# Non-Fermi-Liquid/Marginal-Fermi-Liquid Signatures Induced by Van Hove
Singularity

Yi-Hui Xing1,2 Wu-Ming Liu1,2,3<EMAIL_ADDRESS>1Beijing National Laboratory
for Condensed Matter Physics, Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  2School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100190, China  3Songshan Lake Materials
Laboratory, Dongguan, Guangdong 523808, China

###### Abstract

We theoretically study the two-dimensional metal that is coupled to critical
magnons and features van Hove singularities on the Fermi surface. When there
is only translationally invariant SYK-liked Yukawa interaction, van Hove
points suppress the contribution from the part of the Fermi surface away from
them, dominating and exhibiting non-Fermi-liquid behavior. When introducing
disordered Yukawa coupling, it leads to a crossover from non-Fermi-liquid to
marginal-Fermi-liquid, and the marginal-Fermi-liquid region exhibits the
$T\ln(1/T)$ specific heat and temperature-linear resistivity of strange metal.
By solving the gap equation, we provide the critical temperature for
superconductor induced by van Hove singularities and point out the possible
emergence of pair-density-wave superconductor. Our theory may become a new
mechanism for understanding non-Fermi-liquid or marginal-Fermi-liquid
phenomenons.

Many strongly correlated systems can exhibit phenomenons deviating from the
predictions of the Fermi-liquid theory of metals, namely, non-Fermi-liquid
(NFL) or marginal-Fermi-liquid (MFL) behavior. The most prominent examples
include iron-based superconductors Reiss  _et al._ (2020); Lee and Phillips
(2012), heavy fermion materials Chen  _et al._ (2019a); Seiro  _et al._
(2018), and moiré system Xu  _et al._ (2020); Wang  _et al._ (2022); Cao  _et
al._ (2020), among others. A direct signature of NFL is the frequency $\omega$
and temperature $T$ dependent relationship of self-energy imaginary part
$\mathrm{Im}\Sigma(\omega,T)$ of electrons behaves for $\omega>T$ like
$\sim|\omega|^{\alpha}$, where $\alpha<1$. When $\alpha=1$ Varma  _et al._
(1989); Kakehashi and Fulde (2005), it corresponds to MFL, whereas for
ordinary Fermi-liquid, $\alpha=2$. Because NFL/MFL is defined by deviating
from the Fermi-liquid, this has led to a diverse family of systems, and
lacking universal experimental features akin to the Fermi liquid. The
microscopic origins of the NFL/MFL phenomena have been a fascinating open
issue in condensed matter physics in recent years.

Using the critical Fermi surface CFM  can characterize the low-energy
effective theory of NFL/MFL, where free fermions couple with critical bosons.
However, most of the previous work has been based on Fermi surfaces away from
half-filling Patel  _et al._ (2023); Esterlis  _et al._ (2021); Guo  _et al._
(2022); Lee (2009) or hot spots Holder and Metzner (2014); Rao and Piazza
(2023), i.e., without considering the influence of van Hove singularities
(VHS). Considering that the Fermi surfaces of many materials Polshyn  _et al._
(2019); Cao  _et al._ (2020); Mousatov  _et al._ (2020); Xu  _et al._ (2021a);
Wang  _et al._ (2020) exhibiting NFL/MFL phenomena in experiments include van
Hove points (VHPs) and dopping can modify the Fermi surface to introduce VHS,
our work investigated the impact of VHS on NFL/MFL behavior.

VHPs are points of divergent density of states. Due to the singularities Yudin
_et al._ (2014) of the electron-electron scattering close to VHPs, the Fermi-
liquid picture is violated, leading to the emergence of various phases such as
superconductors (SC) and density waves. The Fermi levels of many materials
Karp  _et al._ (2020); Kang  _et al._ (2022); Wu  _et al._ (2021); Luo  _et
al._ (2023) exhibit VHS, it has been theoretically employed to explain
phenomena such as high-temperature superconductivity Sano  _et al._ (2016);
Irkhin  _et al._ (2002) and pair-density-wave Wu  _et al._ (2023a); Castro
_et al._ (2023). VHS also alters the signatures of critical bosons, modifying
the low-energy and temperature properties, which can, in turn, lead to NFL/MFL
behavior.

In this Letter, we study a system with electron-spin exchange interactions,
where the Fermi surface exhibits VHS. We extending it to a SYK-liked Sachdev
and Ye (1993); Wang (2020); Chowdhury  _et al._ (2018); Wu  _et al._ (2023b)
flavor-random Yukawa interactions, it describes $N$ flavors of fermion couple
with $N^{\prime}$ critical bosons. We show that when the interaction is
translationally invariant, the system will exhibit NFL behavior, and VHS will
suppress the behavior of the regular Fermi surface (i.e., the Fermi surface
away from VHS) and play a dominant role in NFL behavior. When breaking
translational invariance, i.e., introducing impurities, the system will
exhibit MFL behavior. The temperature $T$-linear resistivity and $T\ln(1/T)$
electronic specific heat implies the emergence of a strange metal Mousatov
_et al._ (2020); Patel  _et al._ (2023); Sachdev (2015); Cao  _et al._ (2020);
Chen  _et al._ (2019b); Chowdhury  _et al._ (2022). When both types of
interactions are considered, crossovers will appear and depend on the energy
scale or behavior of $N/N^{\prime}$. For completeness, we numerically identify
the finite-temperature phase boundaries of the boson modes by setting the
effective tuning parameter to zero phi . Using the gap equation, we identify
the critical temperature for the SC. We observed completely different tuning
parameter $\Delta$ dependent behaviors for the critical temperatures of the
two interactions. For real systems, the coexistence of disorder interaction
and translational invariant one, strange metal phase could also appear above
the SC phase. We argue that the pair-density-wave (PDW) Wu  _et al._ (2023a);
Zhou and Wang (2022); Hamidian  _et al._ (2016); Wang  _et al._ (2015) order
will appear to compete with conventional SC when the Fermi surface contains
more than one VHPs.

We consider a two-dimensional normal metal (NM) couple to a ferromagnetic
insulator (TMI). The system is shown in Fig. 1(a) and the interface is
described by the Hamiltonian

$H=-\sum_{i,j,\sigma}t_{ij}c^{\dagger}_{i,\sigma}c_{j,\sigma}+\sum_{k}\omega_{k}a^{\dagger}_{k}a_{k}+H_{int},$
(1)

where $c_{i,\sigma}$ describes annihilating an electron at lattice site $i$
with spin $\sigma$, it constitutes the hopping term with energy $t_{ij}$. It
is worth noting that we do not constrain the types of the lattice and the
range of the hopping. The TMI is modeled by a nearest-neighbor ferromagnetic
exchange interaction of strength $J$ and a tunable easy-axis anisotropy of
strength $K$ Webster and Yan (2018); Mæland and Sudbø (2022) that can be tuned
by applied mechanical strain. The Hamiltonian of TMI is
$H_{FI}=-\bar{J}\sum_{<ij>}\vec{S}_{i}\vec{S}_{j}-K\sum_{i}S_{i}^{z2}$ with
local spin operator $S_{i}$. Performing the Holstein-Primakoff transformation
in term of the magnon operator $a_{k}$, we get the magnon spectrum
$\omega_{k}=S\bar{J}a^{2}k^{2}/2+2SK$ with the lattice constant $a$ of TMI and
the spin $S$. For convenience, we set the magnon energy spectrum to be
$\omega_{k}=k^{2}+\Delta$ (by transformation
$a_{k}\rightarrow\sqrt{2/S\bar{J}a^{2}}a_{k}$), where $\Delta\equiv
k_{z}^{2}+4K/\bar{J}a^{2}$. Note that we are considering a two-dimensional
interface of a three-dimensional system, so the remaining momentum $k_{z}$ in
the z-direction forms the tuning parameter $\Delta$. We neglect high order
magnon-magnon interactions above and the subsequent magnon-electron
interactions as they do not play a crucial role in our calculations.

Figure 1: (a) An illustration of the bilayer system. The itinerant electrons
in the two-dimensional NM interact with the spins (red arrows) in the FMI via
exchange coupling (black Feynman diagram). The NM at the interface is modeled
by lattice system (blue array). (b)-(d) The contributing conductivity
diagrams. Lines represent fermionic propagators, wave lines represent current
operators, dashed lines represent magnon propagators after damped.

Electrons in the normal mental interact with spins in the ferromagnetic
insulator at the interface via exchange
$H_{em}=-\frac{J}{\sqrt{2S}}\sum_{i,\alpha,\beta}c_{i\alpha}^{\dagger}\vec{\sigma}^{\alpha\beta}c_{i\beta}\vec{S}_{i}$,
where $\vec{\sigma}$ is vector of Pauli matrices. Applying the Holstein-
Primakoff transformation and retained to the lowest order, yields
$H_{em}=-J\sum_{i}(c_{i\uparrow}^{\dagger}c_{i\downarrow}a_{i}+\mathrm{h.c.})$.
This interaction can be extended to the situation of quenched disorder after
considering the influence of impurities, i.e. the coupling $J$ depends on the
lattice site $i$. Experimentally, moving the Fermi surface through doping
directly introduces impurities. The identical energy spectrum for electrons
with different spin $\sigma$ ElE  allows for an extension to the coupling
between large-$N$ flavor of electrons. Similarly, we also consider the
large-$N^{\prime}$ flavor of magnons. We contemplate the following two types
of coupling between electrons and magnons:

$\displaystyle H_{int}=$
$\displaystyle-\frac{1}{\sqrt{NN^{\prime}}}\sum_{mnl}J_{mnl}\sum_{i}c^{\dagger}_{i,m}c_{i,n}a_{i,l}$
(2)
$\displaystyle-\frac{1}{\sqrt{NN^{\prime}}}\sum_{mnl}\sum_{i}J^{\prime}_{i,mnl}c^{\dagger}_{i,m}c_{i,n}a_{i,l}+h.c.,$

which includes translationally invariant (i.e. space independent) flavor
random SYK-liked Yukawa coupling and spatially dependent one, respectively.
Where the space independent (dependent) random Yukawa couplings are chosen
from a Gaussian unitary ensemble with zero average and variance
$\overline{J_{mnl}J_{m^{\prime}n^{\prime}l^{\prime}}}=|J|^{2}\delta_{mm^{\prime}}\delta_{nn^{\prime}}\delta_{ll^{\prime}}$
($\overline{J^{\prime}_{i,mnl}J^{\prime}_{i^{\prime},m^{\prime}n^{\prime}l^{\prime}}}=|J^{\prime}|^{2}\delta_{ii^{\prime}}\delta_{mm^{\prime}}\delta_{nn^{\prime}}\delta_{ll^{\prime}}$),
and $J_{mnl}=J^{*}_{nml}$ ($J^{\prime}_{mnl}=J^{\prime*}_{nml}$). We assume
that each sample flows toward a universal low-energy theory.

The electrons and spin couplings lead to the phase transition from $U(1)$
phase to Bose-Einstein condensation (BEC) phase. The critical point of the
phase transition at zero temperature is $D_{0}(0,Q)^{-1}=\Pi(0,Q)$, where
$D_{0}(0,Q)$ is the bare propagator of magnons and $\Pi(0,Q)$ is the self-
energy of magnons introduced by the interactions. Different hopping terms and
crystal lattices in Eq. (1) can lead to rich band structures. We primarily
consider the influence of scattering near VHPs, where the electron’s
dispersion relation exhibits either a maximum or a saddle-point on the Fermi
surface, because VHS can lead to anomalous low-energy behavior. It can be
anticipated that when filled around the Fermi surface with VHS, the physical
properties are primarily determined by VHPs and are not significantly
influenced by the overall shape of the Fermi surface EP . When Taylor
expanding the electron’s energy spectrum at the VHPs, the first-order momentum
term vanishes, and the second-order term appears. The saddle-point nature
tells us that it can always be transformed into
$\varepsilon_{k}=k_{x}^{2}+ak_{y}^{2}$ with $a<0$ through a coordinate
transformation EnE ; Isobe  _et al._ (2018); Yao and Yang (2015), Where
$k_{x}$ and $k_{y}$ are two orthogonal momentum directions after the
coordinate transformation and the coefficient in front of $k_{x}^{2}$ is set
to 1 through a reparameterization of the operator $c_{k}$. We only calculated
the nematic order, which the order parameter momentum condenses at $Q=0$ and
the critical point is $\Delta_{c}=\Pi(0,0)$, and subsequently demonstrate that
our results also apply to density-wave orders (i.e. $Q\neq 0$). The divergence
of the density of state at the VHP allows us to focus only on a small patch
near it, the nematic order compels us to consider interactions within the same
patch only.

When considering only the translationally invariant SYK-liked Yukawa
interaction (i.e. $J^{\prime}=0$), we obtain the self-energy of electrons
($\Sigma$) and magnons ($\Pi$) at zero temperature and low-frequency limit
(detailed derivations are shown in Supplemental Material SM ):

$\displaystyle\Pi(i\Omega_{m},q)-\Pi(0,0)\sim|J|^{2}\frac{N}{N^{\prime}}f(q_{x},q_{y},-a)|\Omega_{m}|,$
(3)
$\displaystyle\Sigma(i\omega_{n},0)-\Sigma(0,0)\sim-i|J|\sqrt{\frac{N^{\prime}}{N}}\mathrm{sgn}(\omega_{n})|\omega_{n}|^{1/2},$

where
$f(q_{x},q_{y},-a)=(|1-\sqrt{-a}\frac{q_{y}}{|q_{x}|}|-|1+\sqrt{-a}\frac{q_{y}}{|q_{x}|}|)/8\pi\sqrt{-a}(q_{x}^{2}+aq_{y}^{2})$.
The quasiparticle decay exhibits the scaling of NFL behavior. Compared to the
quasiparticle decay results
$\mathrm{Im}\Sigma(i\Omega_{n},0)\sim\mathrm{sgn}(\omega_{n})|\omega_{n}|^{2/3}$
Patel  _et al._ (2023); Rao and Piazza (2023); Esterlis  _et al._ (2021); Guo
_et al._ (2022); Holder and Metzner (2014) obtained by considering scattering
between patches around the hot spots in circular Fermi surface, which are far
from the VHPs and the electrons dispersion is $\varepsilon_{k}=\pm
v_{F}k_{\perp}+\kappa k_{\parallel}^{2}/2$, scattering of patch near the VHP
suppresses it and takes a dominant role. So, it is expected that when
considering scattering over the entire Fermi surface with VHS, the
quasiparticle decay rate will exhibits $\sim|\omega_{n}|^{1/2}$ NFL scaling.

When considering only the spatially dependent interaction (i.e. $J=0$), we
obtain the self-energy of electrons and magnons at zero temperature and low-
frequency limit (detailed derivations are shown in Supplemental Material SM ):

$\displaystyle\Pi(i\Omega_{m})-\Pi(0)\sim-|J^{\prime}|^{2}\frac{N}{N^{\prime}}|\Omega_{m}|,$
(4)
$\displaystyle\Sigma(i\omega_{n})-\Sigma(0)\sim-i|J^{\prime}|^{2}\omega_{n}\ln\frac{N^{\prime}}{N|\omega_{n}|}.$

This quasiparticle decay exhibits MFL behavior and the specific heat is $\sim
T\ln(1/T)$ Affleck and Ludwig (1993) with temperature $T$. Then we use the
Kubo formula to calculate the direct current (dc) conductivity at
$\mathcal{O}(|J^{\prime}|^{2})$. The related current-current corrections are
shown in Fig. 1(b)-(d). Retaining up to the lowest order in frequency, the
system resistivity SM  (i.e., the reciprocal of dc conductivity) is

$\mathrm{Re}[\frac{1}{\sigma(\Omega\gg
T)}]=\frac{16\pi^{4}(-a)|\Omega|}{|J^{\prime}|^{2}N\Lambda_{U}^{2}\ln\Lambda},$
(5)

where $\Lambda_{U}$ is the momentum UV cutoff and $\Lambda$ is the ratio of
the momentum UV cutoff to the frequency. Only self-energy correction in Fig.
1(c) contributes in this case. We only considered forward scattering of the
patch near same VHP, which results in one-loop contribution Fig. 1(b) being
zero. Meanwhile, the dispersion relation of electrons at the VHPs only
contains the momentum squared term, ensuring that the vertex correction in
Fig. 1(d) is zero. At the limit of $\Omega\ll T$, this will result in a
$T$-linear resistivity in the dc limit, with no residual constant term. The
$T(\ln 1/T)$ specific heat and $T$-linear resistivity are consistent with the
behavior of strange metals.

We can also obtain the above results of quasiparticle decay through the method
of scaling analysis. After considering the VHS, the effect Lagrangian of our
theory (1) is

$\mathcal{L}=\psi^{\dagger}_{m}[\partial_{\tau}-\partial_{x}^{2}-a\partial_{y}^{2}]\psi+(\partial\phi_{l})^{2}+\mathcal{H}_{int}$
(6)

with
$\mathcal{H}_{int1}=\sum_{mnl}J_{mnl}\psi^{\dagger}_{m}\psi_{n}\phi_{l}/\sqrt{NN^{\prime}}$
and
$\mathcal{H}_{int2}=\sum_{mnl}J^{\prime}_{mnl}(\vec{x})\psi^{\dagger}_{m}\psi_{n}\phi_{l}/\sqrt{NN^{\prime}}$.
If we assume the scaling dimension of space to be $[x]=[y]=1$, and time to be
$[\tau]=z$, respectively. Then the scaling dimensions for the Fermi and Bose
fields are both $[\psi]=[\phi]=z/2$. When we only consider the spatially
independent interaction, we obtained the scaling dimension of the variance $J$
is $2-z/2$. NFL/MFL behavior occurs near the critical point, where we expect
the interaction to be marginal, so $z=4$. This leads to the quasi-particle
decay $\sim\omega^{1/2}$ and consists with the analytical results in Eq. (3).
Similarly, when considering only spatially dependent interaction, the scaling
dimension of $J^{\prime}$ is $1-z/2$, leading to $z=2$ and resulting in quasi-
particle damping $\sim\omega$. This differs slightly from our analytical
result $\omega\ln(1/\omega)$ in Eq. (4), but since the dominant role is played
by the preceding linear term, it is also considered consistent. Our analytical
calculations only considered scattering between the same VHP (i.e. Ising
order). If we consider density-wave order with finite wave vector $\vec{Q}$,
we need to account for scattering between patches around different VHPs,
introducing more fermion modes such as
$\psi^{\dagger}_{m1}[\partial_{\tau}-\partial_{x}^{2}-a\partial_{y}^{2}]\psi_{m1}$
and considering the interactions,
$\psi^{\dagger}_{m}\psi_{m1}\phi_{l}e^{i\vec{Q}\vec{x}}$, between them.
However, the scaling dimensions of the fields and couplings remain unchanged,
so the same dynamic critical exponent, i.e., the same quasi-particle decay,
will be obtained. This has already been verified Rao and Piazza (2023);
Esterlis  _et al._ (2021) in the case considering only scattering at the Fermi
surface away from VHPs.

Figure 2: (a) The coefficients of different scaling as a function of the
variance $|J^{\prime}|$ of the disordered interaction. The black, red, and
blue solid lines represent the coefficients of scaling of $|\omega|^{1/2}$,
$\omega$, and $\ln(|\omega|)$, respectively. The green dashed line represents
the critical $|J^{\prime}|$ of two analytical limits. (b) The imaginary part
of electron self-energy as a function of frequency $\omega_{n}$ when
$|J^{\prime}|=0.2$ and $1$. The other parameters in (a) and (b) are: $a=-1$,
$\Lambda_{U}=300$, $\omega_{n}=21\pi$, $N=N^{\prime}$ and $|J|=2$. (c) A
schematic diagram of SC order and PDW order. The blue region represents the
filled Fermi surface, red dots denote VHPs, and the surrounding red areas
indicate the patches near them. The blue pairings represent conventional SC
pairs formed near the same VHP, while the yellow pairings represent PDW pairs
formed between different VHPs.

In the presence of both types of interactions (i.e. $J\neq 0\neq J^{\prime}$),
the self-energy of magnons at zero temperature and low-frequency is the sum of
the self-energy of magnons in (3) and (4). The self-energy of electrons is
shown in Supplemental Material SM , different scalings are competing with each
other. If we only consider the behavior of large $N$($N^{\prime}$) and ignore
the differences in energy scales. When $N\gg N^{\prime}$, it exhibits
quasiparticle decay rate
$-\frac{|J^{\prime}|^{2}\ln\Lambda}{2\pi^{4}\sqrt{-a}}[\mathrm{ArcTan}(\sqrt{-a})+\mathrm{ArcTan}(\frac{1}{\sqrt{-a}})]\omega_{n}\ln\frac{e\pi^{3}(-a)N^{\prime}\Lambda_{U}^{2}}{N(\ln\Lambda)^{2}|J^{\prime}|^{2}|\omega_{n}|}$,
and it is the scaling of MFL, it will also lead to the $T\ln(1/T)$ specific
heat and the $T$-linear resistivity in the direct current limit; when $N\ll
N^{\prime}$, the scaling of NFL dominates and the quasiparticle decay rate
$\sim|\omega_{n}|^{1/2}$. When $N\approx N^{\prime}$ and considering only the
difference in energy scales, the situation becomes extremely complicated due
to the appearance of logarithmic terms. As shown in Fig. S2(a) and S2(b), when
$|J^{\prime}|\ll|J|$, it exhibits the scaling of NFL. As $|J^{\prime}|$
increases, there is a crossover to the scaling of MFL. Despite the appearance
of the scaling with $\ln|\omega|$ in the analytical expression (appears on the
right side of the green dashed line in Fig. S2(a)), it is always suppressed by
the scaling of MFL.

Figure 3: Finite-temperature phase diagram of model (1) for three different
values of the SYK-liked Yukawa interactions. The phase transition temperature
$T_{c}$ to SC is shown in blue. The transition temperature $T_{BEC}$ from the
$U(1)$ phase to the BEC phase in TMI is indicated in red. Above the zero-
temperature quantum critical point $\Delta_{c}$, a schematic depiction in
purple illustrates the NFL/MFL region induced by Yukawa coupling. The
parameters are: $a=-1$, $\Lambda_{U}=300$ and (a) $|J|/\sqrt{N}=2$,
$|J^{\prime}|/\sqrt{N}=0$, (b) $|J|/\sqrt{N}=0$, $|J^{\prime}|/\sqrt{N}=0.6$
(c) $|J|/\sqrt{N}=2$, $|J^{\prime}|/\sqrt{N}=0.2$.

In experiments/numerical simulations Gerlach  _et al._ (2017); Driskell  _et
al._ (2021), the fluctuations of the critical magnons couple with the Fermi
surface and result in the ground state of NFL/MFL, dominating the properties
of the system within a range of temperature above the critical points. On the
other hand, in the low-temperature region near the critical point, a SC phase
is also generated. The two effects are of comparable strength and it is not
clear which dominates in previous studies. However, in our model, large-$N$
can serve as an effective controllable parameter, and vertex correlations are
suppressed by $1/N$ SM . The phase diagrams of our model (1) are shown in Fig.
3.

When only translationally invariant Yukawa interaction is present as shown in
Fig. 3(a), the critical tuning parameter $\Delta_{c}(T)$ of the $U(1)$ phase
and the BEC phase in TMI at finite temperature decreases as the increasing of
the temperature. A SC transition is established by solving the linearized gap
equation SM . The critical temperature $T_{C}$ of SC increases with the
increasing variance $|J|$ of the translationally invariant Yukawa coupling and
satisfies $T_{C}\sim(|J|/\sqrt{N})^{3}$ (see the Supplemental Material SM ).
The schematic NFL region is also given above the quantum critical point
$\Delta_{c}$. When only disordered Yukawa interaction is present as shown in
Fig. 3(b), the critical tuning parameter $\Delta_{c}(T)\sim T$ exhibits nearly
linear growth with increasing temperature, and the mean-field exponents
$\nu=1/2$ Hertz (1976). It confirms the conclusion that the self-energy of
magnons acquires thermal tuning parameter, $\Delta(T)\sim T$, when analyzing
transport properties SM . This is significantly different from considering
only pervious Yukawa coupling. The critical temperature of SC,
$T_{C}\sim\mathrm{e}^{-\sqrt{N}/|J^{\prime}|}$, exponentially increases with
the increasing variance $|J^{\prime}|$ of the disordered SYK-liked Yukawa
coupling (see the Supplemental Material SM ). When considering both
interactions, as shown in Fig. 3(c), the critical temperature $T_{BEC}$ in the
low-temperature region is determined by the interaction with spatial
dependence, while in the high-temperature region, it is determined by the
quenched disorder component. Therefore, the method of obtaining the
temperature dependence at finite temperatures ($\gg\omega$) from the frequency
$\omega$ dependence at zero temperature (i.e. $T\ll\omega$) is also
applicable. In the case of $|J|=2$ and $|J^{\prime}|=0.2$, the critical
temperature $T_{C}$ of SC is dominated by the behavior of $|J|$. Fluctuations
associated with the nematic phase, particularly near a quantum critical point,
can enhance the superconducting transition temperature $T_{C}$ Lederer  _et
al._ (2017, 2015).

If there are multiple VHPs on the Fermi surface of materials, then the main
contributions also include scattering between patches around different VHPs.
As indicated by the previous scaling analysis, near the critical point, the
frequency-dependent behavior of quasiparticle damping is consistent with the
case when considering the scattering within the same patch. However,
scattering between different patches could result in a constant contribution
to the dc conductivity SM . For real materials, symmetry is broken due to
impurities or frustrations, the energy spectrum after the renormalization of
higher-order diagrams always satisfies $ab^{2}\neq-1$ (the meaning of the
parameters is explained in the Supplemental Material SM ). Therefore, the
constant dc conductivity can always exist and we argue that this can
contribute to the constant components in resistivity. In addition, we can also
consider the superconducting pairing term
$\Delta_{ij}(r)e^{i(K_{1}+K_{2})R}\psi_{1i}^{\dagger}(x)\psi_{2j}^{\dagger}(y)$
[$R=(x+y)/2$ the position of the center of mass, $r=x-y$ the relative
position, and $i$ and $j$ are flavor indices]. $\psi_{1}^{\dagger}(k)$ and
$\psi_{2}^{\dagger}(k)$ are electron creation operators from two patches
around different VHPs with momentum $K_{1}$ and $K_{2}$ (as shown in Fig.
S2(c)), respectively. This PDW order with condensed momentum $K_{1}+K_{2}$
($\neq$ the reciprocal lattice vectors $G$) will compete with the normal SC,
and is believed to exist in the heavy-fermion SC Radovan  _et al._ (2003) and
cuprate SC Agterberg  _et al._ (2020).

The scalar field in theory (1) is considered as magnons, which can also be
viewed as order parameters or fractionalized particles in the two-dimensional
materials. The Yukawa interactions (2) can be obtained by performing a
Hubbard-Stratonovich transformation on the Hubbard term. And our method is
also applicable to any type of lattice and hopping range, as long as there are
VHS on the Fermi surface. The effective theory of these two-dimensional
materials can also be described by Lagrangian (6).

We have shown that in two-dimensional materials/surface with Yukawa
interactions, the VHS on the quasi-particle Fermi surface suppress regions far
from the VHPs, making a significant contribution to NFL. Considering
impurities can explain the $T\ln(1/T)$ specific heat and the linear
temperature dependence of resistivity in strange metals. Moreover, tuning the
strength of two types of interactions can lead to a crossover from NFL to MFL.
In addition, we present the SC phase induced by VHS, revealing that the
exponent of the SC critical temperature depends on the disorder interaction,
and the power-law dependence relies on the translationally invariant one.

Our theory provides a new mechanism to explain NFL behavior and strange metal
phenomena, and can be probed using real material with VHS, which are commonly
present in recent popular twisted bilayer materials Li  _et al._ (2010); Yan
_et al._ (2012); Yin  _et al._ (2016); Xu  _et al._ (2021b). It is noteworthy
that a recent work in Ref. Rao and Piazza (2023) proposed a new experimental
scheme using cavity quantum electrodynamics to detect NFL/MFL, and we
anticipate that our theory can also be probed using similar methods. Future
research will focus on determining the boundaries of the NFL/MFL region and
considering the effects of fluctuations brought about beyond the Migdal-
Eliashberg franework Marsiglio (2020).

###### Acknowledgements.

This work was supported by National Key R&D Program of China under grants No.
2021YFA1400900, 2021YFA0718300, 2021YFA1402100, NSFC under grants Nos.
12174461, 12234012, 12334012, 52327808, Space Application System of China
Manned Space Program. ECM has been partially supported by CNPq, FAPARJ and
CAPES.

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##  S1 Self Energy

###  S1.1 Translationally Invariant SYK-liked Yukawa Interaction

Firstly, we consider translationally invariant flavor-random Yukawa
interaction:

$\mathcal{H}_{int1}=-\sum_{mnl}\frac{J_{mnl}}{\sqrt{NN^{\prime}}}\sum_{i}c^{\dagger}_{i,m}c_{i,n}a_{i,l}+h.c.,$
(S1)

where $i$ is lattice sites, $m,n=1...N$ are the flavors of fermion field and
$l=1...N^{\prime}$ is the flavors of the scalar field. The space independent
coupling $J_{mnl}$ is random in the space of flavors with Gaussian
distribution:

$\overline{J_{mnl}}=0,\ \ \ \
\overline{J_{mnl}J^{*}_{m^{\prime}n^{\prime}l^{\prime}}}=|J|^{2}\delta_{mm^{\prime}}\delta_{nn^{\prime}}\delta_{ll^{\prime}}.$
(S2)

The boson self energy caused by scattering between patch near the van Hove
point is:

$\Pi_{J}(i\Omega_{m},\vec{q})=-|J|^{2}\frac{N}{N^{\prime}}T\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{i\omega_{n}-k_{x}^{2}-ak_{y}^{2}}\frac{1}{i\omega_{n}+i\Omega_{m}-(k_{x}+q_{x})^{2}-a(k_{y}+q_{y})^{2}}.$
(S3)

The scattering between different patches near van Hove points are considered
in the manuscript. Sum over Matsubara frequency and takes $T\rightarrow 0$,
this gives:

$|J|^{2}\frac{N}{N^{\prime}}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{2(q_{x}^{2}+aq_{y}^{2}-2k_{x}q_{x}-2ak_{y}q_{y})}{(q_{x}^{2}+aq_{y}^{2}-2k_{x}q_{y}-2ak_{y}q_{y})^{2}+\Omega_{m}^{2}},$
(S4)

The integral domain of $d^{2}k$ is $-\sqrt{-a}|k_{y}|<k_{x}<\sqrt{-a}|k_{y}|$.
We first integrate with respect to $k_{x}$, and then integrate over $k_{y}$,
get:

$|J|^{2}\frac{N}{N^{\prime}}\frac{1}{4\pi^{2}}[\frac{|\Omega_{m}|}{|q_{x}||aq_{y}-\sqrt{-a}|q_{x}||}\mathrm{ArcTan}(\frac{q_{x}^{2}+aq_{y}^{2}}{|\Omega_{m}|})-\frac{|\Omega_{m}|}{|q_{x}||aq_{y}+\sqrt{-a}|q_{x}||}\mathrm{ArcTan}(\frac{q_{x}^{2}+aq_{y}^{2}}{|\Omega_{m}|})].$
(S5)

In the integral above for $k_{y}$, we only considered contributions near the
Fermi surface and neglect contributions that lead to logarithmic divergences
$\sim\mathrm{ln}\Lambda_{U}$, where $\Lambda_{U}$ is the momentum cutoff. The
contributions far from the Fermi surface can be absorbed by $\Pi(0,0)$, while
the remaining part is expressed as powers of
$|\Omega_{m}|^{2}/(q_{x}^{2}+aq_{y}^{2})$ (similar methods can be found in
Refs.Holder and Metzner (2014); Rao and Piazza (2023)). In the low-frequency
limit $|q_{x}^{2}+aq_{y}^{2}|\gg|\Omega_{m}|$ and the high-frequency limit
$|\Omega_{m}|\gg|q_{x}^{2}+aq_{y}^{2}|$, we have, respectively,

$\begin{matrix}|J|^{2}\frac{N}{N^{\prime}}\frac{1}{8\pi\sqrt{-a}}\frac{|\Omega_{m}|}{aq_{y}^{2}+q_{x}^{2}}(|1-\sqrt{-a}\frac{q_{y}}{|q_{x}|}|-|1+\sqrt{-a}\frac{q_{y}}{|q_{x}|}|),&\mathrm{when}\
|q_{x}^{2}+aq_{y}^{2}|\gg|\Omega_{m}|\\\
|J|^{2}\frac{N}{N^{\prime}}\frac{1}{4\pi^{2}\sqrt{-a}}\frac{\mathrm{sgn}(q_{x}^{2}+aq_{y}^{2})}{|q_{x}|}(||q_{x}|-\sqrt{-a}q_{y}|-||q_{x}|+\sqrt{-a}q_{y}|)&\mathrm{when}\
|q_{x}^{2}+aq_{y}^{2}|\ll|\Omega_{m}|\end{matrix}.$ (S6)

These limits are all achievable. For example, in the low-frequency limit,
although $q_{x}^{2}+aq_{y}^{2}$ can be zero due to the presence of zero
points, as long as the frequency convergence to zero is faster than the
momentum approaching zero point, $|\Omega_{m}|/|q_{x}^{2}+aq_{y}^{2}|\ll 1$
and holds true.

We only take the low-frequency limit in our calculations. The boson self
energy can be expressed in a simpler form based on the division of different
regions in the $q_{x}-q_{y}$ plane:

$\Pi_{J}(i\Omega_{m},\vec{q})=\left\\{\begin{matrix}-|J|^{2}\frac{N}{N^{\prime}}\frac{1}{4\pi\sqrt{-a}}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}&(-\sqrt{-a}q_{y}<q_{x}<\sqrt{-a}q_{y}\
\&\ q_{y}>0)\equiv\mathbb{A}\\\
|J|^{2}\frac{N}{N^{\prime}}\frac{1}{4\pi\sqrt{-a}}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}&(\sqrt{-a}q_{y}<q_{x}<-\sqrt{-a}q_{y}\
\&\ q_{y}<0)\equiv\mathbb{B}\\\
-|J|^{2}\frac{N}{N^{\prime}}\frac{q_{y}}{4\pi|q_{x}|}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}&(-\frac{|q_{x}|}{\sqrt{-a}}<q_{y}<\frac{|q_{x}|}{\sqrt{-a}})\equiv\mathbb{C}\end{matrix}\right.,$
(S7)

where we label the three different regions as $\mathbb{A}$, $\mathbb{B}$ and
$\mathbb{C}$, respectively. We argue that non-Fermi-liquid of marginal-Fermi-
liquid occur above the critical point of bosons, so fermions couple with
critical bosons, i.e., satisfy $\Delta=\Pi(0,0)$ (only consider the Ising
order). We define
$\tilde{J}^{2}\equiv|J|^{2}\frac{N}{N^{\prime}}\frac{1}{4\pi\sqrt{-a}}$ as
simplify.

The fermion self energy can be written as:

$\displaystyle\Sigma_{J}(i\omega_{n},0)=$
$\displaystyle|J|^{2}T\sum_{m}[\int_{\mathbb{A}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}+\int_{\mathbb{B}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}-\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}$
(S8)
$\displaystyle+\int_{\mathbb{C}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{\sqrt{-a}q_{y}}{|q_{x}|}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}].$

We will provide the steps for integrating over regions $\mathbb{A}$ and
$\mathbb{C}$, and the integration over region $\mathbb{B}$, which is similar
to region $\mathbb{A}$, will be omitted:

$\int_{\mathbb{A}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}\approx\frac{1}{2\pi^{2}}\int_{0}^{\infty}dq_{y}\frac{1}{q_{y}^{2}-\tilde{J}^{2}\frac{|\Omega_{m}|}{-aq_{y}^{2}}}\frac{\mathrm{ArcTanh}(\frac{\sqrt{-a}q_{y}}{\sqrt{i\omega_{n}+i\Omega_{m}-aq_{y}^{2}}})}{\sqrt{i\omega_{n}+i\Omega_{m}-aq_{y}^{2}}},$
(S9)

where we neglect the $q_{x}$ component of the damped boson propagator in
region $\mathbb{A}$, because the main contribution of the integral is
concentrated near the positive $q_{y}$-axis. When the integration region
approaches the boundary of the $\mathcal{A}$, the denominator of the boson
propagator diverges. In low-frequency limit $|\omega_{n}+\Omega_{m}|\ll-
aq_{y}^{2}$, we need to define a branch cut on the negative $x$-axis to
calculate the square root,
$\sqrt{i\omega_{n}+i\Omega_{m}-aq_{y}^{2}}\approx\sqrt{-a}q_{y}\mathrm{e}^{i\mathrm{sgn}(\omega_{n}+\Omega_{m})\frac{\eta}{2}}$
with $\eta\equiv|\omega_{n}+\Omega_{m}|/(-a)q_{y}^{2}\rightarrow 0^{+}$, and
logarithmic function. The exponential function
$\mathrm{e}^{i\mathrm{sgn}(\omega_{n}+\Omega_{m})\frac{\eta}{2}}$ is essential
because the integral from $0$ to $\infty$ concerning $1/(q_{y}^{3}-1/q_{y})$
diverges, but transformation
$\sqrt{-a}\rightarrow\sqrt{-a}\mathrm{e}^{i\mathrm{sgn}(\omega_{n}+\Omega_{m})\frac{\eta}{2}}$
can be performed to avoid divergence, resulting in:

$-\frac{1}{32|\tilde{J}||\Omega_{m}|^{\frac{1}{2}}}[1+i\mathrm{sgn}(\omega_{n}+\Omega_{m})\ln\Lambda].$
(S10)

The $\ln\Lambda\approx\ln 4(-a)\Lambda_{U}^{2}/|\omega_{n}+\Omega_{m}|$ at the
low-frequency limit, where $\Lambda_{U}$ is the UV cut off of the momentum.
Similarly, the integral over the domain $\mathbb{B}$ yields:

$-\frac{1}{32|\tilde{J}||\Omega_{m}|^{\frac{1}{2}}}[\mathrm{sgn}(\omega_{n}+\Omega_{m})i-\ln\Lambda].$
(S11)

The integral over the domain $\mathbb{C}$ (a better approximation is the boson
propagator with $q_{y}^{2}\ll q_{x}^{2}$ in domain $\mathbb{C}$):

$\displaystyle\int_{\mathbb{C}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{\sqrt{-a}q_{y}}{|q_{x}|}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}$
(S12) $\displaystyle\approx$
$\displaystyle\frac{1}{\pi^{2}\sqrt{-a}}\int_{0}^{\infty}dq_{x}\frac{\frac{1}{\sqrt{i\omega_{n}+i\Omega_{m}-q_{x}^{2}}}\mathrm{ArcTan}(\frac{q_{x}}{\sqrt{i\omega_{n}+i\Omega_{m}-q_{x}^{2}}})+\frac{\tilde{J}^{2}|\Omega_{m}|}{q_{x}^{5}}\mathrm{ArcTanh}(\frac{\tilde{J}^{2}|\Omega_{m}|}{q_{x}^{4}})}{q_{x}^{2}+\frac{\tilde{J}^{4}|\Omega_{m}|^{2}}{q_{x}^{8}}(i\omega_{n}+i\Omega_{m}-q_{x}^{2})}$
$\displaystyle\approx$
$\displaystyle\frac{1}{32\sqrt{-a}|\tilde{J}||\Omega_{m}|^{\frac{1}{2}}}[-1+i\mathrm{sgn}(\omega_{n}+\Omega_{m})\ln\Lambda]-\frac{1}{32\sqrt{-a}|\tilde{J}||\Omega_{m}|^{\frac{1}{2}}}[\mathrm{sgn}(\omega_{n}+\Omega_{m})i+\ln\Lambda]-\frac{1}{2\pi^{2}\sqrt{-a}\Lambda_{I}},$

where we use the approximation $\tilde{J}|\Omega_{m}|^{1/2}\ll q_{x}^{2}$ when
dealing with $\mathrm{ArcTanh}(\tilde{J}^{2}|\Omega_{m}|/q_{x}^{4})$, which
leads to an infrared divergence. We choose an infrared cutoff
$\Lambda_{I}\gg\tilde{J}|\Omega_{m}|^{1/2}$ and it makes no contribution to
the imaginary part final result after the frequency integration and, under the
minimal subtraction renormalization scheme, has no contribution to the real
part either. Another way to avoid the infrared divergence is
$\tilde{J}|\Omega_{m}|^{1/2}\gg q_{x}^{2}$, which, after frequency
integration, yields the same imaginary part of the fermion self-energy.

At zero temperature, the integral over frequencies yields:

$\Sigma_{J}(i\omega_{n},0)-\Sigma_{J}(0,0)\approx-i\frac{|J|\ln\Lambda}{8\sqrt{\pi}}\sqrt{\frac{N^{\prime}}{N}}[(-a)^{1/4}-(-a)^{-1/4}]\mathrm{sgn}(\omega_{n})|\omega_{n}|^{1/2}-i\frac{|J|}{8\sqrt{\pi}}\sqrt{\frac{N^{\prime}}{N}}[(-a)^{1/4}+(-a)^{-1/4}]\mathrm{sgn}(\omega_{n})|\omega_{n}|^{1/2},$
(S13)

where we use the minimal subtraction renormalization scheme to eliminate the
divergence in the real part. When $a=-1$, Eq. (S13) becomes

$\Sigma_{J}(i\omega_{n},0)-\Sigma_{J}(0,0)\approx-i\frac{|J|}{8\sqrt{\pi}}\sqrt{\frac{N^{\prime}}{N}}\mathrm{sgn}(\omega_{n})|\omega_{n}|^{1/2},$
(S14)

the divergence $\ln\Lambda$ has been offset.

Anyway, we obtained $\mathrm{Im}\Sigma(i\omega_{n},0)\sim|\omega_{n}|^{1/2}$,
just as in the scaling analysis.

###  S1.2 Disordered SYK-liked Yukawa Interaction

The disordered flavor-random Yukawa interaction is

$\mathcal{H}_{int2}=-\frac{1}{\sqrt{NN^{\prime}}}\sum_{mnl}\sum_{i}J^{\prime}_{i,mnl}c^{\dagger}_{i,m}c_{i,n}a_{i,l}+h.c.,$
(S15)

where $i$ is lattice sites, $m,n=1...N$ are the flavors of fermion field and
$l=1...N^{\prime}$ is the flavors of the scalar field. The space dependent
coupling $J^{\prime}_{i,mnl}$ is random in the space of flavors with Gaussian
distribution:

$\overline{J^{\prime}_{i,mnl}=0},\ \ \ \
\overline{J^{\prime}_{i,mnl}J^{\prime}_{i^{\prime},m^{\prime}n^{\prime}l^{\prime}}}=|J^{\prime}|^{2}\delta_{ii^{\prime}}\delta_{mm^{\prime}}\delta_{nn^{\prime}}\delta_{ll^{\prime}}.$
(S16)

The boson self energy caused by scattering between patch near the van Hove
point is:

$\displaystyle\Pi_{J^{\prime}}(i\Omega_{m})-\Pi_{J^{\prime}}(0)$
$\displaystyle=-|J^{\prime}|^{2}\frac{N}{N^{\prime}}T\sum_{n}\int\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{i\omega_{n}-q_{x}^{2}-aq_{y}^{2}}\int\frac{d^{2}k}{(2\pi)^{2}}[\frac{1}{i\omega_{n}+i\Omega_{m}-k_{x}^{2}-ak_{y}^{2}}-\frac{1}{i\omega_{n}-k_{x}^{2}-ak_{y}^{2}}]$
(S17)
$\displaystyle\approx-\frac{|J^{\prime}|^{2}(\ln\Lambda)^{2}N}{(-a)\pi^{3}N^{\prime}}|\Omega_{m}|,$

where we have used an approach $\ln\Lambda\approx\ln
4(-a)\Lambda_{U}^{2}/|\omega_{n}|$ as a constant with the momentum UV cutoff
$\Lambda_{U}$ at the low-frequency limit. The numerical calculation yields
$\ln\Lambda\approx\ln 8(-a)\Lambda_{U}^{2}/|\Omega_{m}|$, and at the linear
order level, we take it as a truncation constant. The critical fluctuations of
bosonic degrees of freedom are used to eliminate electronic quasiparticles, so
non-Fermi-liquid/marginal-Fermi-liquid behavior occurs near the phase
transition critical point $\Pi_{J^{\prime}}(0)=\Delta$ at zero temperature. We
define $\tilde{J}^{\prime
2}\equiv|J^{\prime}|^{2}N(\ln\Lambda)^{2}/(-a)\pi^{3}N^{\prime}$ as simplify.

The fermion self energy can be written as:

$\Sigma_{J^{\prime}}(i\omega_{n})-\Sigma_{J^{\prime}}(0)=|J^{\prime}|^{2}T\sum_{m}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|}\int\frac{d^{2}q}{(2\pi)^{2}}[\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}-\frac{1}{i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}],$
(S18)

where we use the minimal subtraction renormalization scheme to eliminate the
frequency divergence in the real part and only obtain the imaginary part of
the Fermi self-energy:

$\Sigma_{J^{\prime}}(i\omega_{n})-\Sigma_{J^{\prime}}(0)=-i\frac{|J^{\prime}|^{2}\ln\Lambda}{4\pi^{3}\sqrt{-a}}\omega_{n}\ln(\frac{e\Lambda_{U}^{2}}{\tilde{J}^{\prime
2}|\omega_{n}|}),$ (S19)

where $\Lambda_{U}$ is the momentum UV cutoff for the first integral in Eq.
(S18).

###  S1.3 Both translationally invariant and disordered SYK-liked Yukawa
coupling

If we consider both translationally invariant and disordered flavor-random
Yukawa coupling (S1) and (S15). The boson self energy can be obtained from
previous results:

$\Pi(i\Omega_{m},\vec{q})-\Pi(0,0)=|J|^{2}\frac{N}{N^{\prime}}\frac{1}{8\pi\sqrt{-a}}\frac{|\Omega_{m}|}{aq_{y}^{2}+q_{x}^{2}}(|1-\sqrt{-a}\frac{q_{y}}{|q_{x}|}|-|1+\sqrt{-a}\frac{q_{y}}{|q_{x}|}|)-\tilde{J}^{\prime
2}|\Omega_{m}|.$ (S20)

Then the fermion self energy can be written as
$\Sigma(i\omega_{n},0)=\Sigma_{1}(i\omega_{n},0)+\Sigma_{2}(i\omega_{n})$ with

$\displaystyle\Sigma_{1}(i\omega_{n},0)=$
$\displaystyle|J|^{2}T\sum_{m}[\int_{\mathbb{A}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}$ (S21)
$\displaystyle+\int_{\mathbb{B}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}-\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}$
$\displaystyle+\int_{\mathbb{C}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{\sqrt{-a}q_{y}}{|q_{x}|}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}],$

and

$\displaystyle\Sigma_{2}(i\omega_{n},0)=$
$\displaystyle|J^{\prime}|^{2}T\sum_{m}[\int_{\mathbb{A}}\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}+\tilde{J}^{2}\frac{|\Omega_{m}|}{k_{x}^{2}+ak_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}+\int_{\mathbb{B}}\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}-\tilde{J}^{2}\frac{|\Omega_{m}|}{k_{x}^{2}+ak_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}$ (S22)
$\displaystyle+\int_{\mathbb{C}}\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}+\tilde{J}^{2}\frac{\sqrt{-a}k_{y}}{|k_{x}|}\frac{|\Omega_{m}|}{k_{x}^{2}+ak_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}]\int\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}.$

Similar to the treatment in the section 1.1, we can obtain the integrals for
each region:

$\displaystyle\int_{\mathbb{A}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}+\int_{\mathbb{B}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}-\tilde{J}^{2}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}$ (S23)
$\displaystyle\approx$
$\displaystyle\frac{1}{\pi^{2}}\int_{0}^{\infty}dq_{y}\frac{1}{q_{y}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|-\frac{\tilde{J}^{4}|\Omega_{m}|^{2}}{a^{2}q_{y}^{4}(q_{y}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|)}}\frac{\mathrm{ArcTanh}(\frac{\sqrt{-a}q_{y}}{\sqrt{i\omega_{n}+i\Omega_{m}-aq_{y}^{2}}})}{\sqrt{i\omega_{n}+i\Omega_{m}-aq_{y}^{2}}}$
$\displaystyle\approx$
$\displaystyle\left\\{\begin{matrix}\frac{1}{32|\tilde{J}||\Omega_{m}|^{\frac{1}{2}}}[-1-i\mathrm{sgn}(\omega_{n}+\Omega_{m})][1+i\mathrm{sgn}(\omega_{n}+\Omega_{m})\ln\Lambda]&\
\ \ \tilde{J}^{2}\gg(-a\tilde{J}^{\prime 4}|\Omega_{m}|)\\\
\frac{1}{16\sqrt{-a}\tilde{J}^{\prime
2}|\Omega_{m}|}[-1-i\frac{2\mathrm{sgn}(\omega_{n}+\Omega_{m})}{\pi}\ln\frac{\tilde{J}^{2}}{(-a)\tilde{J}^{\prime
4}|\Omega_{m}|}][1+i\mathrm{sgn}(\omega_{n}+\Omega_{m})\ln\Lambda]&\ \ \
\tilde{J}^{2}\ll(-a\tilde{J}^{\prime 4}|\Omega_{m}|).\end{matrix}\right.$

The same low-frequency condition has been applied above and the integral over
the domain $\mathbb{C}$ is

$\displaystyle\int_{\mathbb{C}}\frac{d^{2}q}{(2\pi)^{2}}\frac{1}{q^{2}+\tilde{J}^{2}\frac{\sqrt{-a}q_{y}}{|q_{x}|}\frac{|\Omega_{m}|}{q_{x}^{2}+aq_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}\frac{1}{i\omega_{n}+i\Omega_{m}-q_{x}^{2}-aq_{y}^{2}}$ (S24)
$\displaystyle\approx$
$\displaystyle\frac{1}{\pi^{2}\sqrt{-a}}\int_{0}^{\infty}dq_{x}\frac{\frac{1}{\sqrt{i\omega_{n}+i\Omega_{m}-q_{x}^{2}}}\mathrm{ArcTan}(\frac{q_{x}}{\sqrt{i\omega_{n}+i\Omega_{m}-q_{x}^{2}}})+\frac{\tilde{J}^{2}|\Omega_{m}|}{q_{x}^{3}(q_{x}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|)}\mathrm{ArcTanh}(\frac{\tilde{J}^{2}|\Omega_{m}|}{q_{x}^{2}(q_{x}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|)})}{q_{x}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|+\frac{\tilde{J}^{4}|\Omega_{m}|^{2}}{q_{x}^{6}(q_{x}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|)}(i\omega_{n}+i\Omega_{m}-q_{x}^{2})}$ $\displaystyle\approx$
$\displaystyle\left\\{\begin{matrix}\frac{1}{32\sqrt{-a}|\tilde{J}||\Omega_{m}|^{\frac{1}{2}}}[-1-i\mathrm{sgn}(\omega_{n}+\Omega_{m})][1-i\mathrm{sgn}(\omega_{n}+\Omega_{m})\ln\Lambda]&\
\ \ \tilde{J}^{2}\gg\tilde{J}^{\prime 4}|\Omega_{m}|\\\
\frac{1}{16\sqrt{-a}\tilde{J}^{\prime
2}|\Omega_{m}|}[-1-i\frac{2\mathrm{sgn}(\omega_{n}+\Omega_{m})}{\pi}\ln\frac{\tilde{J}^{2}}{\tilde{J}^{\prime
4}|\Omega_{m}|}][1-i\mathrm{sgn}(\omega_{n}+\Omega_{m})\ln\Lambda]&\ \ \
\tilde{J}^{2}\ll\tilde{J}^{\prime 4}|\Omega_{m}|.\end{matrix}\right.$

Eqs. (S23) and (S24) have almost the same form with the exception of the sign
in front of the $\ln\Lambda$ and it can be exact cancellation when $a=-1$.
Integrals over the domain $\mathbb{A}$ and $\mathbb{B}$ in Eq. (S22) is

Figure S1: Comparison between the approximate analytical results and numerical
results of the integrals for different limits in Eq. (S25). The top two graphs
are the results for $\tilde{J}^{\prime 2}|\Omega_{m}|^{1/2}\gg\tilde{J}$, with
parameters set to $a=-1$,
$\Lambda_{U}/\tilde{J}^{1/2}|\Omega_{m}|^{1/4}=10^{4}$. The graph on the top
right tells us that the analytical approximation fails at $\tilde{J}^{\prime
2}|\Omega_{m}|^{1/2}<5\tilde{J}$. The bottom two graphs are the results for
$\tilde{J}^{\prime 2}|\Omega_{m}|^{1/2}\ll\tilde{J}$, with parameters set to
$a=-1$, $\Lambda_{U}/\tilde{J}^{\prime}|\Omega_{m}|^{1/2}=10^{4}$. The graph
on the top right tells us that the analytical approximation fails at
$\tilde{J}^{\prime 2}|\Omega_{m}|^{1/2}>0.5\tilde{J}$. (When
$\tilde{J}^{\prime 2}|\Omega_{m}|^{1/2}\ll\tilde{J}$, the singularity is in
the large momentum region, and the instability of the integral at the
singularity results in poor accuracy in the two graphs below.)

$\displaystyle\int_{\mathbb{A}}\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}+\tilde{J}^{2}\frac{|\Omega_{m}|}{k_{x}^{2}+ak_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}+\int_{\mathbb{B}}\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}-\tilde{J}^{2}\frac{|\Omega_{m}|}{k_{x}^{2}+ak_{y}^{2}}+\tilde{J}^{\prime
2}|\Omega_{m}|}$ (S25) $\displaystyle\approx$
$\displaystyle\frac{1}{\pi^{2}}\int_{0}^{\infty}dk_{y}\frac{\mathrm{ArcTan}[\frac{\sqrt{-a}k_{y}}{k_{y}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|-\frac{\tilde{J}^{4}|\Omega_{m}|^{2}}{a^{2}k_{y}^{4}(k_{y}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|)}}]}{\sqrt{k_{y}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|-\frac{\tilde{J}^{4}|\Omega_{m}|^{2}}{a^{2}k_{y}^{4}(k_{y}^{2}+\tilde{J}^{\prime
2}|\Omega_{m}|)}}}\approx\left\\{\begin{matrix}\frac{\mathrm{ArcTan}(\sqrt{-a})}{2\pi^{2}}\ln\frac{\Lambda_{U}^{2}}{\tilde{J}|\Omega_{m}|^{1/2}}&\
\ \ \tilde{J}^{2}\gg(-a\tilde{J}^{\prime 4}|\Omega_{m}|)\\\
\frac{\mathrm{ArcTan}(\sqrt{-a})}{\pi^{2}}\ln\frac{\Lambda_{U}}{\tilde{J}^{\prime}|\Omega_{m}|^{1/2}}&\
\ \ \tilde{J}^{2}\ll(-a\tilde{J}^{\prime 4}|\Omega_{m}|).\end{matrix}\right.$

The superficial degree of divergence of Eq. (S25) is $\ln\Lambda_{U}$. In
order to avoid introducing higher-order divergences, we have omitted the
$k_{x}$ component of the second term in the denominator of the first line of
the integral expression and retained only the logarithmic divergence part in
the handling of the second expression (where $\Lambda_{U}$ is the UV cutoff).
The comparison between the analytical and numerical results can be seen in
Fig. (S1), and it can be observed that they agree well under the conditions of
$\tilde{J}^{\prime 2}|\Omega_{m}|^{1/2}<5\tilde{J}$ or $\tilde{J}^{\prime
2}|\Omega_{m}|^{1/2}>0.5\tilde{J}$ when $a=-1$. A similar process applies to
the integration over the domain $\mathrm{C}$ in Eq. (S22), and yields

$\left\\{\begin{matrix}\frac{\mathrm{ArcTan}(1/\sqrt{-a})}{2\pi^{2}}\ln\frac{\Lambda_{U}^{2}}{\tilde{J}|\Omega_{m}|^{1/2}}&\
\ \ \tilde{J}^{2}\gg\tilde{J}^{\prime 4}|\Omega_{m}|\\\
\frac{\mathrm{ArcTan}(1/\sqrt{-a})}{\pi^{2}}\ln\frac{\Lambda_{U}}{\tilde{J}^{\prime}|\Omega_{m}|^{1/2}}&\
\ \ \tilde{J}^{2}\ll\tilde{J}^{\prime 4}|\Omega_{m}|\\\ \end{matrix}\right.$
(S26)

After summing over Matsubara frequency, the fermion self-energy
$\Sigma(i\omega_{n},0)-\Sigma(0,0)$ at zero temperature is found to be

$\displaystyle-i\frac{|J|\ln\Lambda}{8\sqrt{\pi}}\sqrt{\frac{N^{\prime}}{N}}[(-a)^{1/4}-(-a)^{-1/4}]\mathrm{sgn}(\omega_{n})|\omega_{n}|^{1/2}-i\frac{|J|}{8\sqrt{\pi}}\sqrt{\frac{N^{\prime}}{N}}[(-a)^{1/4}+(-a)^{-1/4}]\mathrm{sgn}(\omega_{n})|\omega_{n}|^{1/2}$
(S27)
$\displaystyle-i\frac{|J^{\prime}|^{2}\ln\Lambda}{4\pi^{4}\sqrt{-a}}[\mathrm{ArcTan}(\sqrt{-a})+\mathrm{ArcTan}(\frac{1}{\sqrt{-a}})]\omega_{n}\ln\frac{4e\pi\sqrt{-a}N^{\prime}\Lambda_{U}^{4}}{N|J|^{2}|\omega_{n}|}$

when $\tilde{J}^{2}\gg\max\\{\tilde{J}^{\prime
4}|\Omega_{m}|,(-a)\tilde{J}^{\prime 4}|\Omega_{m}|\\}$, and

$\displaystyle-i\frac{\sqrt{-a}\pi
N^{\prime}\mathrm{sgn}(\omega_{n})}{8(\ln\Lambda)^{2}N|J^{\prime}|^{2}}[\ln(-a)\ln\frac{\pi^{5}(\sqrt{-a})^{3}N^{\prime}|J|^{2}}{4(\ln\Lambda)^{4}N|J^{\prime}|^{4}|\omega_{n}|}-(\ln\frac{\pi^{5}(\sqrt{-a})^{3}N^{\prime}|J|^{2}}{4(\ln\Lambda)^{4}N|J^{\prime}|^{4}|\omega_{n}|})^{2}]$
(S28)
$\displaystyle-i\frac{|J^{\prime}|^{2}\ln\Lambda}{2\pi^{4}\sqrt{-a}}[\mathrm{ArcTan}(\sqrt{-a})+\mathrm{ArcTan}(\frac{1}{\sqrt{-a}})]\omega_{n}\ln\frac{e\pi^{3}(-a)N^{\prime}\Lambda_{U}^{2}}{N(\ln\Lambda)^{2}|J^{\prime}|^{2}|\omega_{n}|}$

when $\tilde{J}^{2}\ll\min\\{\tilde{J}^{\prime
4}|\Omega_{m}|,(-a)\tilde{J}^{\prime 4}|\Omega_{m}|\\}$. Eq. (S21) involves an
IR divergence in the frequency integration when
$\tilde{J}^{2}\ll\min\\{\tilde{J}^{\prime 4}|\Omega_{m}|,(-a)\tilde{J}^{\prime
4}|\Omega_{m}|\\}$, naturally introducing an infrared cutoff
$\max\\{\tilde{J}^{2}/\tilde{J}^{\prime 4},\tilde{J}^{2}/(-a)\tilde{J}^{\prime
4}\\}$. But this treatment is not crucial, as analyzed in the manuscript,
because the scaling of $\ln\omega$ is concealed by the scaling of marginal-
Fermi-liquid.

##  S2 Transport properties

We will calculate the conductivity and frequency dependence at zero
temperature and extend these results to finite temperature. At finite
temperature, the bosonic modes with a dynamic critical exponent $z=2$ (i.e.
$\omega\sim\Delta$), where the magnons’ self-energy correction (S17) is
following a quadratic dispersion relation, acquire a thermal tuning parameter
$\tilde{\Delta}(T)\sim T$ (it can be verified by the temperature $T$-dependent
phase boundaries between the $U(1)$ phase and the BEC phase, as illustrated in
Fig. 3(b) and 3(c) of the main text). This lead to $\omega/T$ scaling in
universal function Patel  _et al._ (2023); Esterlis  _et al._ (2021), the
frequency dependence $\sim\omega^{\alpha}$ at the limit $T\ll|\omega|$, will
transition to a dependence on the temperature $\sim T^{\alpha}$ at limit
$T\gg|\omega|$.

We need to calculate the current-current correlation function to compute the
electrical resistivity. By gauging the global $U(1)$ symmetry to introduce
gauge field, we assume that the gauge potential $A_{k}$ exists only at the
midpoint of the lattice links. Taylor expansion up to the linear order in
$A_{k}$, we obtain the current:

$\vec{j}_{k}=2\sum_{p,m}\sin((\vec{p}+\frac{\vec{k}}{2})\cdot\vec{e})c^{\dagger}_{p+k,m}c_{p,m}\vec{e},$
(S29)

where $\vec{e}$ is the lattice vector. We assume the low-momentum scenario and
consider the gauge potential’s direction to be in the x-direction, then we get
$j_{k}=\sum_{p,m}(2p_{x}+k_{x})c^{\dagger}_{p+k,m}c_{p,m}$. We can also obtain
this electromagnetic current from an effective theory (6) of the main text
with minimal coupling method. Similarly, it requires fixing the gauge to
$A_{y}=0$. All subsequent calculations involve only the interaction term
$H_{int2}$ (i.e. $J=0$) and are at low-frequency limit.

For scattering between the same patch near one van Hove point, the one-loop
diagram as shown in Fig1. (b) is automatically zero, as it involves only
forward scattering processes and lacks Umklapp scattering processes:

$\Xi_{b}(i\Omega_{m},0)=-NT\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{4k_{x}^{2}}{i\omega_{n}+i\Omega_{m}-k_{x}^{2}-ak_{y}^{2}}\frac{1}{i\omega_{n}-k_{x}^{2}-ak_{y}^{2}}=0.$
(S30)

The current-current correction functions in Fig1. (c) and (d) at zero
temperature are

$\displaystyle\Xi_{cJ^{\prime}}(i\Omega_{m},0)=$ $\displaystyle-
NT\sum_{n}\int\frac{dk^{2}}{(2\pi)^{2}}\frac{4k_{x}^{2}}{i\omega_{n}+i\Omega_{m}-k_{x}^{2}-ak_{y}^{2}}\frac{1}{(i\omega_{n}-k_{x}^{2}-ak_{y}^{2})^{2}}\Sigma_{J^{\prime}}(i\omega_{n})$
(S31) $\displaystyle=$ $\displaystyle\frac{2N}{\pi^{2}\sqrt{-a}}T\sum_{n}\int
dk_{y}\frac{\sqrt{i\omega_{n}+i\Omega_{m}+k_{y}^{2}}\mathrm{ArcTanh}(\frac{\Lambda_{UV}}{\sqrt{i\omega_{n}+i\Omega_{m}+k_{y}^{2}}})-\frac{i\omega_{n}+i\Omega_{m}+k_{y}^{2}}{\sqrt{i\omega_{n}+k_{y}^{2}}}\mathrm{ArcTanh}(\frac{\Lambda_{UV}}{\sqrt{i\omega_{n}+k_{y}^{2}}})}{\Omega_{m}^{2}}\Sigma_{J^{\prime}}(i\omega_{n})$
$\displaystyle\approx$ $\displaystyle-\frac{J^{\prime
2}N\Lambda_{U}^{2}\ln\Lambda}{8\pi^{5}(-a)}\int
d\omega\frac{\omega}{\Omega_{m}^{2}}\ln(\frac{e\Lambda_{U}^{2}}{\tilde{J}^{\prime
2}|\omega|})[\mathrm{sgn}(\omega+\Omega_{m})-\mathrm{sgn}(\omega)]$
$\displaystyle=$ $\displaystyle\frac{J^{\prime
2}N\Lambda_{U}^{2}\ln\Lambda}{16\pi^{5}(-a)}\ln\frac{e^{3}\Lambda_{U}^{4}}{\tilde{J}^{\prime
4}\Omega_{m}^{2}},$

and

$\displaystyle\Xi_{dJ^{\prime}}(i\Omega_{m},0)=$
$\displaystyle|J^{\prime}|^{2}N^{\prime}\int\frac{d\omega_{1}}{2\pi}\frac{d\omega_{2}}{2\pi}\int\frac{d^{2}k_{1}}{(2\pi)^{2}}\frac{d^{2}k_{2}}{(2\pi)^{2}}\frac{d^{2}k_{3}}{(2\pi)^{2}}\frac{4k_{1x}k_{2x}}{i\omega_{1}+i\Omega_{m}-k_{1x}^{2}-ak_{1y}^{2}}\frac{1}{i\omega_{1}-k_{1x}^{2}-ak_{1y}^{2}}\frac{1}{i\omega_{2}+i\Omega_{m}-k_{2x}^{2}-ak_{2y}^{2}}$
(S32)
$\displaystyle\frac{1}{i\omega_{2}-k_{2x}^{2}-ak_{2y}^{2}}\frac{1}{k_{3}^{2}+\tilde{J}^{\prime
2}|\omega_{2}-\omega_{1}|}=0,$

respectively. So the main contribution up to two-loop to electrical
conductivity comes from Fig1. (c), that is

$\mathrm{Re}[\sigma(\Omega\gg
T)]=\frac{\mathrm{Im}[\Xi(i\Omega_{m},0)-\Xi(0,0)]_{i\Omega_{m}\rightarrow\Omega+i0^{+}}}{\Omega}=\frac{|J^{\prime}|^{2}N\Lambda_{U}^{2}\ln\Lambda}{16\pi^{4}(-a)}\frac{1}{|\Omega|}.$
(S33)

When $\Omega\ll T$, this leads to temperature-linear resistivity, but without
a residual constant term.

If consider the current-current correlation function arising from scattering
between different patches near the van Hove points, here we make a simple
assumption that the dispersion relations of these two van Hove points are
denoted as $k_{x}^{2}+ak_{y}^{2}$ ($a<0$) and $bk_{x}^{2}+k_{x}k_{y}$. We also
take the Fermi self-energy from Eq. (S19) for simplyfy (just as demonstrated
in scale analysis, the frequency dependence of the imaginary part of the Fermi
self-energy due to scattering between different patch near the van Hove points
remains unchanged, and only the coefficient has changed.). The relevant flow-
flow correlation functions are

$\displaystyle\Xi^{\prime}_{b}(i\Omega_{m},0)-\Xi^{\prime}_{a}(0)=$
$\displaystyle-
NT\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}(\frac{1}{i\omega_{n}+i\Omega_{m}-k_{x}^{2}-ak_{y}^{2}}-\frac{1}{i\omega_{n}-k_{x}^{2}-ak_{y}^{2}})\frac{4k_{x}^{2}}{i\omega_{n}-bk_{x}^{2}-k_{x}k_{y}}$
(S34) $\displaystyle\approx$ $\displaystyle i\pi
NT\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}[\mathrm{sgn}(\omega+\Omega_{m})-\mathrm{sgn}(\omega)]\delta(k_{x}^{2}+ak_{y}^{2})\frac{4k_{x}^{2}}{i\omega_{n}-bk_{x}^{2}-k_{x}k_{y}}$
$\displaystyle=$
$\displaystyle\frac{i\Omega_{m}\sqrt{-a}}{4\pi^{2}}[\frac{1}{ab-\sqrt{-a}}\ln\frac{e^{2}(ab-\sqrt{-a})^{2}\Lambda_{U}^{4}}{-\Omega_{m}^{2}}+\frac{1}{ab+\sqrt{-a}}\ln\frac{e^{2}(ab+\sqrt{-a})^{2}\Lambda_{U}^{4}}{-\Omega_{m}^{2}}],$
$\displaystyle\Xi_{cJ^{\prime}}^{\prime}(i\Omega_{m},0)-\Xi_{bJ^{\prime}}^{\prime}(0,0)=$
$\displaystyle-
NT\sum_{n}\int\frac{dk^{2}}{(2\pi)^{2}}(\frac{1}{i\omega_{n}+i\Omega_{m}-k_{x}^{2}-ak_{y}^{2}}-\frac{1}{i\omega_{n}-k_{x}^{2}-ak_{y}^{2}})\frac{4k_{x}^{2}}{(i\omega_{n}-bk_{x}^{2}-k_{x}k_{y})^{2}}\Sigma_{J^{\prime}}(i\omega_{n})$
(S35) $\displaystyle\approx$ $\displaystyle i\pi
NT\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}[\mathrm{sgn}(\omega+\Omega_{m})-\mathrm{sgn}(\omega)]\delta(k_{x}^{2}+ak_{y}^{2})\frac{4k_{x}^{2}}{(i\omega_{n}-bk_{x}^{2}-k_{x}k_{y})^{2}}\Sigma_{J^{\prime}}(i\omega_{n})$
$\displaystyle=$
$\displaystyle-i\frac{|J^{\prime}|^{2}b\ln\Lambda}{8\pi^{5}(ab^{2}+1)}\Omega_{m}\ln\frac{e^{4}\Lambda_{U}^{4}}{\tilde{J}^{\prime
4}\Omega_{m}^{2}}.$

Let $\vec{k}\rightarrow-\vec{k}$, and you will also find
$\Xi^{\prime}_{dJ^{\prime}}(i\Omega_{m},0)=0$. Results in Eq. (S34) and (S35)
are only valid in $ab^{2}+1\neq 0$. The result dc conductivity is
$\mathrm{Re}[\sigma(\Omega\gg
T)]=-|J^{\prime}|^{2}b\ln(\Lambda)/8\pi^{4}(ab^{2}+1)$ and do not depend on
the frequency $\Omega$.

##  S3 Phase diagrams

Assuming the system exhibits superconductivity, we introduce a superconductor
pairing term
$g\Delta_{ij}(R,r)\psi^{\dagger}_{i}(x)\psi^{\dagger}_{j}(y)+h.c.$
[$R=(x+y)/2$ the position of the center of mass, $r=x-y$ the relative
position, and $i$ and $j$ are flavor indices] into the Hamiltonian. If the
fermi modes $\Psi_{i}$ after the Fourier-transformation primarily come from
the same patch near a van Hove point, then
$\Delta_{ij}(R,r)\equiv\Delta_{ij}(r)e^{i2K_{1}R}$ with momentum $K_{1}$ at
this van Hove point. Interaction (S1) renormalize the superconductor pair up
to $\mathcal{O}(|J|^{2}/N)$. At the critical point of the superconductor-
normal phase transition, the renormalized superconductor gap is required to be
zero. In the large-$N$ limit, the linearized gap equation Wu  _et al._ (2023a)
is obtained as

$\Delta_{ij}(p)=\frac{|J|^{2}}{N}T\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}G(i\omega_{n},k)G(-i\omega_{n},-k)\Delta_{ji}(k)D(k-p)$
(S36)

with fermi propagator $G(i\omega_{n},k)$ and boson propagator $D(k)$.
Similarly, this also applies to the case of interaction (S15), or the joint
action of (S1) and (S15). In the numerical computation part, we simply replace
the boson propagator with $D(p)=1/(p^{2}+\Delta(T))$ to represent the
renormalized boson propagator. Where $\Delta(T)\equiv\Delta+\tilde{\Delta}(T)$
is the square of the effective tuning parameter at finite temperature with
renormalized contributions $\tilde{\Delta}(T)$.

In the main text, we only considered the simplest case of van Hove
singularties at half-filling on a square lattice. Two van Hove points are
located at $(\pm\pi,0)$ and $(0,\pm\pi)$. When electrons involved in the
superconducting pairing come from the same van Hove point, the center-of-mass
momentum is the reciprocal lattice vectors, and corresponds to normal
superconductor. However, when electrons come from two different van Hove
points, the center-of-mass momentum is $(\pm\pi,\pm\pi)$, introducing spatial
modulation and leading to a density wave order. But in bilayer twisted
materials, situations can arise where $2K_{1}$ is not a reciprocal lattice
vector, leading to the appearance of the pair-density-wave order as well.

Figure S2: Relationship of the critical temperature $T_{C}$ of SC with the
variance of random interactions. The left diagram only consider the
interaction (S1) and the right one only consider interaction (S15). The
parameter values are: $a=-1$, $\Lambda_{U}=300$, and $\Delta(T)=0.5$.

Fig. S2, as a supplement to Fig. 3 in the main text, provides the relationship
of the critical temperature $T_{C}$ of superconductor with the variance of
random interactions. The left graph shows a linear fit with slope $2.93$,
indicating $T_{C}\sim(|J|/\sqrt{N})^{3}$, while the right graph indicates
$T_{C}\sim e^{-\sqrt{N}/|J^{\prime}|}$.