlikely to arise from
the normal metal state, but instead from the neighboring IVC state. In
particular, we realized that superconductivity IVC-SC occurs only when in
close vicinity with the band top of the lower energy IVC band, namely near the
phase transition from the IVC state to the normal metal state.

We then constructed a toy model of a graphene-like system, with Dirac points
of Fermi velocity $v$ at the $K$ and $K^{\prime}$ valleys, and open up a mass
gap $m$ at the Dirac points. From a field-theory approach, we determined the
mean-field temperature of s-wave superconductivity $T_{\mathrm{MF}}$, and
constructed the Ginzburg-Landau theory, capturing both the conventional and
quantum metric contributions to the superconducting coherence length. In
particular, we notice when the chemical potential is near the mass gap, the
mean-field temperature is more suppressed, and the conventional coherence
length is no longer proportional to $v/T_{\mathrm{MF}}$, but instead
$v/\sqrt{mT_{\mathrm{MF}}}$. We also note that the quantum metric could have a
comparable contribution depending on the mean-field temperature, in comparison
with the mass gap.

Having built up the framework for calculating various physical quantities
using field theory, we perform numerical calculation on $T_{\mathrm{MF}}$ and
$H_{c2}$ of the IVC-SC state, which matches well with the experimental
measurement on SC1 from [14], act as concrete evidence supporting that SC1
arises from the pairings of IVC quasiparticles. We should remark that the IVC-
SC state extends the IVC order into the normal metal state, as contrary to the
bare electron picture where IVC and SC orders are competing orders. Before
closing, we note that the quantum metric only has a correction effect, on the
overall coherence length, for most ranges of doping of RTG. However, near the
phase transition from SC-IVC to IVC state, the quantum metric contribution
increases rapidly and diverges at the phase boundary. As such, the quantum
metric could play an important role in the transition from the IVC-SC state
into the IVC state. This can be verified experimentally by extracting the
zero-temperature Ginzburg-Landau coherence length in proximity of the phase
boundary using the measurement of $H_{c2}$ near the transition temperature.

## Acknowledgments

K.T.L. acknowledges the support of the Ministry of Science and Technology,
China, and the Hong Kong Research Grants Council through Grants No.
2020YFA0309600, No. RFS2021-6S03, No. C6025-19G, No. AoE/P-701/20, No.
16310520, No. 16310219 and No. 16307622.

## References

  * Cao  _et al._ [2018a] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018a), arXiv:1802.00553 [cond-mat.mes-hall] .
  * Serlin  _et al._ [2020] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020), arXiv:1907.00261 [cond-mat.str-el] .
  * Sharpe  _et al._ [2019] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science 365, 605 (2019), arXiv:1901.03520 [cond-mat.mes-hall] .
  * Arora  _et al._ [2020] H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi, H. Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Superconductivity in metallic twisted bilayer graphene stabilized by WSe2, Nature (London) 583, 379 (2020).
  * Cao  _et al._ [2018b] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018b), arXiv:1803.02342 [cond-mat.mes-hall] .
  * Lu  _et al._ [2019] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold, A. H. MacDonald, and D. K. Efetov, Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature (London) 574, 653 (2019), arXiv:1903.06513 [cond-mat.str-el] .
  * Yankowitz  _et al._ [2019a] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363, 1059 (2019a), arXiv:1808.07865 [cond-mat.mes-hall] .
  * Yankowitz  _et al._ [2019b] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363, 1059 (2019b), arXiv:1808.07865 [cond-mat.mes-hall] .
  * Kerelsky  _et al._ [2019] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean, A. Rubio, and A. N. Pasupathy, Maximized electron interactions at the magic angle in twisted bilayer graphene, Nature (London) 572, 95 (2019).
  * Jiang  _et al._ [2019] Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule, J. Mao, and E. Y. Andrei, Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene, Nature (London) 573, 91 (2019), arXiv:1904.10153 [cond-mat.mes-hall] .
  * Xie  _et al._ [2019] Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene, Nature (London) 572, 101 (2019), arXiv:1906.09274 [cond-mat.mes-hall] .
  * Lin  _et al._ [2018] X. Lin, D. Liu, and D. Tománek, Shear instability in twisted bilayer graphene, Phys. Rev. B 98, 195432 (2018).
  * Lin  _et al._ [2021] X. Lin, K. Su, and J. Ni, Misalignment instability in magic-angle twisted bilayer graphene on hexagonal boron nitride, 2D Materials 8, 025025 (2021), arXiv:2011.01541 [cond-mat.mes-hall] .
  * Zhou  _et al._ [2021a] H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F. Young, Superconductivity in rhombohedral trilayer graphene, Nature (London) 598, 434 (2021a), arXiv:2106.07640 [cond-mat.mes-hall] .
  * Zhou  _et al._ [2021b] H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M. Serbyn, and A. F. Young, Half- and quarter-metals in rhombohedral trilayer graphene, Nature (London) 598, 429 (2021b), arXiv:2104.00653 [cond-mat.mes-hall] .
  * Arp  _et al._ [2023] T. Arp, O. Sheekey, H. Zhou, C. L. Tschirhart, C. L. Patterson, H. M. Yoo, L. Holleis, E. Redekop, G. Babikyan, T. Xie, J. Xiao, Y. Vituri, T. Holder, T. Taniguchi, K. Watanabe, M. E. Huber, E. Berg, and A. F. Young, Intervalley coherence and intrinsic spin-orbit coupling in rhombohedral trilayer graphene, arXiv e-prints , arXiv:2310.03781 (2023), arXiv:2310.03781 [cond-mat.mes-hall] .
  * Zhang  _et al._ [2010] F. Zhang, B. Sahu, H. Min, and A. H. MacDonald, Band structure of $abc$-stacked graphene trilayers, Phys. Rev. B 82, 035409 (2010).
  * Koshino and McCann [2009] M. Koshino and E. McCann, Trigonal warping and Berry’s phase N$\pi$ in ABC-stacked multilayer graphene, Phys. Rev. B 80, 165409 (2009), arXiv:0906.4634 [cond-mat.mes-hall] .
  * Huang  _et al._ [2023] C. Huang, T. M. R. Wolf, W. Qin, N. Wei, I. V. Blinov, and A. H. MacDonald, Spin and orbital metallic magnetism in rhombohedral trilayer graphene, Phys. Rev. B 107, L121405 (2023), arXiv:2203.12723 [cond-mat.mes-hall] .
  * Zhumagulov  _et al._ [2023] Y. Zhumagulov, D. Kochan, and J. Fabian, Emergent correlated phases in rhombohedral trilayer graphene induced by proximity spin-orbit and exchange coupling, arXiv e-prints , arXiv:2305.14277 (2023), arXiv:2305.14277 [cond-mat.str-el] .
  * Koh  _et al._ [2024] J. M. Koh, J. Alicea, and E. Lantagne-Hurtubise, Correlated phases in spin-orbit-coupled rhombohedral trilayer graphene, Phys. Rev. B 109, 035113 (2024).
  * Wang  _et al._ [2023] T. Wang, M. Vila, M. P. Zaletel, and S. Chatterjee, Electrical control of spin and valley in spin-orbit coupled graphene multilayers, arXiv e-prints , arXiv:2303.04855 (2023), arXiv:2303.04855 [cond-mat.str-el] .
  * Lui  _et al._ [2011] C. H. Lui, Z. Li, K. F. Mak, E. Cappelluti, and T. F. Heinz, Observation of an electrically tunable band gap in trilayer graphene, Nature Physics 7, 944 (2011), arXiv:1105.4658 [cond-mat.mes-hall] .
  * Chou  _et al._ [2022] Y.-Z. Chou, F. Wu, J. D. Sau, and S. Das Sarma, Acoustic-phonon-mediated superconductivity in moiréless graphene multilayers, Phys. Rev. B 106, 024507 (2022).
  * Viñas Boström  _et al._ [2023] E. Viñas Boström, A. Fischer, J. B. Hauck, J. Zhang, D. M. Kennes, and A. Rubio, Phonon-mediated unconventional $s$\- and $f$-wave pairing superconductivity in rhombohedral stacked multilayer graphene, arXiv e-prints , arXiv:2311.02494 (2023), arXiv:2311.02494 [cond-mat.supr-con] .
  * You and Vishwanath [2022] Y.-Z. You and A. Vishwanath, Kohn-luttinger superconductivity and intervalley coherence in rhombohedral trilayer graphene, Phys. Rev. B 105, 134524 (2022).
  * Ghazaryan  _et al._ [2021] A. Ghazaryan, T. Holder, M. Serbyn, and E. Berg, Unconventional Superconductivity in Systems with Annular Fermi Surfaces: Application to Rhombohedral Trilayer Graphene, Phys. Rev. Lett. 127, 247001 (2021), arXiv:2109.00011 [cond-mat.supr-con] .
  * Cea  _et al._ [2022] T. Cea, P. A. Pantaleón, V. T. Phong, and F. Guinea, Superconductivity from repulsive interactions in rhombohedral trilayer graphene: A Kohn-Luttinger-like mechanism, Phys. Rev. B 105, 075432 (2022), arXiv:2109.04345 [cond-mat.mes-hall] .
  * Jimeno-Pozo  _et al._ [2023] A. Jimeno-Pozo, H. Sainz-Cruz, T. Cea, P. A. Pantaleón, and F. Guinea, Superconductivity from electronic interactions and spin-orbit enhancement in bilayer and trilayer graphene, Phys. Rev. B 107, L161106 (2023), arXiv:2210.02915 [cond-mat.mes-hall] .
  * Lu  _et al._ [2022] D.-C. Lu, T. Wang, S. Chatterjee, and Y.-Z. You, Correlated metals and unconventional superconductivity in rhombohedral trilayer graphene: A renormalization group analysis, Phys. Rev. B 106, 155115 (2022), arXiv:2206.01213 [cond-mat.str-el] .
  * Qin  _et al._ [2023] W. Qin, C. Huang, T. Wolf, N. Wei, I. Blinov, and A. H. MacDonald, Functional Renormalization Group Study of Superconductivity in Rhombohedral Trilayer Graphene, Phys. Rev. Lett. 130, 146001 (2023), arXiv:2203.09083 [cond-mat.supr-con] .
  * Dai  _et al._ [2023] H. Dai, R. Ma, X. Zhang, T. Guo, and T. Ma, Quantum Monte Carlo study of superconductivity in rhombohedral trilayer graphene under an electric field, Phys. Rev. B 107, 245106 (2023), arXiv:2204.06222 [cond-mat.str-el] .
  * Dong and Levitov [2021] Z. Dong and L. Levitov, Superconductivity in the vicinity of an isospin-polarized state in a cubic Dirac band, arXiv e-prints , arXiv:2109.01133 (2021), arXiv:2109.01133 [cond-mat.supr-con] .
  * Chatterjee  _et al._ [2022] S. Chatterjee, T. Wang, E. Berg, and M. P. Zaletel, Inter-valley coherent order and isospin fluctuation mediated superconductivity in rhombohedral trilayer graphene, Nature Communications 13, 6013 (2022), arXiv:2109.00002 [cond-mat.supr-con] .
  * Szabó and Roy [2022] A. L. Szabó and B. Roy, Metals, fractional metals, and superconductivity in rhombohedral trilayer graphene, Phys. Rev. B 105, L081407 (2022), arXiv:2109.04466 [cond-mat.str-el] .
  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Superconductivity, Physical Review 108, 1175 (1957).
  * [37] See Supplemental Material.
  * Yankowitz  _et al._ [2014] M. Yankowitz, J. I. J. Wang, A. G. Birdwell, Y.-A. Chen, K. Watanabe, T. Taniguchi, P. Jacquod, P. San-Jose, P. Jarillo-Herrero, and B. J. Leroy, Electric field control of soliton motion and stacking in trilayer graphene, Nature Materials 13, 786 (2014), arXiv:1401.7663 [cond-mat.mtrl-sci] .
  * Chen and Law [2024] S. A. Chen and K. T. Law, Ginzburg-landau theory of flat-band superconductors with quantum metric, Phys. Rev. Lett. 132, 026002 (2024).
  * Hu  _et al._ [2023] J.-X. Hu, S. A. Chen, and K. T. Law, Anomalous Coherence Length in Superconductors with Quantum Metric, arXiv e-prints , arXiv:2308.05686 (2023), arXiv:2308.05686 [cond-mat.supr-con] .
  * Han  _et al._ [2023] X. Han, Q. Liu, Y. Wang, R. Niu, Z. Qu, Z. Wang, Z. Li, C. Han, K. Watanabe, T. Taniguchi, Z. Song, J. Mao, Z. V. Han, Z. Gan, and J. Lu, Chemical Potential Characterization of Symmetry-Breaking Phases in a Rhombohedral Trilayer Graphene, Nano Letters 23, 6875 (2023).
  * Winterer  _et al._ [2023] F. Winterer, F. R. Geisenhof, N. Fernandez, A. M. Seiler, F. Zhang, and R. T. Weitz, Ferroelectric and anomalous quantum Hall states in bare rhombohedral trilayer graphene, arXiv e-prints , arXiv:2305.04950 (2023), arXiv:2305.04950 [cond-mat.mes-hall] .
  * Bao  _et al._ [2011] W. Bao, L. Jing, J. Velasco, Y. Lee, G. Liu, D. Tran, B. Standley, M. Aykol, S. B. Cronin, D. Smirnov, M. Koshino, E. McCann, M. Bockrath, and C. N. Lau, Stacking-dependent band gap and quantum transport in trilayer graphene, Nature Physics 7, 948 (2011), arXiv:1103.6088 [cond-mat.mes-hall] .

Appendix for “Origin of Superconductivity in Rhombohedral Trilayer Graphene:
Quasiparticle Pairing within the Inter-Valley Coherent Phase”

##  Appendix A Details of 6-band Hamiltonian for RTG

Following the definition of the sub-lattice A and B in the single layer of
graphene, we can denote the structure of RTG by stating which pairs of sites
coincide from the top view:

$B_{1}\leftrightarrow A_{2},\ B_{2}\leftrightarrow A_{3},\
B_{3}\leftrightarrow A_{1}$ (25)

As such we can classify our hopping [17] ($S_{i}\in\\{A_{i},B_{i}\\}$):

$\displaystyle\gamma_{0}:$ $\displaystyle A_{i}\leftrightarrow B_{i},\quad
i=1,2,3$ $\displaystyle\gamma_{1}:$ $\displaystyle B_{i}\leftrightarrow
A_{i+1},\quad i=1,2$ $\displaystyle\gamma_{2}:$ $\displaystyle
A_{1}\leftrightarrow B_{3}$ $\displaystyle\gamma_{3}:$ $\displaystyle
A_{i}\leftrightarrow B_{i+1},\quad i=1,2$ $\displaystyle\gamma_{4}:$
$\displaystyle S_{i}\leftrightarrow S_{i+1},\quad i=1,2$
$\displaystyle\gamma_{5}:$ $\displaystyle B_{1}\leftrightarrow A_{3}$
$\displaystyle\gamma_{6}:$ $\displaystyle S_{1}\leftrightarrow S_{3}$

Due to direct interlayer hopping $\gamma_{1}$ between $B_{i}$ and $A_{i+1}$,
sites not involved, namely $A_{1}$ and $B_{3}$, would have lower energy. We
denote the energy difference of $A_{1}(B_{3})$ from $B_{1}(A_{3})$ as
$\delta<0$. Note that these hopping are listed in decreasing magnitude, with
$\gamma_{3}$ and $\gamma_{4}$ having similar order. $\gamma_{5}$ and
$\gamma_{6}$ having similar order as well, but should be much weaker in
comparison.

We should also note the massless Dirac fermion contribution near the valleys,
for a single layer graphene takes the form:

$H=v_{0}\begin{pmatrix}0&\pi^{\dagger}\\\ \pi&0\end{pmatrix},$ (26)

where we have defined $\pi=\tau p_{x}+ip_{y}$, with $\tau=+(-)$ correspond to
the $K(K^{\prime})$ valley. With the single layer picture and list of possible
hopping in mind, we can construct our basis as
$(A_{1},B_{3},B_{1},A_{2},B_{2},A_{3})$ for convenience:

$\mathcal{H}=\begin{pmatrix}u_{1}+\delta&\gamma_{2}/2&v_{4}\pi^{\dagger}&v_{4}\pi^{\dagger}&v_{3}\pi&v_{6}\pi\\\
\gamma_{2}/2&u_{3}+\delta&v_{6}\pi^{\dagger}&v_{3}\pi^{\dagger}&v_{4}\pi&v_{0}\pi\\\
v_{0}\pi&v_{6}\pi&u_{1}&\gamma_{1}&v_{4}\pi^{\dagger}&v_{5}\pi^{\dagger}\\\
v_{4}\pi&v_{3}\pi&\gamma_{1}&u_{2}&v_{0}\pi^{\dagger}&v_{4}\pi^{\dagger}\\\
v_{3}\pi^{\dagger}&v_{4}\pi^{\dagger}&v_{4}\pi&v_{0}\pi&u_{2}&\gamma_{1}\\\
v_{6}\pi^{\dagger}&v_{0}\pi^{\dagger}&v_{5}\pi&v_{4}\pi&\gamma_{1}&u_{3}\end{pmatrix},$
(27)

where $v_{i}=\sqrt{3}a\gamma_{i}/2\hbar$, as motivated by single layer
graphene. ($a=0.246\mathrm{nm}$)

Note that for convenience, we will take $v_{i}=\sqrt{3}a\gamma_{i}/2$. And
define $\mathbf{q}$ as the wavevector instead of using momentum $\mathbf{p}$
to compensate for the factor of $\hbar$. The low-energy effective Hamiltonian
is:

$\displaystyle H_{\mathrm{eff}}$
$\displaystyle=H_{\mathrm{ch}}+H_{\mathrm{s}}+H_{\mathrm{tr}}+H_{\mathrm{gap}}+H_{\mathrm{s}}^{\prime},$
(28) $\displaystyle H_{\mathrm{ch}}$
$\displaystyle=\frac{(v_{0}q)^{3}}{\gamma_{1}^{2}}[\cos(3\varphi_{\mathbf{q}})\sigma_{x}+\sin(3\varphi_{\mathbf{q}})\sigma_{y}],$
(29) $\displaystyle H_{\mathrm{s}}$
$\displaystyle=\left(\delta-\frac{2v_{0}v_{4}q^{2}}{\gamma_{1}}\right)\sigma_{0},$
(30) $\displaystyle H_{\mathrm{tr}}$
$\displaystyle=\left(\frac{\gamma_{2}}{2}-\frac{2v_{0}v_{3}q^{2}}{\gamma_{1}}\right)\sigma_{x},$
(31) $\displaystyle H_{\mathrm{gap}}$
$\displaystyle=u_{d}\left[1-\left(\frac{v_{0}q}{\gamma_{1}}\right)^{2}\right]\sigma_{z},$
(32) $\displaystyle H_{\mathrm{s}}^{\prime}$
$\displaystyle=\frac{u_{a}}{3}\left[1-3\left(\frac{v_{0}q}{\gamma_{1}}\right)^{2}\right]\sigma_{0},$
(33)

where we have defined $\tan\varphi_{\mathbf{q}}=q_{y}/q_{x}$,
$u_{d}=(u_{1}-u_{3})/2$ and $u_{a}=(u_{1}+u_{3})/2-u_{2}$. In our scope of
discussion, the contribution from $\gamma_{5}$ and $\gamma_{6}$ are assumed to
be small thus neglected.

We will use parameters fitted in [15], with the parameters listed in Table 1.
The parameter $u_{d}$, which is the energy difference between each layer, can
be induced via an external displacement field [20]:
$u_{d}=\frac{ez}{\epsilon_{r}}D$, where $z=0.33$nm is the interlayer distance,
and $\epsilon_{r}=4.4$ is the dielectric constant hexagonal boron nitride. In
our study, we will focus on superconductivity of the valence band when
$D=0.46$V/nm, which corresponds to $u_{d}=34.5$meV.

Table 1: Tight-binding Parameters for Rhombohedral Trilayer Graphene. Retrieved from [15]. Hopping parameters | Values (eV)
---|---
$\delta$ | -0.00105
$\gamma_{0}$ | 3.1
$\gamma_{1}$ | 0.38
$\gamma_{2}$ | -0.015
$\gamma_{3}$ | -0.29
$\gamma_{4}$ | -0.141
$u_{a}$ | -0.0023

##  Appendix B Derivation of free energy

Consider the case of two massive Dirac valleys, centering at $\pm\mathbf{K}$,
corresponding to valley index of $\tau=\pm 1$ respectively. The Hamiltonian of
each valley is given by:

$\displaystyle h_{\tau}(\mathbf{q})$
$\displaystyle=v\mathbf{q}\cdot(\tau\sigma_{x}\hat{\mathbf{e}}_{x}+\sigma_{y}\hat{\mathbf{e}}_{y})+m\sigma_{z},$
(34)

where $v$ is the fermi velocity before gap-opening, $\tau=+/-$ correspond to
$K$ and $K^{\prime}$ valley respectively, $m$ is the gap size, and
$\mathbf{q}$ is the momentum difference with respect to the center of the
valley, and $\sigma_{i}$ Pauli matrices that act on the sublattice basis. The
corresponding Bloch states in sublattice basis are given by:

$\displaystyle u_{c,\tau}(\mathbf{q})$
$\displaystyle=\frac{1}{\sqrt{2\epsilon_{0}(\mathbf{q})}}\begin{pmatrix}\tau
e^{-i\tau\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})+m}\\\
e^{i\tau\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})-m}\end{pmatrix}$ $\displaystyle
E$ $\displaystyle=\epsilon_{0}(\mathbf{q}),$ (35) $\displaystyle
u_{v,\tau}(\mathbf{q})$
$\displaystyle=\frac{1}{\sqrt{2\epsilon_{0}(\mathbf{q})}}\begin{pmatrix}-\tau
e^{-i\tau\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})-m}\\\
e^{i\tau\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})+m}\end{pmatrix}$ $\displaystyle
E$ $\displaystyle=-\epsilon_{0}(\mathbf{q}),$ (36)

where the dispersion is given by
$\epsilon_{0}(\mathbf{q})=\sqrt{v^{2}q^{2}+m^{2}}$, index $c$ ($v$) denotes
the conduction (valence) band, and have rewritten our momentum in polar
coordinate as $qe^{i\phi}=q_{x}+iq_{y}$. To induce superconductivity in the
system, we introduce an effective attractive interaction between electrons:

$H_{\mathrm{int}}=-U\int
d^{2}\mathbf{r}a_{+}^{\dagger}(\mathbf{r})a_{-}^{\dagger}(\mathbf{r})a_{-}(\mathbf{r})a_{+}(\mathbf{r}).$
(37)

Where $a_{\tau}(\mathbf{r})$ is the fermionic annihilation operator of states
in valley of momentum $\tau\mathbf{K}$ at position $\mathbf{r}$. As such we
can write down a partition function involving only fermionic fields $a$:

$\displaystyle Z$ $\displaystyle=\int
D[a,\bar{a}]e^{-\int_{0}^{\beta}d\tau\int
d\mathbf{r}\mathcal{L}_{a}[a,\bar{a}]},$ (38)
$\displaystyle\mathcal{L}_{a}[a,\bar{a}]$
$\displaystyle=\mathcal{L}_{\mathrm{Dirac}}[a,\bar{a}]+(\partial_{\tau}-\mu)[\bar{a}_{-}(\mathbf{r})a_{-}(\mathbf{r})+\bar{a}_{+}(\mathbf{r})a_{+}(\mathbf{r})]$
$\displaystyle-U\bar{a}_{-}(\mathbf{r})\bar{a}_{+}(\mathbf{r})a_{+}(\mathbf{r})a_{-}(\mathbf{r}),$
(39)

where $\bar{a}$ and $a$ are Grassmann field. Using Hubbard-Stratonovich (HS)
transformation, we can introduce a Bosonic field:

$1=\int D[\Delta,\bar{\Delta}]e^{-U\int_{0}^{\beta}d\tau\int
d\mathbf{r}[\bar{\Delta}(\mathbf{r})-\bar{a}_{-}(\mathbf{r})\bar{a}_{+}(\mathbf{r})][\Delta(\mathbf{r})-a_{+}(\mathbf{r})a_{-}(\mathbf{r})]}.$
(40)

As such we can rewrite our partition function as:

$\displaystyle Z$ $\displaystyle=\int
D[\Delta,\bar{\Delta}]e^{-U\int_{0}^{\beta}d\tau\int
d\mathbf{r}|\bm{\Delta}(\mathbf{r})|^{2}}\mathcal{Z}[\Delta,\bar{\Delta}],$
(41) $\displaystyle\mathcal{Z}[\Delta,\bar{\Delta}]$ $\displaystyle=\int
D[a,\bar{a}]e^{-\int_{0}^{\beta}d\tau\int
d\mathbf{r}\mathcal{L}[\Delta,\bar{\Delta},a,\bar{a}]},$ (42)
$\displaystyle\mathcal{L}[\Delta,\bar{\Delta},a,\bar{a}]$
$\displaystyle=\mathcal{L}_{\mathrm{Dirac}}[a,\bar{a}]+(\partial_{\tau}-\mu)[\bar{a}_{-}(\mathbf{r})a_{-}(\mathbf{r})+\bar{a}_{+}(\mathbf{r})a_{+}(\mathbf{r})]$
$\displaystyle\
-U[\bar{\Delta}(\mathbf{r})a_{+}(\mathbf{r})a_{-}(\mathbf{r})+\mathrm{h.c.}].$
(43)

We can introduce the projection of an annihilation operator onto the relevant
band as:

$a_{\tau}(\mathbf{r})\rightarrow\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}e^{i\mathbf{q}\cdot\mathbf{r}}u_{c,\tau}^{*}(\mathbf{q})c_{\tau}(\mathbf{q}),$
(44)

where $c_{\tau}(\mathbf{q})$ is the the band electron annihilation operator of
momentum $\mathbf{q}$, and $\mathcal{A}$ is the unit cell area of the lattice.
As such we can project our Lagrangian density to
$\mathcal{L}[\Delta,\bar{\Delta},c,\bar{c}]$:

$\displaystyle\mathcal{L}[\Delta,\bar{\Delta},a,\bar{a}]\rightarrow$
$\displaystyle\mathcal{L}[\Delta,\bar{\Delta},c,\bar{c}]$ $\displaystyle=$
$\displaystyle[\epsilon_{0}(\mathbf{q})+\partial_{\tau}-\mu][\bar{c}_{\uparrow,-\mathbf{K}}(\mathbf{q})c_{\uparrow,-\mathbf{K}}(\mathbf{q})+\bar{c}_{\downarrow,+\mathbf{K}}(\mathbf{q})c_{\downarrow,+\mathbf{K}}(\mathbf{q})]$
$\displaystyle-U\mathcal{A}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\left[\bar{\Delta}(\mathbf{k})\Gamma(\mathbf{q},\mathbf{k})c_{+}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)c_{-}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\mathrm{h.c.}\right],$
(45)

where we have redefined $c_{\tau}(\mathbf{q})$ as the corresponding Grassmann
field. Note that the form factor is defined by
$\Gamma(\mathbf{q},\mathbf{k})=u_{c,+}^{T}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)u_{c,-}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)$.
We have also assume the bosonic field $\Delta(\mathbf{q)}$ transforms
trivially as a spin 0 boson
$\Delta(\mathbf{r)}\rightarrow\mathcal{A}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}e^{i\mathbf{k}\cdot\mathbf{r}}\Delta(\mathbf{k)}$.
We can rewrite our Lagrangian density into a matrix form using the Nambu
spinors:

$\mathcal{L}[\Delta,\bar{\Delta},c,\bar{c}]=\mathcal{A}\int\frac{d^{2}\mathbf{q}^{\prime}}{(2\pi)^{2}}\begin{pmatrix}\bar{c}_{-}(\mathbf{q^{\prime})}&c_{+}(\mathbf{-q^{\prime})}\end{pmatrix}G_{\mathbf{q}^{\prime},\mathbf{q}}\begin{pmatrix}c_{-}(\mathbf{q})\\\
\bar{c}_{+}(\mathbf{-q})\end{pmatrix}.$ (46)

Defining the free energy of the system as $Z=\int
D[\Delta,\bar{\Delta}]e^{-\beta F[\Delta,\bar{\Delta}]}$, or equivalently as:

$\displaystyle F[\Delta,\bar{\Delta}]$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}U|\bm{\Delta}(\mathbf{k})|^{2}-T\ln\int
D[c,\bar{c}]\
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathcal{L}[\Delta,\bar{\Delta},c,\bar{c}]}$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}U|{\Delta}(\mathbf{k})|^{2}-T\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\int\frac{d^{2}\mathbf{q}^{\prime}}{(2\pi)^{2}}\ln\det
G_{\mathbf{q},\mathbf{q}^{\prime}}.$ (47)

To calculate the free energy, we expand the Bosonic field $\Delta(\mathbf{k})$
around the extremum of the free energy as:

$\Delta(\mathbf{k})=\Delta_{0}\delta(\mathbf{k})+\delta\Delta(\mathbf{k}),$
(48)

where $\Delta_{0}$ is the mean-field solution, real when properly gauged, and
$\delta\Delta(\mathbf{k})$ is the fluctuation. We can thus decompose our
Lagrangian density $\mathcal{L}[\Delta,\bar{\Delta},c,\bar{c}]$ as:

$\displaystyle\mathcal{L}[\Delta,\bar{\Delta},c,\bar{c}]$
$\displaystyle=\mathcal{L}_{0}+\delta\mathcal{L}$ (49)
$\displaystyle\mathcal{L}_{0}$
$\displaystyle=[\epsilon_{0}(\mathbf{q})+\partial_{\tau}-\mu][\bar{c}_{-}(\mathbf{q})c_{-}(\mathbf{q})+\bar{c}_{+}(\mathbf{q})c_{+}(\mathbf{q})]-U[\Gamma(\mathbf{q},0)\Delta_{0}c_{+}(\mathbf{q})c_{-}(-\mathbf{q})+\mathrm{h.c.}]$
(50) $\displaystyle\delta\mathcal{L}$
$\displaystyle=-U\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\left[\Gamma(\mathbf{q},\mathbf{k})\delta\bar{\Delta}(\mathbf{k})c_{+}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)c_{-}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\mathrm{h.c.}\right].$
(51)

Correspondingly, we can decompose the free energy $F[\Delta,\bar{\Delta}]$ as:

$F[\Delta,\bar{\Delta}]=F_{0}+\delta F,$ (52)

where $F_{0}$ would be given by:

$\displaystyle F_{0}$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{U|\bm{\Delta}_{0}|^{2}-T\sum_{n}\ln\left[\beta^{2}\det\begin{pmatrix}\epsilon_{0}(\mathbf{q})-i\omega_{n}-\mu&-U\Gamma(\mathbf{q},0)\Delta_{0}\\\
-(U\Gamma(\mathbf{q},0)\Delta_{0})^{*}&-\epsilon_{0}(\mathbf{q})-i\omega_{n}+\mu\end{pmatrix}\right]\right\\}$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{U|\Delta_{0}|^{2}-T\sum_{n}\ln\beta^{2}[-\omega_{n}^{2}-(\epsilon_{0}(\mathbf{q})-\mu)^{2}-|U\Gamma(\mathbf{q},0)\Delta_{0}|^{2}]\right\\}$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{U|\Delta_{0}|^{2}-T\sum_{n}\left\\{\ln\beta[-i\omega_{n}+\epsilon(\mathbf{q})]+\ln\beta[-i\omega_{n}-\epsilon(\mathbf{q})]\right\\}\right\\}$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{U|\Delta_{0}|^{2}-T\left[\ln\left(1+e^{-\beta\epsilon(\mathbf{q})}\right)+\ln\left(1+e^{\beta\epsilon(\mathbf{q})}\right)\right]\right\\}$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left[U|\Delta_{0}|^{2}-\epsilon(\mathbf{q})-2T\ln\left(1+e^{-\beta\epsilon(\mathbf{q})}\right)\right].$
(53)

Here
$\epsilon(\mathbf{q})=\sqrt{(\epsilon_{0}(\mathbf{q})-\mu)^{2}+|U\Gamma(\mathbf{q},0)\Delta_{0}|^{2}}$
and $\omega_{n}=2n\pi T$ is the Matsubara frequencies. Note that $F_{0}$
recovers the form of the grand potential. We should also mention the self-
consistency condition for $\Delta_{0}$, namely that $\frac{\partial
F_{0}}{\partial\Delta_{0}}=0$ gives us:

$\displaystyle 0$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left[2U\Delta_{0}-\frac{\partial\epsilon(\mathbf{q})}{\partial\Delta_{0}}\left(1-\frac{2e^{-\beta\epsilon(\mathbf{q})}}{1+e^{-\beta\epsilon(\mathbf{q})}}\right)\right]$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left[2U\Delta_{0}-\frac{U^{2}|\Gamma(\mathbf{q},0)|^{2}\Delta_{0}}{\epsilon(\mathbf{q})}\tanh\frac{\beta\epsilon(\mathbf{q})}{2}\right]$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left[1-\frac{U|\Gamma(\mathbf{q},0)|^{2}}{2\epsilon(\mathbf{q})}\tanh\frac{\beta\epsilon(\mathbf{q})}{2}\right]$
$\displaystyle 1$
$\displaystyle=\mathcal{\mathcal{A}}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{U|\Gamma(\mathbf{q},0)|^{2}}{2\epsilon(\mathbf{q})}\tanh\frac{\beta\epsilon(\mathbf{q})}{2}.$
(54)

The higher order contribution $\delta F$ is given by:

$\displaystyle\delta F$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\left[U|\delta\Delta|^{2}-T\ln\frac{\int
D[c,\bar{c}]\
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}(\mathcal{L}_{0}+\delta\mathcal{L})}}{\int
D[c,\bar{c}]\
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathcal{L}_{0}}}\right]$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\left[U|\delta\Delta|^{2}-T\left\langle
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\delta\mathcal{L}}-1\right\rangle\right],$
(55)

where we have defined the thermal average $\langle f\rangle=\left(\int
D[c,\bar{c}]\
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathcal{L}_{0}}\right)^{-1}\int
D[c,\bar{c}]\
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathcal{L}_{0}}f$.
To facilitate the calculation of Gor’kov’s Green function thus $\delta F$, we
should introduce generating functional:

$\mathcal{Z}[\eta,\bar{\eta}]=\int D[\psi,\bar{\psi}]\
e^{-\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left[\bar{\psi}(\mathbf{q})G_{0}(\mathbf{q})\psi(\mathbf{q})-\bar{\eta}(\mathbf{q})\psi(\mathbf{q})-\bar{\psi}(\mathbf{q})\eta(\mathbf{q})\right]},$
(56)

where
$\bar{\psi}(\mathbf{q)}=\begin{pmatrix}\bar{c}_{-}(\mathbf{q})&c_{+}(\mathbf{-q})\end{pmatrix}$
is the Nambu spinor, and
$\mathcal{L}_{0}=\bar{\psi}(\mathbf{q})G_{0}(\mathbf{q})\psi(\mathbf{q})$. By
performing shifting on the field
$\psi(\mathbf{q})\rightarrow\psi(\mathbf{q})+G_{0}^{-1}(\mathbf{q})\eta(\mathbf{q})$
and
$\bar{\psi}(\mathbf{q})\rightarrow\bar{\psi}(\mathbf{q})+\bar{\eta}(\mathbf{q})G_{0}^{-1}(\mathbf{q})$,
we arrive at:

$\mathcal{Z}[\eta,\bar{\eta}]=\mathcal{Z}[0,0]\
e^{\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\bar{\eta}(\mathbf{q})G_{0}^{-1}(\mathbf{q})\eta(\mathbf{q})}.$
(57)

As such thermal average
$\langle\psi_{\mathbf{q}}\bar{\psi}_{\mathbf{q}^{\prime}}\rangle$ takes the
form:

$\displaystyle\langle\psi_{\mathbf{q}}\bar{\psi}_{\mathbf{q}^{\prime}}\rangle$
$\displaystyle=\left.\frac{\delta}{\delta\eta(\mathbf{q}^{\prime})}\frac{\delta}{\delta\bar{\eta}(\mathbf{q})}e^{\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\bar{\eta}(\mathbf{k})G_{0}^{-1}(\mathbf{k})\eta(\mathbf{k})}\right|_{\eta,\bar{\eta}=0}$
$\displaystyle=\left.\frac{\delta}{\delta\eta(\mathbf{q}^{\prime})}\left[G_{0}^{-1}(\mathbf{q})\eta(\mathbf{q})e^{\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\bar{\eta}(\mathbf{k})G_{0}^{-1}(\mathbf{k})\eta(\mathbf{k})}\right]\right|_{\eta,\bar{\eta}=0}$
$\displaystyle=G_{0}^{-1}(\mathbf{q})\delta_{\mathbf{q},\mathbf{q^{\prime}}}.$
(58)

where $G_{0}^{-1}(\mathbf{q})$ has the expression:

$G_{0}^{-1}(\mathbf{q})=\frac{1}{\omega_{n}^{2}+\epsilon^{2}(\mathbf{q})}\begin{pmatrix}\epsilon_{0}(\mathbf{q})+i\omega_{n}-\mu&-U\Gamma(\mathbf{q},0)\Delta_{0}\\\
-U\Gamma^{*}(\mathbf{q},0)\Delta_{0}&-\epsilon_{0}(\mathbf{q})+i\omega_{n}+\mu\end{pmatrix}.$
(59)

As such, we have Gor’kov’s Green functions:

$\displaystyle\mathcal{G}(\mathbf{q})$ $\displaystyle=\langle
c_{-}(\mathbf{q})\bar{c}_{-}(\mathbf{q})\rangle=\langle
c_{+}(\mathbf{q})\bar{c}_{+}(\mathbf{q})\rangle=\frac{i\omega_{n}+\epsilon_{0}(\mathbf{q})-\mu}{\omega_{n}^{2}+\epsilon^{2}(\mathbf{q})},$
(60) $\displaystyle\mathcal{F}(\mathbf{q})$ $\displaystyle=\langle
c_{+}(-\mathbf{q})c_{-}(\mathbf{q})\rangle=-\frac{U\Gamma^{*}(\mathbf{q},0)\Delta_{0}}{\omega_{n}^{2}+\epsilon^{2}(\mathbf{q})}.$
(61)

Note that due to time reversal invariant, $\Gamma(\mathbf{q},0)=1$, thus
$\epsilon(\mathbf{q})=\sqrt{(\epsilon_{0}(\mathbf{q})-\mu)^{2}+U^{2}\Delta_{0}^{2}}$.
With all information, we can solve for $\delta F$. Due to stability of the
mean-field solution, a linear correction has to vanish, and the lowest order
term is thus second order in $\delta\Delta(\mathbf{k})$, namely $F_{2}$ the
Gaussian fluctuation of the free energy, given by:

$\displaystyle F_{2}$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}U|\delta\Delta(\mathbf{k})|^{2}-\frac{T}{2}\left\langle\left(\int_{0}^{\beta}d\tau\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\delta\mathcal{L}\right)^{2}\right\rangle$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}U|\delta\Delta(\mathbf{k})|^{2}$
$\displaystyle\
-\frac{T}{2}\sum_{n}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\left[2|\Gamma(\mathbf{q},\mathbf{k})|^{2}|\delta\Delta(\mathbf{k})|^{2}\left\langle
c_{-}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\bar{c}_{-}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right\rangle\left\langle
c_{+}\left(\mathbf{-q}+\frac{\mathbf{k}}{2}\right)\bar{c}_{+}\left(\mathbf{-q}+\frac{\mathbf{k}}{2}\right)\right\rangle\right.$
$\displaystyle\
+\Gamma(\mathbf{q},\mathbf{k})\Gamma(\mathbf{q},-\mathbf{k})\delta\Delta(\mathbf{k})\delta\Delta(-\mathbf{k})\left\langle\bar{c}_{-}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\bar{c}_{+}\left(\mathbf{-q}-\frac{\mathbf{k}}{2}\right)\right\rangle\left\langle\bar{c}_{+}\left(\mathbf{-q}+\frac{\mathbf{k}}{2}\right)\bar{c}_{-}\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right\rangle$
$\displaystyle\
\left.+\Gamma^{*}(\mathbf{q},\mathbf{k})\Gamma^{*}(\mathbf{q},-\mathbf{k})\delta\Delta^{*}(\mathbf{k})\delta\Delta^{*}(-\mathbf{k})\left\langle
c_{+}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)c_{-}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right\rangle\left\langle
c_{-}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)c_{+}\left(-\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right\rangle\right]$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}|\delta\Delta(\mathbf{k})|^{2}\left[U-U^{2}\chi(\mathbf{k})\right]$
(62) $\displaystyle\chi(\mathbf{q})$
$\displaystyle=T\sum_{n}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}|\Gamma(\mathbf{q},\mathbf{k})|^{2}\left[\mathcal{G}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\mathcal{G}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\mathcal{F}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\mathcal{F}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right].$
(63)

##  Appendix C Quantum metric of massive Dirac cones

Assuming time reversal invariant, form factor
$|\Gamma(\mathbf{q},\mathbf{k})|^{2}$ can be expanded given that $\mathbf{k}$
is small as:

$\displaystyle|\Gamma(\mathbf{q},\mathbf{k})|^{2}$
$\displaystyle=\left|\left\langle
u_{\mathbf{q}+\mathbf{k}/2}\right|\left.u_{\mathbf{q}-\mathbf{k}/2}\right\rangle\right|^{2}$
$\displaystyle=\mathrm{Tr}\left[P\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)P\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]$
$\displaystyle=\mathrm{Tr}\left\\{\left[P(\mathbf{q})+\sum_{i}\frac{k_{i}}{2}\partial_{i}P(\mathbf{q})+\sum_{i,j}\frac{k_{i}k_{j}}{8}\partial_{i}\partial_{j}P(\mathbf{q})\right]\left[P(\mathbf{q})-\sum_{a}\frac{k_{a}}{2}\partial_{a}P(\mathbf{q})+\sum_{a,b}\frac{k_{a}k_{b}}{8}\partial_{a}\partial_{b}P(\mathbf{q})\right]\right\\}$
$\displaystyle=1-\sum_{i,j}\frac{k_{i}k_{j}}{8}\mathrm{Tr}\left[2\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})-P(\mathbf{q})\partial_{i}\partial_{j}P(\mathbf{q})-\partial_{i}\partial_{j}P(\mathbf{q})P(\mathbf{q})\right]$
$\displaystyle=1-\sum_{i,j}\frac{k_{i}k_{j}}{8}\mathrm{Tr}\left[4\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})-\partial_{i}\partial_{j}P(\mathbf{q})\right]$
$\displaystyle=1-\sum_{i,j}\frac{1}{2}k_{i}k_{j}\mathrm{Tr}\left[\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})\right]+\frac{k_{i}k_{j}}{8}\partial_{i}\partial_{j}\mathrm{Tr}P(\mathbf{q})$
$\displaystyle=1-\sum_{i,j}\frac{1}{2}k_{i}k_{j}\mathrm{Tr}\left[\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})\right]$
$\displaystyle=1-\sum_{i,j}k_{i}k_{j}g_{ij}(\mathbf{q}),$ (64)

where
$g_{ij}(\mathbf{q})=\frac{1}{2}\mathrm{Tr}\left[\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})\right]$
is the quantum metric. For our toy model, in terms of elementary matrices, the
projector matrix is given by:

$\displaystyle P$
$\displaystyle=\frac{1}{2\epsilon_{0}(\mathbf{q})}\begin{pmatrix}e^{-i\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})+m}\\\
e^{i\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})-m}\end{pmatrix}\begin{pmatrix}e^{i\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})+m}&e^{-i\phi/2}\sqrt{\epsilon_{0}(\mathbf{q})-m}\end{pmatrix}$
$\displaystyle=\frac{1}{2\epsilon_{0}(\mathbf{q})}\begin{pmatrix}\epsilon_{0}(\mathbf{q})+m&e^{-i\phi}\sqrt{\epsilon_{0}(\mathbf{q})^{2}-m^{2}}\\\
e^{i\phi}\sqrt{\epsilon_{0}(\mathbf{q})^{2}-m^{2}}&\epsilon_{0}(\mathbf{q})-m\end{pmatrix}$
$\displaystyle=\frac{1}{2\epsilon_{0}(\mathbf{q})}\begin{pmatrix}\epsilon_{0}(\mathbf{q})+m&vqe^{-i\phi}\\\
vqe^{i\phi}&\epsilon_{0}(\mathbf{q})-m\end{pmatrix}$
$\displaystyle=\frac{1}{2}\left(\mathds{1}_{2}+\frac{m}{\epsilon_{0}(\mathbf{q})}\sigma_{z}+\frac{v}{\epsilon_{0}(\mathbf{q})}\mathbf{q}\cdot\bm{\sigma}\right).$
(65)

As such the derivative is given by:

$\displaystyle\partial_{i}P(\mathbf{q})$
$\displaystyle=\frac{1}{2\epsilon_{0}(\mathbf{q})}\left[v\sigma_{i}-\frac{\partial_{i}\epsilon_{0}(\mathbf{q})}{\epsilon_{0}(\mathbf{q})}(m\sigma_{z}+v\mathbf{q}\cdot\bm{\sigma})\right]$
$\displaystyle=\frac{1}{2\epsilon_{0}(\mathbf{q})}\left[v\sigma_{i}-\frac{v^{2}q_{i}}{\epsilon_{0}(\mathbf{q})^{2}}(m\sigma_{z}+v\mathbf{q}\cdot\bm{\sigma})\right]$
$\displaystyle=\frac{v}{2\epsilon_{0}(\mathbf{q})^{3}}\left[\epsilon_{0}(\mathbf{q})^{2}\sigma_{i}-vq_{i}(m\sigma_{z}+v\mathbf{q}\cdot\bm{\sigma})\right]$
$\displaystyle=\frac{v}{2\epsilon_{0}(\mathbf{q})^{3}}\left[\left(m^{2}+v^{2}q_{j}^{2}+iv^{2}q_{i}q_{j}\sigma_{z}\epsilon_{ij}\right)\sigma_{i}-mvq_{i}\sigma_{z}\right]$
(66)
$\displaystyle=\frac{v}{2\epsilon_{0}(\mathbf{q})^{3}}\left[\sigma_{i}\left(m^{2}+v^{2}q_{j}^{2}-iv^{2}q_{i}q_{j}\sigma_{z}\epsilon_{ij}\right)-mvq_{i}\sigma_{z}\right],$
(67)

where $j\neq i$, and $\epsilon_{xy}=-\epsilon_{yx}=1$. Correspondingly quantum
geometry is:

$\displaystyle g_{ii}(\mathbf{q})$
$\displaystyle=\frac{1}{2}\mathrm{Tr}[\partial_{i}P(\mathbf{q})\partial_{i}P(\mathbf{q})]$
$\displaystyle=\frac{v^{2}}{8\epsilon_{0}(\mathbf{q})^{6}}\left[\left(m^{2}+v^{2}q_{j}^{2}\right)^{2}+v^{4}q_{i}^{2}q_{j}^{2}+m^{2}v^{2}q_{i}^{2}\right]\mathrm{Tr}\mathds{1}_{2}$
$\displaystyle=\frac{v^{2}}{4\epsilon_{0}(\mathbf{q})^{6}}\left(m^{4}+2m^{2}v^{2}q_{j}^{2}+v^{4}q_{j}^{4}+v^{4}q_{i}^{2}q_{j}^{2}+m^{2}v^{2}q_{i}^{2}\right)$
$\displaystyle=\frac{v^{2}}{4\epsilon_{0}(\mathbf{q})^{6}}\left(m^{4}+m^{2}v^{2}q_{j}^{2}+v^{4}q^{2}q_{j}^{2}+m^{2}v^{2}q^{2}\right)$
$\displaystyle=\frac{v^{2}}{4\epsilon_{0}(\mathbf{q})^{4}}\left(m^{2}+v^{2}q_{j}^{2}\right)$
(68) $\displaystyle g_{ij}(\mathbf{q},i\neq j)$
$\displaystyle=\frac{1}{2}\mathrm{Tr}[\partial_{i}P(\mathbf{q})\partial_{i}P(\mathbf{q})]$
$\displaystyle=\frac{v^{2}}{8\epsilon_{0}(\mathbf{q})^{6}}\left\\{\mathrm{Tr}\left[i\epsilon_{ij}\sigma_{z}\left(m^{2}+v^{2}q_{i}^{2}+iv^{2}q_{i}q_{j}\sigma_{z}\epsilon_{ij}\right)\left(m^{2}+v^{2}q_{j}^{2}+iv^{2}q_{i}q_{j}\sigma_{z}\epsilon_{ij}\right)\right]+2m^{2}v^{2}q_{i}q_{j}\right\\}$
$\displaystyle=\frac{v^{2}}{8\epsilon_{0}(\mathbf{q})^{6}}\left\\{\mathrm{Tr}\left[i\epsilon_{ij}\sigma_{z}\left(2m^{2}+v^{2}q^{2}\right)iv^{2}q_{i}q_{j}\sigma_{z}\epsilon_{ij}\right]+2m^{2}v^{2}q_{i}q_{j}\right\\}$
$\displaystyle=\frac{v^{2}}{4\epsilon_{0}(\mathbf{q})^{6}}\left[-\left(2m^{2}+v^{2}q^{2}\right)v^{2}q_{i}q_{j}+m^{2}v^{2}q_{i}q_{j}\right]$
$\displaystyle=\frac{v^{2}}{4\epsilon_{0}(\mathbf{q})^{6}}\left[-\left(2m^{2}+v^{2}q^{2}\right)v^{2}q_{i}q_{j}+m^{2}v^{2}q_{i}q_{j}\right]$
$\displaystyle=-\frac{v^{2}}{4\epsilon_{0}(\mathbf{q})^{4}}v^{2}q_{i}q_{j}$
(69) $\displaystyle g(\mathbf{q})$
$\displaystyle=\frac{v^{2}}{4(m^{2}+v^{2}q^{2})^{2}}\begin{pmatrix}m^{2}+v^{2}q_{y}^{2}&-v^{2}q_{x}q_{y}\\\
-v^{2}q_{x}q_{y}&m^{2}+v^{2}q_{x}^{2}\end{pmatrix}.$ (70)

We will use this result to compare between the conventional and the
geometrical contribution in the later session.

##  Appendix D Ginzburg-Landau Theory of Massive Dirac Cone

At the upper critical field $H_{c2}$, the order parameter is suppressed,
namely $\Delta_{0}=0$. In this situation,
$\epsilon(\mathbf{q})=|\epsilon_{0}(\mathbf{q})-\mu|$, and Gor’kov’s anomalous
Green Function (61) vanishes, with the normal Green Function (60) simplifies
to:

$\mathcal{G}(\mathbf{q})=\frac{1}{-i\omega_{n}+\epsilon_{0}(\mathbf{q})-\mu}.$
(71)

Recall Eq. (62) where the free energy density takes the form
$F_{2}=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}|\delta\Delta(\mathbf{k})|^{2}\left[U-U^{2}\chi(\mathbf{k})\right]$,
where:

$\displaystyle\chi(\mathbf{k})$
$\displaystyle=T\sum_{n}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}|\Gamma(\mathbf{q},\mathbf{k})|^{2}\mathcal{G}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\mathcal{G}\left(-\mathbf{q}+\frac{\mathbf{k}}{2}\right)$
$\displaystyle=T\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}|\Gamma(\mathbf{q},\mathbf{k})|^{2}\sum_{n}\frac{1}{-i\omega_{n}+\epsilon_{0}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)-\mu}\frac{1}{i\omega_{n}+\epsilon_{0}\left(\mathbf{-q}+\frac{\mathbf{k}}{2}\right)-\mu}$
$\displaystyle=\frac{1}{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}|\Gamma(\mathbf{q},\mathbf{k})|^{2}\frac{\tanh\frac{\beta}{2}\left[\epsilon_{0}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)-\mu\right]+\tanh\frac{\beta}{2}\left[\epsilon_{0}\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)-\mu\right]}{\epsilon_{0}\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\epsilon_{0}\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)-2\mu}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}|\Gamma(\mathbf{q},\mathbf{k})|^{2}\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right]+\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}{2\left[\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}.$
(72)

In the last line, we have defined
$\varepsilon(\mathbf{q})=\epsilon_{0}(\mathbf{q})-\mu$ for convenience. Note
that the conduction band dispersion is invariant under rotational symmetry,
thus isotropic. As such, we can perform expansion around $\mathbf{k}=0$ to
obtain:

$\displaystyle\chi(\mathbf{k})$
$\displaystyle=\chi_{0}-\left(\chi_{2,\mathrm{con}}+\chi_{2,\mathrm{qm}}\right)k^{2}+\mathcal{O}(k^{4}),$
(73) $\displaystyle\chi_{0}$
$\displaystyle=\mathcal{\mathcal{A}}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{1}{2\varepsilon(\mathbf{q})}\tanh\frac{\beta\varepsilon(\mathbf{q})}{2},$
(74) $\displaystyle\chi_{2,\mathrm{qm}}k^{2}$
$\displaystyle=\frac{k^{2}}{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\mathrm{tr}g(\mathbf{q})\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}\right)\right]}{2\varepsilon\left(\mathbf{q}\right)},$
(75) $\displaystyle\chi_{2,\mathrm{con}}k^{2}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}\right)\right]}{2\varepsilon\mathbf{(q})}-\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right]+\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}{2\left[\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}\right\\},$
(76)

where $\chi_{2,\mathrm{con}}$ is contribution due to dispersion of the band,
and $\chi_{2,\mathrm{qm}}$ is due to the form factor, thus the quantum metric.
Note the zeroth order term can be rewritten as:

$\displaystyle\chi_{0}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}\right)\right]}{2\varepsilon\left(\mathbf{q}\right)}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}\right)\right]-\tanh\left[\frac{\beta_{\mathrm{MF}}}{2}\varepsilon\left(\mathbf{q}\right)\right]}{2\varepsilon\left(\mathbf{q}\right)}+\frac{1}{U},$
(77)

where $T_{\mathrm{MF}}=\frac{1}{\beta_{\mathrm{MF}}}$is the mean-field
critical temperature, defined by the continuous version of the self-
consistency condition (54) when $\Delta_{0}=0$:

$\displaystyle\frac{1}{U}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{1}{2\varepsilon(\mathbf{q})}\tanh\left[\frac{\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})\right].$
(78)

In the following, we will perform analytical calculations for the nearly
massless regime, namely the mass gap of the Dirac bands, is much smaller than
the Debye energy, which defines the energy range where we have effective
attractive interaction. In particular, we will study the case where the mass
gap $m\gg T_{\mathrm{MF}}$ and $m\ll T_{\mathrm{MF}}$.

###  D.1 Case 1: $m\gg T_{\mathrm{MF}}$

In continuous model, we can treat the chemical potential as a free parameter.
In particular, we will set it to $\mu=m+\kappa\epsilon_{D}$, and with
$0\leq\kappa\leq 1$. To begin with we should write down our integral over
momentum space in terms of energy:

$\displaystyle\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}=\int_{m}^{m+(1+\nu)\epsilon_{D}}\frac{\epsilon_{0}d\epsilon_{0}}{2\pi
v^{2}}=\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{(\varepsilon+\mu)d\varepsilon}{2\pi
v^{2}}.$ (79)

Considering the contribution from quantum metric, we have:

$\displaystyle\chi_{2,\mathrm{qm}}k^{2}$
$\displaystyle=\frac{k^{2}}{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}(g_{xx}+g_{yy})\frac{\tanh\left[\frac{\beta}{2}\varepsilon(\mathbf{q})\right]}{2\varepsilon(\mathbf{q})}$
$\displaystyle=\frac{1}{2}k^{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{v^{2}(2m^{2}+v^{2}q^{2})}{4(m^{2}+v^{2}q^{2})^{2}}\frac{\tanh\left[\frac{\beta}{2}\varepsilon(\mathbf{q})\right]}{2\varepsilon(\mathbf{q})}$
$\displaystyle=\frac{1}{4}k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{m^{2}+\left(\varepsilon+\mu\right)^{2}}{\left(\varepsilon+\mu\right)^{3}}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon}$
$\displaystyle=\frac{1}{4}k^{2}\mathcal{A}\left[\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{m^{2}}{\left(\varepsilon+\mu\right)^{3}}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon}+\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon\left(\varepsilon+\mu\right)}\right].$
(80)

For $\kappa\beta\epsilon_{D}\gg 1$, the latter term dominates:

$\displaystyle\chi_{2,\mathrm{qm}}k^{2}$
$\displaystyle=\frac{1}{4}k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon\left(\varepsilon+\mu\right)}$
$\displaystyle=\frac{1}{32\pi}k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\left(\frac{1}{\varepsilon}-\frac{1}{\varepsilon+\mu}\right)\tanh\left(\frac{\beta}{2}\varepsilon\right)$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\frac{1}{\varepsilon}\tanh\left(\frac{\beta}{2}\varepsilon\right)-\frac{1}{16\pi}k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\frac{1}{\varepsilon+\mu}$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}\frac{1}{\mu}\left[2\ln\left(\frac{4e^{\gamma}}{\pi}\frac{\beta\epsilon_{D}}{2}\right)+\ln\kappa-\ln\frac{(\epsilon_{D}+\mu)m}{\mu^{2}}\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi\mu}\left[\ln\left(\frac{\beta^{2}\mu^{3}\epsilon_{D}}{m(\mu+\epsilon_{D})}\right)+2\ln\left(\frac{2e^{\gamma}}{\pi}\right)\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi\mu}\ln\left(\frac{\beta^{2}\mu^{3}}{m}\right).$
(81)

where we have used the approximation
$\int_{0}^{a}dx\tanh(x)/x\approx\ln\left(4e^{\gamma}a/\pi\right)$ for $a\gg
1$.

For $\kappa\beta\epsilon_{D}\ll 1$, both term contributes. We can appoximate
the values of each term by setting $\kappa=0$. The first term is:

$\displaystyle\frac{1}{4}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{m^{2}}{\left(\varepsilon+\mu\right)^{3}}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon}$
$\displaystyle=\frac{1}{32\pi}k^{2}\mathcal{A}m^{2}\int_{0}^{\epsilon_{D}}d\varepsilon\frac{\tanh\left(\frac{\beta}{2}\varepsilon\right)}{\varepsilon(\varepsilon+m)^{3}}$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}m^{2}\left[\int_{0}^{2/\beta}d\varepsilon\frac{\beta}{2(\varepsilon+m)^{3}}+\int_{2/\beta}^{\epsilon_{D}}d\varepsilon\frac{1}{\varepsilon(\varepsilon+m)^{3}}\right]$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}m^{2}\left\\{\frac{\beta}{4}\left[\frac{1}{m^{2}}-\frac{1}{(m+2T)^{2}}\right]\right.$
$\displaystyle\left.+\frac{1}{2m^{3}}\left[\frac{2m\epsilon_{D}}{\epsilon_{D}^{2}}+2\ln\left(\frac{\epsilon_{D}}{\epsilon_{D}+m}\right)-\frac{m(3m+4T)}{(m+2T)^{2}}-2\ln\left(\frac{2T}{m+2T}\right)\right]\right\\}$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{64\pi m}\left[-1-2\ln
2+2\ln\left(\beta m\right)\right].$

The latter term is:

$\displaystyle\frac{1}{4}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon\left(\varepsilon+\mu\right)}$
$\displaystyle=\frac{1}{32\pi}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\left(\frac{1}{\varepsilon}-\frac{1}{\varepsilon+\mu}\right)\tanh\left(\frac{\beta}{2}\varepsilon\right)$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\frac{1}{\varepsilon}\tanh\left(\frac{\beta}{2}\varepsilon\right)-\frac{1}{16\pi}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\frac{1}{\varepsilon+\mu}$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}\frac{1}{m}\left[\ln\left(\frac{2e^{\gamma}}{\pi}\beta\epsilon_{D}\right)-\ln\left(1+\frac{\epsilon_{D}}{m}\right)\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi
m}\left[\ln\left(\frac{2e^{\gamma}}{\pi}\right)+\ln\left(\beta
m\right)\right].$

Thus the total contribution at $\kappa=0$ (namely $\mu=m$) is:

$\chi_{\mathrm{qm}}(\kappa=0)k^{2}=\frac{k^{2}\mathcal{A}}{64\pi
m}\left[-1+2\ln\left(\frac{e^{\gamma}}{\pi}\right)+4\ln\left(\beta
m\right)\right].$ (82)

For conventional contribution, we have:

$\displaystyle\chi_{2,\mathrm{con}}k^{2}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\left\\{\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}\right)\right]}{2\varepsilon\mathbf{(q})}-\frac{\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right]+\tanh\left[\frac{\beta}{2}\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}{2\left[\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)+\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]}\right\\}$
$\displaystyle=\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{1}{2}\left\\{\frac{1-2n_{F}\left[\varepsilon\left(\mathbf{q}\right)\right]}{2\varepsilon(\mathbf{q})}+\frac{n_{F}\left[\varepsilon\left(\mathbf{q}+\frac{\mathbf{k}}{2}\right)\right]+n_{F}\left[\varepsilon\left(\mathbf{q}-\frac{\mathbf{k}}{2}\right)\right]-1}{2\varepsilon(\mathbf{q})}\right\\}$
$\displaystyle\approx\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{1}{2\varepsilon(\mathbf{q})}\partial_{\varepsilon}^{2}n_{F}[\varepsilon(\mathbf{q})]\left\\{\frac{v^{2}(\mathbf{q}\cdot\mathbf{k})}{2[\varepsilon(\mathbf{q})+\mu]}\right\\}^{2}$
$\displaystyle\approx\mathcal{A}\int\frac{qdq}{2\pi}\frac{1}{2\varepsilon(\mathbf{q})}\partial_{\varepsilon}^{2}n_{F}[\varepsilon(\mathbf{q})]\frac{v^{4}q^{2}k^{2}}{8[\varepsilon(\mathbf{q})+\mu]^{2}}$
$\displaystyle=\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\frac{1}{\varepsilon}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)\frac{k^{2}}{\varepsilon+\mu}\left[(\varepsilon+\mu)^{2}-m^{2}\right]$
$\displaystyle=k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)\left[1+\frac{\mu}{\varepsilon}-\frac{m^{2}}{\varepsilon(\varepsilon+\mu)}\right]$
$\displaystyle=k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)\left[1+\frac{\mu}{\varepsilon}\left(1-\frac{m^{2}}{\mu^{2}}\right)+\frac{m^{2}}{\mu(\varepsilon+\mu)}\right].$
(83)

For case where $\kappa\beta\epsilon_{D}\gg 1$, we have:

$\displaystyle\chi_{2,\mathrm{con}}k^{2}$ $\displaystyle\approx
k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)\left[1+\frac{\mu}{\varepsilon}\left(1-\frac{m^{2}}{\mu^{2}}\right)+\frac{m^{2}}{\mu(\varepsilon+\mu)}\right]$
$\displaystyle\approx
k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)+k^{2}\mu\mathcal{A}\left(1-\frac{m^{2}}{\mu^{2}}\right)\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\frac{\partial_{\varepsilon}^{2}n_{F}(\varepsilon)}{\varepsilon}$
$\displaystyle\approx
k^{2}\mathcal{A}\frac{1}{32\pi}\beta\left(e^{-\kappa\beta\epsilon_{D}}-e^{-\beta\epsilon_{D}}\right)+k^{2}\mu\mathcal{A}\frac{1}{32\pi}\left(1-\frac{m^{2}}{\mu^{2}}\right)\int_{-\infty}^{\infty}d\varepsilon\frac{\partial_{\varepsilon}^{2}n_{F}(\varepsilon)}{\varepsilon}$
$\displaystyle\approx
k^{2}\mu\mathcal{A}\frac{1}{32\pi}\left(1-\frac{m^{2}}{\mu^{2}}\right)\beta^{2}\frac{7\zeta(3)}{2\pi^{2}}$
$\displaystyle\approx
k^{2}\mathcal{A}\frac{7\zeta(3)}{64\pi^{3}}\frac{\kappa\epsilon_{D}}{T^{2}}.$
(84)

For case where $|\kappa|\beta\epsilon_{D}\ll 1$, we have:

$\displaystyle\chi_{2,\mathrm{con}}k^{2}$ $\displaystyle\approx
k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)\left[1+\frac{\mu}{\varepsilon}\left(1-\frac{m^{2}}{\mu^{2}}\right)+\frac{m^{2}}{\mu(\varepsilon+\mu)}\right]$
$\displaystyle\approx
k^{2}\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{32\pi}\partial_{\varepsilon}^{2}n_{F}(\varepsilon)\left(1+\frac{m}{\varepsilon}\frac{2\kappa\epsilon_{D}}{m}+\frac{m}{\varepsilon+m+\kappa\epsilon_{D}}\right)$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi}\left[\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\partial_{\varepsilon}^{2}n_{F}(\varepsilon)+2\kappa\epsilon_{D}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{\partial_{\varepsilon}^{2}n_{F}(\varepsilon)}{\varepsilon}+m\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{\partial_{\varepsilon}^{2}n_{F}(\varepsilon)}{\varepsilon+m+\kappa\epsilon_{D}}\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi}\left[\beta\frac{e^{-\kappa\beta\epsilon_{D}}}{(1+e^{-\kappa\beta\epsilon_{D}})^{2}}+2\kappa\epsilon_{D}\beta^{2}\frac{7\zeta(3)}{4\pi^{2}}+\beta\frac{e^{-\kappa\beta\epsilon_{D}}}{(1+e^{-\kappa\beta\epsilon_{D}})^{2}}\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi}\left[\frac{\beta}{2}+2\kappa\epsilon_{D}\beta^{2}\frac{7\zeta(3)}{4\pi^{2}}\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{64\pi
T}\left[1+\kappa\beta\epsilon_{D}\frac{7\zeta(3)}{\pi^{2}}\right].$ (85)

As for the mean-field temperature, we have:

$\displaystyle\frac{1}{U}$
$\displaystyle=\mathcal{A}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{(\varepsilon+\mu)d\varepsilon}{2\pi
v^{2}}\frac{1}{2\varepsilon}\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}$
$\displaystyle=\frac{\mathcal{A}}{4\pi
v^{2}}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\left(\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}+\frac{\mu}{\varepsilon}\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right)$
$\displaystyle=\frac{\mathcal{A}}{4\pi
v^{2}}\left[\frac{2}{\beta_{MF}}\ln\frac{\cosh(\beta_{\mathrm{MF}}\epsilon_{D}/2)}{\cosh(\kappa\beta_{\mathrm{MF}}\epsilon_{D}/2)}+\mu\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon}\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right].$
(86)

Finally we should calculate $U-U^{2}\chi_{0}$:

$\displaystyle U-U^{2}\chi_{0}$
$\displaystyle=-U^{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\tanh\left[\frac{\beta}{2}\varepsilon(\mathbf{q})\right]-\tanh\left[\frac{\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})\right]}{2\varepsilon(\mathbf{q})}$
$\displaystyle\approx-U^{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\mathrm{sech}^{2}\left[\frac{\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})\right]}{2\varepsilon(\mathbf{q})}\frac{\beta-\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})$
$\displaystyle=-U^{2}\mathcal{A}\frac{1}{v^{2}}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}(\varepsilon+\mu)\frac{\mathrm{sech}^{2}\left(\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right)}{4}(\beta-\beta_{\mathrm{MF}})$
$\displaystyle=-U^{2}\mathcal{A}\frac{1}{8\pi
v^{2}}(\beta-\beta_{\mathrm{MF}})\frac{2}{\beta_{\mathrm{MF}}}\left\\{\left.\left[(\varepsilon+m+\kappa\epsilon_{D})\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right]\right|_{-\kappa\epsilon_{D}}^{\epsilon_{D}}-\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right\\}$
$\displaystyle=\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}\left\\{\left[m+(1+\kappa)\epsilon_{D}\right]\tanh\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}+m\tanh\frac{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}{2}\right.$
$\displaystyle\left.-\frac{2}{\beta_{MF}}\left(\ln\cosh\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}-\ln\cosh\frac{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}{2}\right)\right\\},$
(87)

where for convenience we have defined $\alpha=4\pi v^{2}/U\mathcal{A}$. For
$\kappa\beta\epsilon_{D}\gg 1$, we have:

$\displaystyle\alpha$
$\displaystyle\approx\frac{2}{\beta_{MF}}\ln\frac{\cosh(\beta_{\mathrm{MF}}\epsilon_{D}/2)}{\cosh(\kappa\beta_{\mathrm{MF}}\epsilon_{D}/2)}+\mu\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon}\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}$
$\displaystyle\approx(1-\kappa)\epsilon_{D}+2(m+\kappa\epsilon_{D})\ln\left(\frac{2e^{\gamma}}{\pi}\beta_{\mathrm{MF}}\epsilon_{D}\right)+(m+\kappa\epsilon_{D})\ln\kappa$
$\displaystyle T_{\mathrm{MF}}$
$\displaystyle\approx\frac{2\sqrt{\kappa}}{\pi}\epsilon_{D}\exp\left(\gamma+\frac{1-\kappa}{2\kappa}-\frac{\alpha}{2\kappa\epsilon_{D}}\right).$
(88)

For $\kappa\beta\epsilon_{D}\ll 1$, we have:

$\displaystyle\alpha$
$\displaystyle\approx\frac{2}{\beta_{MF}}\ln\frac{\cosh(\beta_{\mathrm{MF}}\epsilon_{D}/2)}{\cosh(\kappa\beta_{\mathrm{MF}}\epsilon_{D}/2)}+\mu\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon}\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}$
$\displaystyle\approx\epsilon_{D}+(m+\kappa\epsilon_{D})\ln\left(\frac{2e^{\gamma}}{\pi}\beta_{\mathrm{MF}}\epsilon_{D}\right)+(m+\kappa\epsilon_{D})\frac{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}{2}$
$\displaystyle\approx\epsilon_{D}+(m+\kappa\epsilon_{D})\ln\left(\frac{2e^{\gamma}}{\pi}\beta_{\mathrm{MF}}\epsilon_{D}\right)$
$\displaystyle T_{\mathrm{MF}}$
$\displaystyle\approx\frac{2}{\pi}\epsilon_{D}\exp\left(\gamma+\frac{\epsilon_{D}-\alpha}{\mu}\right).$
(89)

We should also calculate $U-U^{2}\chi_{0}$ to determine the coherence length:

$\displaystyle U-U^{2}\chi_{0}$
$\displaystyle=-U^{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\tanh\left[\frac{\beta}{2}\varepsilon(\mathbf{q})\right]-\tanh\left[\frac{\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})\right]}{2\varepsilon(\mathbf{q})}$
$\displaystyle\approx-U^{2}\mathcal{A}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\frac{\mathrm{sech}^{2}\left[\frac{\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})\right]}{2\varepsilon(\mathbf{q})}\frac{\beta-\beta_{\mathrm{MF}}}{2}\varepsilon(\mathbf{q})$
$\displaystyle=-U^{2}\mathcal{A}\frac{1}{v^{2}}\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}(\varepsilon+\mu)\frac{\mathrm{sech}^{2}\left(\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right)}{4}(\beta-\beta_{\mathrm{MF}})$
$\displaystyle=-U^{2}\mathcal{A}\frac{1}{8\pi
v^{2}}(\beta-\beta_{\mathrm{MF}})\frac{2}{\beta_{\mathrm{MF}}}\left\\{\left.\left[(\varepsilon+m+\kappa\epsilon_{D})\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right]\right|_{-\kappa\epsilon_{D}}^{\epsilon_{D}}-\int_{-\kappa\epsilon_{D}}^{\epsilon_{D}}d\varepsilon\tanh\frac{\beta_{\mathrm{MF}}\varepsilon}{2}\right\\}$
$\displaystyle=\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}\left\\{\left[m+(1+\kappa)\epsilon_{D}\right]\tanh\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}+m\tanh\frac{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}{2}\right.$
$\displaystyle\left.-\frac{2}{\beta_{MF}}\left(\ln\cosh\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}-\ln\cosh\frac{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}{2}\right)\right\\}.$
(90)

Specifying to the case where $\epsilon_{D}\gg m,\
\kappa\beta_{\mathrm{MF}}\epsilon_{D}\gg 1$, we have:

$\displaystyle U-U^{2}\chi_{0}$
$\displaystyle\approx\frac{U}{\alpha}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}\left\\{(1+\kappa)\epsilon_{D}+m-\frac{2}{\beta_{MF}}\left(\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}-\frac{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}{2}\right)\right\\}$
$\displaystyle\approx
2\kappa\epsilon_{D}\frac{U}{\alpha}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}.$
(91)

For $\kappa\beta\epsilon_{D}\ll 1$, we have:

$\displaystyle U-U^{2}\chi_{0}$
$\displaystyle\approx\frac{U}{\alpha}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}\left\\{\left[m+(1+\kappa)\epsilon_{D}\right]-\frac{2}{\beta_{MF}}\left(\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}-\ln
2\right)\right\\}$
$\displaystyle\approx\mu\frac{U}{\alpha}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}.$
(92)

Note that we can write down an effective Lagrangian for our bosonic field:

$\displaystyle\mathcal{L}[\delta\Delta,\delta\bar{\Delta}]$
$\displaystyle=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}\ F_{2}(\mathbf{k})$
$\displaystyle=(U-U^{2}\chi_{0})|\delta\Delta|^{2}+(\chi_{2,\mathrm{qm}}+\chi_{2,\mathrm{con}})|\partial_{\mathbf{k}}\delta\Delta|^{2}.$
(93)

As such, the coherence length should be given by:

$\displaystyle\xi$
$\displaystyle=\sqrt{-U^{2}\frac{\chi_{2,\mathrm{qm}}+\chi_{2,\mathrm{con}}}{U-U^{2}\chi_{0}}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{qm}}}{U-U^{2}\chi_{0}}-\frac{U\chi_{2,\mathrm{con}}}{U-U^{2}\chi_{0}}}$
$\displaystyle=\sqrt{\xi_{\mathrm{qm}}^{2}+\xi_{\mathrm{con}}^{2}}.$ (94)

At the $\kappa\beta_{\mathrm{MF}}\epsilon_{D}\gg 1$ regime, the coherence
lengths are given by:

$\displaystyle\xi_{\mathrm{qm}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{qm}}}{U-U^{2}\chi_{0}}}$
$\displaystyle\approx\sqrt{-\frac{\frac{U^{2}\mathcal{A}}{32\pi\kappa\epsilon_{D}}\left[\ln\left(\frac{\epsilon_{D}^{3}}{mT_{\mathrm{MF}}^{2}}\right)+2\ln\left(\frac{2e^{\gamma}}{\pi}\right)+\ln\left(\frac{\kappa^{3}}{1+\kappa}\right)\right]}{2\kappa\epsilon_{D}\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}}}$
$\displaystyle=\frac{v}{4\mu}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{\ln\left(\frac{\mu^{3}}{mT_{\mathrm{MF}}^{2}}\right)}$
(95) $\displaystyle\xi_{\mathrm{con}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{con}}}{U-U^{2}\chi_{0}}}$
$\displaystyle\approx\sqrt{-\frac{U^{2}\mathcal{A}\frac{7\zeta(3)}{64\pi^{3}}\frac{\kappa\epsilon_{D}}{T_{\mathrm{MF}}^{2}}}{2\kappa\epsilon_{D}\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}}}$
$\displaystyle=\frac{v}{T_{\mathrm{MF}}}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{\frac{7\zeta(3)}{32\pi^{2}}}.$
(96)

As such, the conventional contribution dominates, by a factor of
$\epsilon_{D}/T_{\mathrm{MF}}\gg 1$. For $\kappa=0$, the coherence lengths are
given by:

$\displaystyle\xi_{\mathrm{qm}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{qm}}}{U-U^{2}\chi_{0}}}$
$\displaystyle\approx\sqrt{-\frac{\frac{U^{2}\mathcal{A}}{64\pi
m}\left[-1+2\ln\left(\frac{e^{\gamma}}{\pi}\right)+4\ln\left(\beta_{\mathrm{MF}}m\right)\right]}{m\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}}}$
$\displaystyle=\frac{v}{4m}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{-1+2\ln\left(\frac{e^{\gamma}}{\pi}\right)+4\ln\left(\beta_{\mathrm{MF}}m\right)}$
(97) $\displaystyle\xi_{\mathrm{con}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{con}}}{U-U^{2}\chi_{0}}}$
$\displaystyle\approx\sqrt{-\frac{\frac{U^{2}\mathcal{A}}{64\pi
T_{\mathrm{MF}}}}{m\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}}}$
$\displaystyle=\frac{1}{4}\frac{v}{\sqrt{mT_{\mathrm{MF}}}}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}.$
(98)

Note that in this case the quantum metric contribution still smaller, but is
comparable with the conventional contribution by a factor of
$2\sqrt{\ln(\beta_{\mathrm{MF}}m)/\beta_{\mathrm{MF}}m}$.

####  D.1.1 Beyond band bottom

In this part, we aim to provide a brief discussion on the case beyond the band
bottom. We assume $|\mu-m|\ll m$ such that
$\chi_{2,\mathrm{qm}}=\chi_{2,\mathrm{qm}}(\kappa=0)$, corresponding to
$\mu=m$ with $\chi_{2,qm}(\kappa)$ defined in Eq. (82). We should calculate
$U-U^{2}\chi_{0}$ in the vicinity where superconductivity vanishes, namely
when $|\beta_{\mathrm{MF}}(\mu-m)|\gg 0$:

$\displaystyle\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}\left[\left(\mu+\epsilon_{D}\right)\tanh\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}+m\tanh\frac{\beta_{\mathrm{MF}}(\mu-m)}{2}-\frac{2}{\beta_{MF}}\left(\ln\cosh\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}+\ln\cosh\frac{\beta_{\mathrm{MF}}(\mu-m)}{2}\right)\right]$
$\displaystyle\approx\ $ $\displaystyle\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}\left[\mu+\epsilon_{D}-m-\frac{2}{\beta_{MF}}\left(\frac{\beta_{\mathrm{MF}}\epsilon_{D}}{2}+\frac{\beta_{\mathrm{MF}}(\mu-m)}{2}\right)\right]+\frac{4}{\beta_{\mathrm{MF}}}\ln
2$ $\displaystyle=\ $ $\displaystyle\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}4T_{\mathrm{MF}}\ln
2.$ (99)

which vanishes as the mean-field transition temperature goes to zero due to
vanishing density of state. As such the quantum metric coherence length
diverges at boundary of phase transition. Note that the conventional
contribution remains finite until $T_{\mathrm{MF}}\rightarrow 0$ due to both
$\chi_{2,\mathrm{con}}\propto\beta_{\mathrm{MF}}e^{\kappa\beta_{\mathrm{MF}}\epsilon_{D}}$
and $U-U^{2}\chi_{0}$ vanishing as $T_{\mathrm{MF}}\rightarrow 0$. It is
important to note that the superconductivity exists due to a finite
temperature effect, and is expected to vanish as temperature goes to 0.

###  D.2 Case 2: $T_{\mathrm{MF}}\gg m$

In this scenario, most of the calculation is identical to the previous case,
with the exception of $\chi_{\mathrm{qm}}$ when $\mu\approx m$:

$\displaystyle\frac{1}{4}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}\frac{d\varepsilon}{2\pi}\frac{1}{2}\frac{\tanh\left({\frac{\beta}{2}\varepsilon}\right)}{2\varepsilon\left(\varepsilon+\mu\right)}$
$\displaystyle=\frac{1}{32\pi}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\left(\frac{1}{\varepsilon}-\frac{1}{\varepsilon+\mu}\right)\tanh\left(\frac{\beta}{2}\varepsilon\right)$
$\displaystyle=\frac{1}{32\pi}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\frac{1}{\varepsilon}\tanh\left(\frac{\beta}{2}\varepsilon\right)-\frac{1}{16\pi}k^{2}\mathcal{A}\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\mu}\frac{1}{\varepsilon+\mu}\tanh\left(\frac{\beta}{2}\varepsilon\right)$
$\displaystyle\approx\frac{1}{32\pi}k^{2}\mathcal{A}\frac{1}{m}\left[\int_{0}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon}\tanh\left(\frac{\beta}{2}\varepsilon\right)-\int_{m}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon}\left(1-\frac{m}{\varepsilon}\right)\tanh\left(\frac{\beta}{2}\varepsilon\right)\right]$
(100)
$\displaystyle=\frac{1}{32\pi}k^{2}\mathcal{A}\int_{m}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon^{2}}\tanh\left(\frac{\beta}{2}\varepsilon\right)$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi}\left[\int_{m}^{2/\beta}d\varepsilon\
\frac{\beta}{2\varepsilon}+\int_{2/\beta}^{\epsilon_{D}}d\varepsilon\
\frac{1}{\varepsilon^{2}}\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{32\pi}\left[\frac{\beta}{2}\ln\frac{2}{\beta
m}-\frac{1}{\epsilon_{D}}+\frac{1}{2T}\right]$
$\displaystyle\approx\frac{k^{2}\mathcal{A}}{64\pi T}\left(1-\ln\frac{\beta
m}{2}\right).$ (101)

As such, the coherence length at $\kappa=0$ should change accordingly:

$\displaystyle\xi_{\mathrm{qm}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{qm}}}{U-U^{2}\chi_{0}}}$
$\displaystyle\approx\sqrt{-\frac{\frac{U^{2}\mathcal{A}}{64\pi
T_{\mathrm{MF}}}\left(1-\ln\frac{\beta_{\mathrm{MF}}m}{2}\right)}{m\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}}}$
$\displaystyle=\frac{1}{4}\frac{v}{\sqrt{mT_{\mathrm{MF}}}}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}\sqrt{1-\ln\frac{\beta_{\mathrm{MF}}m}{2}},$
(102) $\displaystyle\xi_{\mathrm{con}}$
$\displaystyle=\sqrt{-\frac{U^{2}\chi_{2,\mathrm{con}}}{U-U^{2}\chi_{0}}}$
$\displaystyle\approx\sqrt{-\frac{\frac{U^{2}\mathcal{A}}{64\pi
T_{\mathrm{MF}}}}{m\frac{U^{2}\mathcal{A}}{4\pi
v^{2}}\frac{\beta_{\mathrm{MF}}-\beta}{\beta_{\mathrm{MF}}}}}$
$\displaystyle=\frac{1}{4}\frac{v}{\sqrt{mT_{\mathrm{MF}}}}\left(\frac{T_{\mathrm{MF}}-T}{T_{\mathrm{MF}}}\right)^{-1/2}.$
(103)

Note the two contributions share a similar form, with the quantum metric
contribution having an extra factor of
$\sqrt{1-\ln(\beta_{\mathrm{MF}}m/2)}>1$. The quantum metric, thus plays a
dominating role in this regime, when superconductivity occurs near the band
bottom. The results for the toy model is summarized in Table 2, alongside an
illustration of the energy scale and band structure in Fig. 4.

Figure 4: Illustration of the energy scale and band structure for the toy
model. The dashed area denote the energy range ($\epsilon_{D}$) where the
interaction is effectively attractive. In the figure we have assumed the
chemical potential $\mu=m$, and $\epsilon_{D}\gg m,\ T_{\mathrm{MF}}$. Figure
(a) correspond to the case where $m\gg T_{\mathrm{MF}}$, while figure (b)
correspond to the case where $m\ll T_{\mathrm{MF}}$

##  Appendix E Discussion on pairing and time-reversal symmetry in IVC

In this section, we aim to discuss the pairing of the IVC state, in order to
induce superconductivity. To begin with, we can write down our IVC state in
the valley basis $\begin{pmatrix}K\\\ K^{\prime}\end{pmatrix}$:

$\displaystyle\psi_{-,\mathbf{q}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{-i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}(\mathbf{q})}{|\bm{\Delta}(\mathbf{q})|}}\\\
\sqrt{1+\frac{\Delta_{z}(\mathbf{q})}{|\bm{\Delta}(\mathbf{q})|}}\end{pmatrix},$
(104) $\displaystyle\psi_{-,\mathbf{-q}}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{-i\phi_{-\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}(-\mathbf{q})}{|\bm{\Delta}(-\mathbf{q})|}}\\\
\sqrt{1+\frac{\Delta_{z}(-\mathbf{q})}{|\bm{\Delta}(-\mathbf{q})|}}\end{pmatrix}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{-i\phi_{\mathbf{q}}}\sqrt{1+\frac{\Delta_{z}(\mathbf{q})}{|\bm{\Delta}(\mathbf{q})|}}\\\
\sqrt{1-\frac{\Delta_{z}(\mathbf{q})}{|\bm{\Delta}(\mathbf{q})|}}\end{pmatrix}.$
(105)

Recall the IVC order parameter’s phase can be matched with the phase of the
inter-valley form factor $\left\langle
u_{K,\mathbf{k}}\right.\left|u_{K^{\prime},\mathbf{k}}\right\rangle$[34],
inducing a six-fold symmetry in the phase, resulting in
$\phi_{\mathbf{q}}=\phi_{-\mathbf{q}}$. As such the form factor is given by:

$\displaystyle\Gamma(\mathbf{q},0)$
$\displaystyle=\psi_{-,\mathbf{-q}}(K)\psi_{-,\mathbf{q}}(K^{\prime})+\psi_{-,\mathbf{-q}}(K^{\prime})\psi_{-,\mathbf{q}}(K)$
$\displaystyle=\psi_{-,\mathbf{-q}}^{T}\sigma_{x}\psi_{-,\mathbf{q}}$
$\displaystyle=\frac{1}{2}\left[\left(1+\frac{\Delta_{z}}{|\bm{\Delta}|}\right)e^{-i\phi_{\mathbf{q}}}+\left(1-\frac{\Delta_{z}}{|\bm{\Delta}|}\right)e^{-i\phi_{\mathbf{q}}}\right]$
$\displaystyle=e^{-i\phi_{\mathbf{q}}}$ (106)
$\displaystyle|\Gamma(\mathbf{q},0)|^{2}$ $\displaystyle=1.$ (107)

Table 2: Summary on the toy model, where $\frac{1}{U^{\prime}}=\frac{1}{U}-\frac{\epsilon_{D}\mathcal{A}}{4\pi v^{2}}$ | $\mu$ | $T_{\mathrm{MF}}$ | $\xi_{\mathrm{con}}$ | $\xi_{\mathrm{qm}}$
---|---|---|---|---
$m\gg T_{\mathrm{MF}}$ | $\gg m$ | $\propto\sqrt{\rho\epsilon_{D}}\exp\left(-\frac{1}{\rho U^{\prime}}\right)$ | $\propto\frac{v}{T_{\mathbf{MF}}}$ | $\propto\frac{v}{\mu}\sqrt{\ln{(\beta_{\mathrm{MF}}\mu})}$
$m\ll T_{\mathrm{MF}}$
$m\gg T_{\mathrm{MF}}$ | $\rightarrow m$ | $\propto\epsilon_{D}\exp\left(-\frac{2}{\rho U^{\prime}}\right)$ | $\propto\frac{v}{\sqrt{mT_{\mathrm{MF}}}}$
$m\ll T_{\mathrm{MF}}$ | $\propto\frac{v}{\sqrt{mT_{\mathrm{MF}}}}\sqrt{1-\ln\frac{\beta_{\mathrm{MF}}m}{2}}$

As such we have recovered time reversal invariant for the IVC state. With the
invariant being intact after introducing the IVC order parameter, our previous
calculation on the conventional coherence length remains valid. As for the
projector matrices, we note the symmetry:

$\displaystyle P(\mathbf{q})$
$\displaystyle=\psi_{-,\mathbf{q}}\psi_{-,\mathbf{q}}^{\dagger}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}e^{-i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}}{|\bm{\Delta}|}}\\\
\sqrt{1+\frac{\Delta_{z}}{|\bm{\Delta}|}}\end{pmatrix}\begin{pmatrix}e^{i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}}{|\bm{\Delta}|}}&\sqrt{1+\frac{\Delta_{z}}{|\bm{\Delta}|}}\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1-\frac{\Delta_{z}}{|\bm{\Delta}|}&e^{-i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}\\\
e^{i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}&1+\frac{\Delta_{z}}{|\bm{\Delta}|}\end{pmatrix},$
(108) $\displaystyle\tilde{P}(-\mathbf{q})$
$\displaystyle=\psi_{-,-\mathbf{q}}^{*}\psi_{-,-\mathbf{q}}^{T}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}e^{i\phi_{-\mathbf{q}}}\sqrt{1+\frac{\Delta_{z}}{|\bm{\Delta}|}}\\\
\sqrt{1-\frac{\Delta_{z}}{|\bm{\Delta}|}}\end{pmatrix}\begin{pmatrix}e^{-i\phi_{-\mathbf{q}}}\sqrt{1+\frac{\Delta_{z}}{|\bm{\Delta}|}}&\sqrt{1-\frac{\Delta_{z}}{|\bm{\Delta}|}}\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1+\frac{\Delta_{z}}{|\bm{\Delta}|}&e^{i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}\\\
e^{-i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}&1-\frac{\Delta_{z}}{|\bm{\Delta}|}\end{pmatrix},$
(109) $\displaystyle\sigma_{x}\tilde{P}(-\mathbf{q})\sigma_{x}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}e^{-i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}&1-\frac{\Delta_{z}}{|\bm{\Delta}|}\\\
1+\frac{\Delta_{z}}{|\bm{\Delta}|}&e^{i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}\end{pmatrix}\sigma_{x}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1-\frac{\Delta_{z}}{|\bm{\Delta}|}&e^{-i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}\\\
e^{i\phi_{\mathbf{q}}}\sqrt{1-\frac{\Delta_{z}^{2}}{|\bm{\Delta}|^{2}}}&1+\frac{\Delta_{z}}{|\bm{\Delta}|}\end{pmatrix}$
$\displaystyle=P(\mathbf{q}).$ (110)

Note that the expansion of form factor around $\mathbf{k}=0$ retrieves the
quantum metric:

$\displaystyle\left|\psi_{-,\mathbf{-q}}^{T}\sigma_{x}\psi_{-,\mathbf{q}+\mathbf{k}}\right|{}^{2}$
$\displaystyle=\mathrm{Tr}{\left[\sigma_{x}\tilde{P}(-\mathbf{q})\sigma_{x}P(\mathbf{q}+\mathbf{k})\right]}$
$\displaystyle=\mathrm{Tr}{\left[P(\mathbf{q})P(\mathbf{q}+\mathbf{k})\right]}$
$\displaystyle=1-\frac{1}{2}\sum_{i,j}k_{i}k_{j}\mathrm{Tr}[\partial_{i}P(\mathbf{q})\partial_{j}P(\mathbf{q})]$
$\displaystyle=1-\sum_{i,j}k_{i}k_{j}g_{ij}(\mathbf{q}),$ (111)

where the intermediate step is similar to (C) thus are not shown explicitly.
As such, our results obtained for quantum metric coherence length from bare
electrons, is also applicable for superconductivity originating from the IVC
state, besides a change in the Bloch state, thus the quantum metric. This
justifies the application of the Ginzburg-Landau theory developed in the toy
model section, to study the superconductivity originating from the IVC state
in RTG.