# Emergent space-time supersymmetry at disordered quantum critical points

Xue-Jia Yu  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  Peng-Lu Zhao  Shenzhen Institute for
Quantum Science and Engineering and Department of Physics, Southern University
of Science and Technology, Shenzhen  Shenzhen Key Laboratory of Quantum
Science and Engineering, Shenzhen 518055, China  Shao-Kai Jian  Department of
Physics, Brandeis University, Waltham, Massachusetts 02453, USA  Zhiming Pan
<EMAIL_ADDRESS>Institute of Natural Sciences, Westlake Institute
for Advanced Study, Hangzhou 310024, China  Department of Physics, School of
Science, Westlake University, Hangzhou 310024, China

###### Abstract

We study the effect of disorder on the spacetime supersymmetry that is
proposed to emerge at the quantum critical point of pair density wave
transition in (2+1)D Dirac semimetals and (3+1)D Weyl semimetals. In the
(2+1)D Dirac semimetal, we consider three types of disorder, including random
scalar potential, random vector potential and random mass potential, while the
random mass disorder is absent in the (3+1)D Weyl semimetal. Via a systematic
renormalization group analysis, we find that any type of weak random disorder
is irrelevant due to the couplings between the disorder potential and the
Yukawa vertex. The emergent supersymmetry is thus stable against weak random
potentials. Our work will pave the way for exploration supersymmetry in
realistic condensed matter systems.

##  I Introduction

About five decades ago, the spacetime supersymmetry (SUSY) was proposed as a
possible way of solving the hierarchy problem of the standard model [1, 2, 3,
4] and the cosmological constant problem[5]. Later, some supersymmetric
theories have been studied as toy models to understand strong coupling physics
rigorously [6, 7]. Due to these attractive features, SUSY has been studied
intensively in past fifty years and there is some expectation before that SUSY
may be revealed in the large hadron collider (LHC). Unfortunately, the recent
experiments at the LHC have found no evidence of SUSY and/or its spontaneous
breaking in particle physics.

Three-dimensional (3D) Weyl fermions [8, 9, 10] in noncentrosymmetric
materials [11, 12, 13, 14, 15] provide an opportunity to test and investigate
important concepts developed in the context of high-energy physics in
realistic condensed matter systems. It has been suggested that SUSY can emerge
in the low-energy limit of a number of non-supersymmetric models [16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26]. In particular, SUSY is proposed to emerge at
quantum critical points (QCPs) in Bose-Fermi lattice models [27, 28], in the
(2+1)D surface states of topological insulators [29, 30, 31, 32, 33], as well
as at multicritical points in some low-dimensional systems [34, 35, 36].
Moreover, an interesting recent suggestion [37] is that SUSY can be realized
at certain pair-density-wave (PDW) superconducting quantum critical points of
ideal Weyl semimetals [38, 39](WSMs).

The realization of SUSY at QCPs relies crucially on the fact that the infrared
fixed point is stable against small perturbations. In particular, for the
emergent SUSY to be realized, it must be robust when the fermions are subject
to small perturbations from quenched disorder and other dissipation effects.
Here, we are particularly interested in the impact of quenched disorder on the
emergent SUSY, because disorder unavoidably exists in all realistic materials.
It is well known that disorder plays an essential role in condensed matter
systems [40, 41, 42, 43, 44, 45, 46, 47, 48]and may lead to a plenty of
prominent phenomena, such as Anderson localization and metal-insulator
transition. In graphene-like Dirac semimetals (DSMs), depending on the
specific type, disorder can either enhance or reduce the effective Coulomb
interaction strength [49, 50, 51, 52, 53, 54, 55], which in turn drastically
modifies the phase diagram obtained in the clean limit [49, 50, 51, 52, 53,
54, 55]. Moreover, disorder may have a significant impact on the low-
temperature properties of various Dirac or Weyl semimetals, such as the
conductivity of graphene [56, 50, 57, 58], the optical conductivity of WSMs
[59], and the low-energy spectral, thermodynamic, and transport behaviors of
$d$-wave cuprate superconductors [60, 45, 61, 62, 63]. Disorder also plays a
vital role in quantum Hall systems [64, 65, 66, 67] and topological insulators
[29, 30].

In this paper, we investigate the stability of emergent SUSY against the
disorder scattering. We focus on the disorder-induced unusual renormalization
of the fermion velocity [53, 62, 68], and examine whether such a
renormalization effect causes a substantial difference between the velocities
of fermions and bosons at low energies, and ruins the emergent SUSY. Based on
this analysis, we are able to identify the influence of non-magnetic disorder
on the particular fixed point that is argued to display an emergent SUSY at
the QCP of pair density wave (PDW) transition in (3+1)D WSMs and (2+1)D DSMs
[37]. In the case of (2+1)D DSMs, we consider three types of disorder,
including random scalar potential (RSP), random vector potential (RVP), and
random mass (RM). In (2+1)D, our systematic RG analysis reveals that weak RVP,
RMP and RSP are irrelevant at the QCPs where fermion velocity and boson
velocity flow into same value under renormalization, which certainly does not
breaks the emergent SUSY. In (3+1)D WSMs, the disorder potential becomes more
irrelevant and the effective SUSY is robust against weak disorder. The
schematic RG flow diagram for emergent SUSY is shown in Fig.1

Figure 1: Schematic RG flow near SUSY fixed point as a function the Yukawa
coupling $g^{2}$ and quench disorder strength $\Delta$. The emergent SUSY
fixed point is stable for weak random quench disorders from our one-loop level
RG calculations.

This paper is structured as follows. In Sec. II, we present the effective
model for (2+1)D disordered DSMs and perform the RG calculations. In Sec. III,
the same analysis is carried out in (3+1)D WSMs. We briefly summarize the
results of this work in Sec. IV. Further RG details for our calculations are
provided in Appendix.

##  II (2+1)D Dirac semimetals

As demonstrated in Ref. [37], a spactime SUSY could emerge in the low-energy
region at the PDW QCP of (2+1)D DSMs only when the number of massless Dirac
fermions is $N_{f}=2$. In this case, the low-energy effective action at the
PDW criticality in a clean system is given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{f}+S_{b}+S_{I},$ (1)
$\displaystyle S_{f}$ $\displaystyle=$ $\displaystyle\int
d^{2}xd\tau\sum_{n=\pm}\psi_{n}^{\dagger}\Big{[}\partial_{\tau}+iv_{f}\sum_{j=1}^{2}\gamma_{j}\partial_{j}\Big{]}\psi_{n},$
(2) $\displaystyle S_{b}$ $\displaystyle=$ $\displaystyle\int
d^{2}xd\tau\bigg{\\{}\sum_{n=\pm}\Big{[}|\partial_{\tau}\phi_{n}|^{2}+v_{b}^{2}\sum_{j=1}^{2}|\partial_{j}\phi_{n}|^{2}$
$\displaystyle+r|\phi_{n}|^{2}+u|\phi_{n}|^{4}\Big{]}+u_{+-}|\phi_{1}|^{2}|\phi_{2}|^{2}\bigg{\\}},$
$\displaystyle S_{I}$ $\displaystyle=$ $\displaystyle\int d^{2}xd\tau
g\sum_{n=\pm}\Big{[}\phi_{n}\psi_{n}^{T}\sigma_{y}\psi_{n}+h.c.\Big{]},$ (4)

where $\gamma_{j}=(\sigma_{x},\sigma_{y})$. $\sigma_{\alpha},\alpha=x,y$ is
Pauli matrix with spin indices. $S_{f}$ corresponds to the action for two non-
interacting two-component Dirac fermions $\psi_{\pm}$ at two Dirac points
$\bm{Q}_{\pm}$ [69, 70], with quartic and higher order self-coupling terms
being irrelevant [27] at low energies. $S_{b}$ describes the quantum
fluctuation and the self-coupling of the PDW order parameter $\phi_{n}$ near
the QCP, where $\phi_{\pm}$ is the superconducting order with momentum
$2\bm{Q}_{\pm}$, respectively. Terms with higher powers of $\phi_{n}$ are all
irrelevant, whereas $\phi_{n}^{\ast}\partial_{\tau}\phi_{n}$ is excluded by
particle-hole symmetry [37]. $S_{I}$ represents the Yukawa coupling between
Dirac fermions and bosons. The terms of the form
$\phi_{-}^{\ast}\psi_{+}{\sigma}_{y}\psi_{+}$ and
$\phi_{-}^{\ast}\psi_{+}{\sigma}_{y}\psi_{-}$ are not allowed because they do
not satisfy momentum conservation [27]. Therefore, the effective action given
above is of the most general form. It has been shown through renormalization
group analysis that an emergent spacetime SUSY occurs at the low energy limit.
A necessary condition for the emergent SUSY is that velocities of fermions and
bosons flow to the same value under RG, which renders the emergent Lorentz
symmetry. It was claimed that such an emergent Lorentz symmetry can be
naturally realized in a number of correlated electron systems [27, 28, 37, 29,
30, 31, 32, 33].

Figure 2: Feynman diagrams for all the relevant one-loop diagrams that survive
within replica limit. Here, the solid line represents the free fermion
propagator, the wavy line the free boson propagator, and the dashed line the
disorder.

The aim of the present work is to examine whether the emergent SUSY is robust
against disorder scattering. For this purpose, we now introduce a direct
fermion-disorder coupling term to the system via the standard form, also see
Appendix A [67, 71, 45, 49, 50, 51, 63, 53],

$\displaystyle S_{\mathrm{dis}}=\int
d^{2}xd\tau\sum_{n=\pm}\psi_{n}^{\dagger}\left(\sum_{\Gamma}V_{\Gamma}(\mathbf{x})\Gamma\right)\psi_{n},$
(5)

where $V_{\Gamma}(\mathbf{x})$ stands for the random potential and $\Gamma$
labels the type of the disorder potential. We assume $V_{\Gamma}(\mathbf{x})$
to be a quenched, Gaussian white noise potential characterized by the
following identities:

$\displaystyle\langle V_{\Gamma}(\mathbf{x})\rangle=0,\quad\langle
V_{\Gamma}(\mathbf{x})V_{\Gamma^{\prime}}(\mathbf{x}^{\prime})\rangle=\Delta_{\Gamma}\delta_{\Gamma\Gamma^{\prime}}\delta(\mathbf{x}-\mathbf{x}^{\prime}),$
(6)

where $\langle...\rangle$ denotes average over disorder distribution and
$\Delta_{\Gamma}$ is introduced to characterize the strength of random
potential.

We consider three different types of disorder classified by the different
matrices $\Gamma$. In particular, $\Gamma=\mathbb{I}_{2\times 2}$ for RSP,
$\Gamma=\sigma_{z}$ for RM, and $\Gamma=(\sigma_{x},\sigma_{y})$ for RVP.
These three types are most frequently studied in the literature and they can
be induced by some specific mechanisms in realistic materials [71, 72, 73, 74,
75, 76, 77]. These three types of random potential might exist individually,
or coexist in the same material. To be general, we assume that they coexist in
the system and analyze their impact by performing RG calculations.

The random potential $V(\mathbf{x})$ can be properly averaged by employing the
replica trick [40, 78, 52, 54, 79, 59], which leads us to an interacting
effective action of short-range fermion-fermion interaction:

$\displaystyle S_{\mathrm{dis}}=$ $\displaystyle-\frac{1}{2}\int d^{2}xd\tau
d\tau^{\prime}\Big{\\{}\sum_{n=\pm}\Big{[}\Delta_{S}\big{(}\psi_{n}^{\dagger\alpha}\psi_{n}^{\alpha}\big{)}_{x}\big{(}\psi_{n}^{\dagger\beta}\psi_{n}^{\beta}\big{)}_{x^{\prime}}+\Delta_{M}\big{(}\psi_{n}^{\dagger\alpha}{\sigma}_{z}\psi_{n}^{\alpha}\big{)}_{x}\big{(}\psi_{n}^{\dagger\beta}{\sigma}_{z}\psi_{n}^{\beta}\big{)}_{x^{\prime}}$
$\displaystyle+\Delta_{V}\sum_{j}\big{(}\psi_{n}^{\dagger\alpha}{\sigma}_{j}\psi_{n}^{\alpha}\big{)}_{x}\big{(}\psi_{n}^{\dagger\beta}{\sigma}_{j}\psi_{n}^{\beta}\big{)}_{x^{\prime}}\Big{]}+2\Delta_{S}^{\prime}\big{(}\psi_{+}^{\dagger\alpha}\psi_{+}^{\alpha}\big{)}_{x}\big{(}\psi_{-}^{\dagger\beta}\psi_{-}^{\beta}\big{)}_{x^{\prime}}$
$\displaystyle+2\Delta_{M}^{\prime}\big{(}\psi_{+}^{\dagger\alpha}{\sigma}_{z}\psi_{+}^{\alpha}\big{)}_{x}\big{(}\psi_{-}^{\dagger\beta}{\sigma}_{z}\psi_{-}^{\beta}\big{)}_{x^{\prime}}+2\Delta_{V}^{\prime}\sum_{j}\big{(}\psi_{+}^{\dagger\alpha}{\sigma}_{j}\psi_{+}^{\alpha}\big{)}_{x}\big{(}\psi_{-}^{\dagger\beta}{\sigma}_{j}\psi_{-}^{\beta}\big{)}_{x^{\prime}}\Big{\\}}$
(7)

where $j=(x,y)$, $\alpha$ and $\beta$ are the replica indices,
$x\equiv(\mathbf{x},\tau)$ and $x^{\prime}\equiv(\mathbf{x},\tau^{\prime})$
are the space-time coordinate. The repeated indices $\alpha$ and $\beta$ are
summed automatically. In the replica theory, the replica limit
$\sum_{\alpha}=N\rightarrow 0$ is implemented in the following RG calculation.
Three parameters $\Delta_{S}$, $\Delta_{M}$, and $\Delta_{V}$ characterize the
effective strength of quartic couplings of Dirac fermions induced by averaging
over RS, RM, and RVP, respectively. The two pieces of Dirac fermions share the
same random potential. Three cross terms characterized by
$\Delta_{S}^{\prime}$, $\Delta_{M}^{\prime}$, and $\Delta_{V}^{\prime}$ are
induced in the replica limit. In the RG analysis, the bare values of the
parameters are the same, $\Delta_{S}^{\prime 0}=\Delta_{S}^{0}$,
$\Delta_{M}^{\prime 0}=\Delta_{M}^{0}$, and $\Delta_{V}^{\prime
0}=\Delta_{V}^{0}$, moreover, the RG equations for $\Delta_{\Gamma}$ and
$\Delta_{\Gamma^{\prime}}$ are the same (see Appendix B for details), so we
focus on $\Delta_{\Gamma}$ in the following.

As shown in previous calculations [53, 62, 68], disorder can strongly affect
the RG flow of fermion velocity as the energy is lowered. If the disorder
coupling is relevant that flow to a finite value at low energy limit, it will
drive the fermion velocity to vanish at sufficiently low energies, which then
spoils the Lorentz symmetry for the fermion sector, but not for the boson
sector. As a result, the emergent Lorentz symmetry, and thus the emergent SUSY
will be ruined by disorder. However, whether this takes place relies crucially
on the scale dependence of disorder coupling parameter. In the case of (2+1)D
DSMs, naive power-counting, according to Eqs. (2) and (7), shows that disorder
is marginal. A careful analysis of the marginal disorder effect is helpful to
tell us whether a irrelevant and a relevant coupling need to investigate
further.

To this end, we carry out a detailed RG analysis starting from the critical
action with $r=0$, represented by Eq. (1) along with Eq. (7), by considering
the leading order of the $\epsilon$-expansion, where $\epsilon=4-D=3-d$, $D$
and $d$ are the spacetime dimension and the spatial dimension, respectively.
The pertinent one-loop Feynman diagrams are shown in Fig. 2. After integrating
out the fast modes defined within the momentum shell
$e^{-l}\Lambda<|\mathbf{p}|<\Lambda$ and then performing RG transformations
[80], we obtain the following RG equations (the detailed results are presented
in appendix)

$\displaystyle\frac{dv_{f}}{dl}$
$\displaystyle=v_{f}\Big{[}g^{2}(G_{1}-G_{0})-2\sum_{\Gamma}\Delta_{\Gamma}\Big{]},$
(8) $\displaystyle\frac{da}{dl}$
$\displaystyle=g^{2}\Big{(}\frac{1-a^{2}}{2a}+a(G_{0}-G_{1})\Big{)}+2a\sum_{\Gamma}\Delta_{\Gamma},$
(9) $\displaystyle\frac{dg^{2}}{dl}$ $\displaystyle=\epsilon
g^{2}-g^{4}\big{(}1-G_{0}+3G_{1}\big{)}+\big{(}4\Delta_{S}+2\sum_{\Gamma}\Delta_{\Gamma}\big{)}g^{2},$
(10) $\displaystyle\frac{d\Delta_{S}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{S}+2\Delta_{S}\big{(}\Delta_{S}+\Delta_{M}+2\Delta_{V}\big{)}+4\Delta_{M}\Delta_{V}+(G_{0}-3G_{2}-2G_{1})\Delta_{S}g^{2},$
(11) $\displaystyle\frac{d\Delta_{M}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{M}-2\Delta_{M}\big{(}\Delta_{S}+\Delta_{M}-2\Delta_{V}\big{)}+4\Delta_{S}\Delta_{V}-(G_{0}+3G_{2}+2G_{1})\Delta_{M}g^{2},$
(12) $\displaystyle\frac{d\Delta_{V}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{V}+2\Delta_{M}\Delta_{S}-(G_{0}+2G_{1})\Delta_{V}g^{2},$
(13)

where $a=v_{b}/v_{f}$, $G_{0}=\frac{4}{a(a+1)^{2}}$,
$G_{1}=\frac{4(2a+1)}{3a(a+1)^{2}}$, and $G_{2}=\frac{4(2+a)}{3a(a+1)^{2}}$.
In the above calculations, we have rescaled all the couplings as follows:
$g^{2}\Lambda^{-\epsilon}S_{D-1}/[2(2\pi)^{D-1}v_{f}^{D-1}]\rightarrow g^{2},$
and
$\Delta_{\Gamma}\Lambda^{1-\epsilon}S_{D-1}/[(2\pi)^{D-1}v_{f}^{2}]\rightarrow\Delta_{\Gamma}$,
with $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ as the area of the unit sphere in $d$
dimensions. By setting all $\Delta_{\Gamma}=0$, Eqs. (9) - (10) recover the RG
equations for $v_{f}$, $a$, and $g^{2}$ previously obtained in Refs. [27] and
[37]. In the case of disordered Dirac fermion systems with $g=0$, our RG
results for $\Delta_{\Gamma}$ are in accordance with that previously obtained
in [56, 57, 81]. The RG equations for $u$ and $u_{+-}$, which are not shown
here, are exactly the same as those presented in Refs. [27] and [37], since
there is no direct coupling between boson and fermion disorder potential. In
the case of clean system, as demonstrated in Refs. [27] and [37], $a^{\ast}=1$
is the only stable infrared fixed point for $a$, which means that the bosons
and fermions have the same velocity at low energies. Moreover, the coupling
constant $g$, $u$ and $u_{+-}$ will flow to a strongly coupled fixed point
that preserves SUSY. In the following, we first analyze the effects of single
disorder, and then consider the interplay between different types of disorder.

We now consider the case in which RVP exists by itself by taking
$\Delta_{M}=\Delta_{S}=0$. Noting that the physical case of (2+1)D corresponds
to $\epsilon\rightarrow 1$, Eq. (13) becomes

$\displaystyle\frac{d\Delta_{V}}{dl}=$
$\displaystyle-(G_{0}+2G_{1})\Delta_{V}g^{2}$ (14)

Thus the effective coupling strength for RVP, namely $\Delta_{V}$, is
irrelevant and flows to zero. Without the fermion-boson coupling $g$, RVP is
marginal, which originates from the existence of a time-independent gauge
transformation that ensures RVP unrenormalized and is valid at any order of
loop expansion [67, 50, 51, 81, 68]. Nevertheless, near the emergent SUSY
fixed point, where the coupling constant $g$ remains finite, RVP is
irrelevant, and thus the emergent SUSY is stable.

We then assume that RM exists alone, which means $\Delta_{S}=\Delta_{V}=0$ in
Eq. (12), and we have

$\displaystyle\frac{d\Delta_{M}}{dl}$ $\displaystyle=$
$\displaystyle-2\Delta_{M}^{2}-(G_{0}+3G_{2}+2G_{1})\Delta_{M}g^{2}.$ (15)

From the RG function, we see that ${\Delta}_{M}$ is always irrelevant. We thus
can infer that the emergent SUSY is also robust again RM.

The RSP can be similarly analyzed. The simplified RG equations for RSP are

$\displaystyle\frac{da}{dl}=$ $\displaystyle
g^{2}\Big{(}\frac{1-a^{2}}{2a}+a(G_{0}-G_{1})\Big{)}+2a\Delta_{S},$ (16)
$\displaystyle\frac{dg^{2}}{dl}=$ $\displaystyle
g^{2}-g^{4}\big{(}1-G_{0}+3G_{1}\big{)}+6g^{2}\Delta_{S},$ (17)
$\displaystyle\frac{d\Delta_{S}}{dl}=$ $\displaystyle
2\Delta_{S}^{2}+(G_{0}-3G_{2}-2G_{1})\Delta_{S}g^{2}.$ (18)

There exist two fixed points, Gaussian fixed point ($g^{2}=\Delta_{s}=0$) and
the SUSY fixed point. The Gaussian fixed point is unstable. With only random
scalar potential ($\Delta_{s}\neq 0,g^{2}=0$), the system will flow into
strong disorder regime and explicitly breaks the Lorentz symmetry. On the
contrary, $g^{2}$ has a finite critical point $g_{c}^{2}$,

$\displaystyle g_{c}^{2}=\frac{1+6\Delta_{S}}{1-G_{0}+3G_{1}},$ (19)

where the RG function of $\Delta_{S}$ near $\Delta_{S}=0$ is negative
($G_{0}-3G_{2}-2G_{1}<0$), and we expect small $\Delta_{S}$ is irrelevant. The
RG flow of the equations within the critical plane $g^{2}=g_{c}^{2}$ is shown
in Fig. 3.

Figure 3: Flow diagram in the $\Delta_{S}-a$ plane with $g^{2}=g_{c}^{2}$ when
the system contains only RSP. There is an stable fixed point(red point)
$(g^{2},\Delta_{S}^{\ast},a^{\ast})=(g_{c}^{2},0,1)$. Within $g_{c}^{2}$
plane, no other fixed point exists and any RSP $\Delta_{S}$ is irrelevant at
one-loop level. The system always flows into weak disorder regime.

Finite $g^{2}$ will always pull back the RG flow and RSP $\Delta_{S}$ is
irrelevant eventually at one-loop level for any initial values. We find that
SUSY is stable: an arbitrarily strong RSP flows to the weak coupling regime in
the lowest energy limit. This means that the RSP is an irrelevant
perturbation, which is consistent with the one-loop RG result in [82].
Previous results show that the RSP is marginally relevant for Dirac fermions,
and will induce an instability of the system, leading to a diffusive motion of
the Dirac fermions [42, 45, 56, 57, 52, 81, 54]. However, our result shows
that, the RSP is rendered irrelevant by the critical fluctuations at the PDW
QCP through the finite Yukawa coupling $g^{2}$ between fermion and boson.
There exists no diffusive behaviors as long as $g^{2}$ is finite.

When more than one type of disorder exist, from the properties of the RG
equation, Eqs. (12)-(14) that the coexistence of any two types disorder
dynamically generate the third one. Thus we need to analyze the full set of RG
equations given by Eqs. (8)-(13). It is easy to see that there is a fixed
point given by
$(v_{f},a^{*},g^{2\ast},\Delta_{S}^{\ast},\Delta_{M}^{\ast},\Delta_{V}^{\ast})=(v_{f},1,\frac{\epsilon}{3},0,0,0)$,
where $v_{f}$ can take any values. Combining with the RG equation for coupling
constants $u$ and $u_{+-}$, it turns out that this is the SUSY fixed point.
Now we examine whether this fixed point is stable by expanding the RG equation
at the fixed point, and calculate eigenvalues of the stability matrix (see
Appendix C). The eigenvalues of the stability matrix are all negative except
for one marginal direction at $v_{f}$. So we can conclude that the emergent
SUSY is robust against weak disorders.

##  III (3+1)D Weyl semimetals

In this section, we examine the disorder effects on the emergent SUSY in
(3+1)D WSMs [37]. The effective action of the disordered system in the
vicinity of PDW QCP is given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{f}+S_{b}+S_{I}+S_{\text{dis}},$ (20) $\displaystyle S_{f}$ $\displaystyle=$
$\displaystyle\int
d^{4}x\sum_{n=\pm}\Big{[}\psi_{n}^{\dagger}\partial_{\tau}\psi_{n}+\sum_{j=1}^{3}iv_{fj}\psi_{n}^{\dagger}\gamma_{n}^{j}\partial_{j}\psi_{n}\Big{]},\qquad$
(21) $\displaystyle S_{b}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\bigg{\\{}\sum_{n=\pm}\Big{[}|\partial_{\tau}\phi_{n}|^{2}+\sum_{j=1}^{3}v_{bj}^{2}|\partial_{j}\phi_{n}|^{2}$
$\displaystyle+r|\phi_{n}|^{2}+u|\phi_{n}|^{4}\Big{]}+u_{+-}|\phi_{+}|^{2}|\phi_{-}|^{2}\bigg{\\}},$
$\displaystyle S_{I}$ $\displaystyle=$ $\displaystyle\int
d^{4}xg\sum_{n=\pm}\Big{[}\phi_{n}\psi_{n}\sigma^{y}\psi_{n}+h.c.\Big{]},$
(23) $\displaystyle S_{\text{dis}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int d^{3}xd\tau
d\tau^{\prime}\Big{[}\Delta_{S}\big{(}\psi^{\dagger}_{\alpha}{\sigma}_{0}\psi_{\alpha}\big{)}_{x}\big{(}\psi^{\dagger}_{\beta}{\sigma}_{0}\psi_{\beta}\big{)}_{x^{\prime}}$
$\displaystyle+\sum_{i=x,y,z}\Delta_{i}\big{(}\psi^{\dagger}_{\alpha}{\sigma}_{i}\psi_{\alpha}\big{)}_{x}\big{(}\psi^{\dagger}_{\beta}{\sigma}_{i}\psi_{\beta}\big{)}_{x^{\prime}}\Big{]},$

where $\gamma_{\pm}^{j}=({\sigma}^{x},{\sigma}^{y},\pm{\sigma}^{z})$. Now,
$\psi_{\pm}$ denotes the two-component Weyl fermions at two Weyl points
$\bm{Q}_{\pm}$, and $\phi_{\pm}$ is the superconducting order with momentum
$2\bm{Q}_{\pm}$, respectively. In (3+1)D Weyl semimetal, RVP has three
components, i.e., RM becomes the third component [83, 84]. In general, the
fermion velocity is anisotropic, with unequal values along different
directions. As shown in Ref. [37], even in the extremely anisotropic case, an
emergent Lorentz symmetry can be established in the lowest energy limit. Our
current concern is whether this emergent Lorentz symmetry can be broken by
quenched disorder, thus it suffices to consider the isotopic case. We now can
assume that $v_{fx}=v_{fy}=v_{fz}=v_{f}$, and also make the same assumption
for the bosonic field. We employ the same symbol $a$ to identify the ratio
between boson and fermion, and the same definition of $G_{0}$, $G_{1}$ and the
rescaled couplings in Sec. II. Calculating the same diagrams in Fig. 2, we
obtain the following RG equations:

$\displaystyle\frac{d\ln v_{f}}{dl}$ $\displaystyle=$ $\displaystyle
g^{2}(G_{1}-G_{0})-2\left(\Delta_{S}+\sum_{i}\Delta_{i}\right),$ (25)
$\displaystyle\frac{d\ln a^{2}}{dl}$ $\displaystyle=$
$\displaystyle\frac{g^{2}(1-a^{2})}{a^{2}}-2g^{2}(G_{1}-G_{0})+4\left(\Delta_{S}+\sum_{i}\Delta_{i}\right),$
(26) $\displaystyle\frac{d\ln g^{2}}{dl}$ $\displaystyle=$
$\displaystyle\epsilon-g^{2}(3G_{1}-G_{0}+1)+\left(6\Delta_{S}+2\sum_{i}\Delta_{i}\right),$
(27) $\displaystyle\frac{d\Delta_{S}}{dl}$ $\displaystyle=$
$\displaystyle(\epsilon-1)\Delta_{S}+2\Delta_{S}\left(\Delta_{S}+\sum_{i}\Delta_{i}\right)+\frac{2}{3}\sum_{i}\sum_{j\neq
i}\Delta_{i}\Delta_{j}+g^{2}(G_{0}-3G_{2}-2G_{1})\Delta_{S},$ (28)
$\displaystyle\frac{d\Delta_{i}}{dl}$ $\displaystyle=$
$\displaystyle(\epsilon-1)\Delta_{i}+\frac{4}{3}\sum_{j\neq
i}\Delta_{S}\Delta_{j}-\frac{2}{3}\Delta_{i}\left(\Delta_{S}+2\Delta_{i}-\sum_{j}\Delta_{j}\right)-\Delta_{i}g^{2}(G_{0}+G_{2}+2G_{1}),$
(29)

in these equations, the index $i$ is summed over $x,y,z$. The analysis of
these RG equations follows similarly as done in Sec. II. Firstly, we consider
only a single vector component disorder exists, which means

$\displaystyle\Delta_{i}\neq 0,\quad\Delta_{j\neq i}=0,\quad\Delta_{S}=0,$
(30)

by substituting this conditions to Eqs. (25-29), three simplified RG equations
for $a$, $g^{2}$ and $\Delta_{i}$ are obtained, we just exhibit the result for
disorder coupling as

$\displaystyle\frac{d\Delta_{i}}{dl}$
$\displaystyle=(\epsilon-1)\Delta_{i}-\frac{2}{3}\Delta_{i}^{2}-\Delta_{i}g^{2}(G_{0}+G_{2}+2G_{1}).$
(31)

Therefore, for an exact (3+1)D system corresponding to $\epsilon=0$, any
component of RVP is irrelevant. As a result, in (3+1)D WSMs, the emergent SUSY
is robust against any single component of RVP.

For there is only RSP in the system, we have $\Delta_{i}=0$. Now Eqs. (25) -
(29) are simplified to

$\displaystyle\frac{d\ln a^{2}}{dl}=$
$\displaystyle\frac{g^{2}(1-a^{2})}{a^{2}}-2g^{2}(G_{1}-G_{0})+4\Delta_{S},$
(32) $\displaystyle\frac{d\ln g^{2}}{dl}=$
$\displaystyle\epsilon-g^{2}(3G_{1}-G_{0}+1)+6\Delta_{S},$ (33)
$\displaystyle\frac{d\Delta_{S}}{dl}=$
$\displaystyle(\epsilon-1)\Delta_{S}+2\Delta_{S}^{2}$
$\displaystyle\qquad+g^{2}(G_{0}-3G_{2}-2G_{1})\Delta_{S},$ (34)

according to Eq. (33), and noting the fact $3G_{1}-G_{0}+1>0$, the stable
fixed point for $g^{2}$ is located at
$g_{c}^{2}=6\Delta_{S}/(3G_{1}-G_{0}+1)$. Weak RSP itself is irrelevant as
indicated in because $G_{0}-3G_{2}-2G_{1}<0$. Whereas for strong RSP, the
scenario is similar to the case of (2+1) DSMs, namely, the RSP becomes
irrelevant due to the interplay between the Weyl fermion and the PDW order
parameter. We thus conclude that the the emergent SUSY is stable against RSP.

Next, we consider the coexisting case. According to Eq. (28), the coexistence
of two components of RVP can dynamically generate RSP even when RSP does not
exist at the beginning. We also learn from Eq. (29) that the coexistence of
RSP and any component of RVP produce the other two components. Therefore, we
need to consider the generic case in which all three components of RVP coexist
with RSP. Now the disorder effects should be analyzed by solving the complete
set of equations given by Eq. (25) - (29). It is hard to solve these coupled
equations, but fortunately, it is simple to show that in the weak disorder
regime the SUSY fixed point is stable against all random potentials, similar
to the case of (2+1) DSMs.

##  IV Summary and discussion

In summary, we perform a standard perturbative RG to study the effect of
disorder on the emergent SUSY in (2+1)D DSMs [37, 27] and (3+1)D WSMs [37].
According to our RG results, the effective SUSY fixed point is robust against
any weak disorder irrespective of the type of disorder potentials. Our RG
analysis of the disorder effects on the emergent SUSY appearing in (2+1)D DSMs
[37, 27] and (3+1)D WSMs [37] can be directly extended to other analogous
models, which may be more realistic to detect emergent SUSY in quantum
materials. In our one-loop RG analysis, we have omitted new vertex that could
be generated from the disorder potential and the Yukawa coupling. It will be
interesting to include their effects although at tree-level they are
irrelevant. Our work can shed new light on the understanding and exploring the
emergent SUSY in realistic condensed matter systems.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China (No.
12147104). The work of SKJ is also supported in part by the Simons Foundation
via the It From Qubit Collaboration.

## Appendix A: Disorder potentials

In this section, we briefly consider the possible disorder potential in the
two-component Dirac (Weyl) system. The original spinful fermion annihilation
operator $\psi$ can be expand around two Dirac (Weyl) point $\bm{Q}_{\pm}$,

$\displaystyle\psi(\bm{x})=e^{-i\bm{Q}_{+}\cdot\bm{x}}\psi_{+}(\bm{x})+e^{-i\bm{Q}_{-}\cdot\bm{x}}\psi_{-}(\bm{x}),$
(35)

where $\psi_{\pm}$ corresponds to the two low-energy Dirac (Weyl) fermion. For
the fermion system $\psi$, a general disorder potential takes the form,

$\displaystyle H_{\mathrm{dis}}=\int
d^{d}\bm{x}\psi^{\dagger}(\bm{x})\left(\sum_{\Gamma}V_{\Gamma}(\mathbf{x})\Gamma\right)\psi(\bm{x}),$
(36)

where the $V_{\Gamma}(\mathbf{x})$ stands for the randomly distributed
potential and $\Gamma=I,\sigma_{\mu}$ ($\mu=x,y,z$) labels type of the
disorder potential. We focus on the quenched disorder potential
$V_{\Gamma}(\mathbf{x})$, with Gaussian white noise potential characterized by
the following identities:

$\displaystyle\langle V_{\Gamma}(\mathbf{x})\rangle=0,\,\langle
V_{\Gamma}(\mathbf{x})V_{\Gamma^{{}^{\prime}}}(\mathbf{x}^{\prime})\rangle=\Delta_{\Gamma}\delta_{\Gamma\Gamma^{{}^{\prime}}}\delta^{d}(\mathbf{x}-\mathbf{x}^{\prime}).$
(37)

Substituting the fermion field Eq. (35) into the disordered Hamiltonian
$H_{\text{dis}}$, we can arrive at the disorder Hamiltonian for the two low-
energy Dirac (Weyl) fermion $\psi_{\pm}$. Notice that for the coupling between
the two Dirac field, e.g. $\psi_{+}^{\dagger}(\bm{x})\Gamma\psi_{-}(\bm{x})$,
there is an overall oscillating factor $e^{\pm
i(\bm{Q}_{+}+\bm{Q}_{-})\cdot\bm{x}}$. Such disorder potential coupled the two
Dirac (Weyl) fermion $\psi_{\pm}$ will be small in general. We only need to
consider the disorder Hamiltonian Eq. (5) with respect to a piece of Dirac
fermion. It should be emphasized that the two pieces of Dirac (Weyl) fermion
share the same statistical distribution of random potential.

## Appendix B: RG details

We present here the detailed calculation of Fig. 2(a)-Fig. 2(h) as well as the
RG equations in $(2+1)$D, the calculation for $(3+1)$D is directly followed,
which is not detailed shown here.

From the free action of fermions and bosons,

$\displaystyle S_{f0}$ $\displaystyle=\int
d^{2}xd\tau\sum_{n=\pm}\psi_{n}^{\dagger}\Big{[}\partial_{\tau}+iv_{f}\sum_{j=1}^{2}\gamma_{j}\partial_{j}\Big{]}\psi_{n},$
(38) $\displaystyle S_{b0}$ $\displaystyle=\int
d^{2}xd\tau\bigg{\\{}\sum_{n=\pm}\Big{[}|\partial_{\tau}\phi_{n}|^{2}+v_{b}^{2}\sum_{j=1}^{2}|\partial_{j}\phi_{n}|^{2}\bigg{\\}},$
(39)

the free propagator for fermions and bosons are

$\displaystyle G_{0}(k)$
$\displaystyle=\frac{1}{ik_{\tau}-v_{f}\bm{\gamma}\cdot\mathbf{k}};$
$\displaystyle D_{0}(k)$
$\displaystyle=\frac{1}{k_{\tau}^{2}+v_{b}^{2}\mathbf{k}^{2}}.$

with $k=(k_{\tau},\bm{k})$ in the momentum space through replacement
$(\partial_{\tau},\partial_{i})\rightarrow(ik_{\tau},ik_{i})$. The free
propagators for the two pieces of Dirac fermion take the same form. For Fig.
2(a), it corresponds

$\displaystyle\Pi(k)$
$\displaystyle=-2g^{2}\int_{p}\text{Tr}\left[\sigma^{y}G_{0}^{T}(p)\sigma^{y}G_{0}(-p-k)\right]$
$\displaystyle=\frac{g^{2}S_{d}\Lambda^{-\epsilon}}{2(2\pi)^{d}v_{f}^{3}}l\Big{(}k_{\tau}^{2}+\Big{(}2-\frac{3}{d}\Big{)}v_{f}^{2}\bm{k}^{2}\Big{)}$
(40)

where $\int_{p}\equiv\int dp_{\tau}d^{d}\bm{p}/(2\pi)^{D}$ is the $D=d+1$
dimensional momentum integral and $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the area
of the unit sphere in $d$ dimensions. For Fig. 2(c), it gives

$\displaystyle\Sigma_{f}^{(b)}(k)=-4g^{2}(-)\int_{p}\sigma^{y}G_{0}^{T}(p)\sigma^{y}D_{0}(-k-p)$
$\displaystyle=\frac{g^{2}S_{d}\Lambda^{-\epsilon}}{2(2\pi)^{d}v_{f}^{3}}l\Big{(}\frac{4(ik_{\tau})}{a(a+1)^{2}}-\frac{4(2a+1)}{ad(a+1)^{2}}(v_{f}\bm{\gamma}\cdot\mathbf{k})\Big{)}$
$\displaystyle=\frac{g^{2}S_{d}\Lambda^{-\epsilon}}{2(2\pi)^{d}v_{f}^{3}}l\Big{(}G_{0}(ik_{\tau})-G_{1}(v_{f}\bm{\gamma}\cdot\mathbf{k})\Big{)}$
(41)

The diagram of Fig. 2(d) is ,

$\displaystyle\Sigma_{f}^{(c)}(k)$
$\displaystyle=-\sum_{\Gamma}(\Delta_{\Gamma}+\Delta_{\Gamma}^{\prime})\int\frac{d^{d}\mathbf{k}}{(2\pi)^{d}}\Gamma
G_{0}(k)\Gamma$
$\displaystyle=\sum_{\Gamma}(\Delta_{\Gamma}+\Delta_{\Gamma}^{\prime})\frac{\Lambda^{1-\epsilon}S_{d}}{(2\pi)^{d}v_{f}^{2}}(ik_{\tau})\ell,$
(42)

which only contribute to the velocity renormalization at one-loop. The diagram
of Fig. 2(e) is given by

$\displaystyle\delta\Delta_{\Gamma}^{(d)}=$
$\displaystyle-8\Delta_{\Gamma}{g}^{2}\int_{k}\sigma^{y}D(-k)G^{T}(k)\Gamma^{T}G^{T}(k)\sigma^{y}$
(43) $\displaystyle\delta\Delta_{\Gamma}^{\prime(d)}=$
$\displaystyle-8\Delta_{\Gamma}^{\prime}{g}^{2}\int_{k}\sigma^{y}D(-k)G^{T}(k)\Gamma^{T}G^{T}(k)\sigma^{y}$
(44)

Calculating out these integrals for different $\Gamma_{a}$ one by one, we have

$\displaystyle\delta\Delta_{S}^{(d)}=$
$\displaystyle\Delta_{S}\frac{g^{2}S_{d}\Lambda^{-\epsilon}}{2(2\pi)^{d}v_{f}^{3}}(G_{0}-dG_{2}),$
(45) $\displaystyle\delta\Delta_{M}^{(d)}=$
$\displaystyle\Delta_{M}\frac{g^{2}S_{d}\Lambda^{-\epsilon}}{2(2\pi)^{d}v_{f}^{3}}(-G_{0}-dG_{2}),$
(46) $\displaystyle\delta\Delta_{V}^{(d)}=$
$\displaystyle\Delta_{V}\frac{g^{2}S_{d}\Lambda^{-\epsilon}}{2(2\pi)^{d}v_{f}^{3}}(-G_{0}),$
(47)

and same for $\Delta\rightarrow\Delta^{\prime}$. The diagram of Fig. 2(b)
involves $\Delta_{\gamma}$-vertex and is given by

$\displaystyle\delta
g^{(h)}=g\sum_{\Gamma}\Delta_{\Gamma}\int_{\bm{k}}\Gamma^{T}G_{+}^{T}(k)\sigma^{y}G_{+}(-k)\Gamma$
$\displaystyle=$ $\displaystyle
gl\sum_{\Gamma}\frac{\Delta_{\Gamma}\Lambda^{1-\epsilon}S_{d}}{(2\pi)^{d}v_{f}^{2}}\Gamma^{T}\sigma^{y}\Gamma$
(48)

The remaining diagrams Fig. 2(f,g,h) correspond to pure coupling between
disorder potentials [67], which can be obtained as follows,

$\displaystyle\delta\Delta_{S}^{(e,f,g)}=$
$\displaystyle\Big{(}\frac{S_{d}}{(2\pi)^{d}v_{f}^{2}\Lambda^{2-d}}\Big{)}l\Big{[}+2\Delta_{S}\big{(}\Delta_{S}+\Delta_{M}+2\Delta_{V}\big{)}+4\Delta_{M}\Delta_{V}\Big{]}$
(49) $\displaystyle\delta\Delta_{M}^{(e,f,g)}=$
$\displaystyle\Big{(}\frac{S_{d}}{(2\pi)^{d}v_{f}^{2}\Lambda^{2-d}}\Big{)}l\Big{[}-2\Delta_{M}\big{(}\Delta_{S}+\Delta_{M}-2\Delta_{V}\big{)}+4\Delta_{S}\Delta_{V}\Big{]}$
(50) $\displaystyle\delta\Delta_{V}^{(e,f,g)}=$
$\displaystyle\Big{(}\frac{S_{d}}{(2\pi)^{d}v_{f}^{2}\Lambda^{2-d}}\Big{)}l\Big{[}2\Delta_{M}\Delta_{S}\Big{]}$
(51)

and similarly for the other three terms,

$\displaystyle\delta\Delta_{S}^{\prime(e,f,g)}=$
$\displaystyle\Big{(}\frac{S_{d}}{(2\pi)^{d}v_{f}^{2}\Lambda^{2-d}}\Big{)}l\Big{[}+2\Delta_{S}^{\prime}\big{(}\Delta_{S}+\Delta_{M}+2\Delta_{V}\big{)}+4\Delta_{M}^{\prime}\Delta_{V}^{\prime}\Big{]}$
(52) $\displaystyle\delta\Delta_{M}^{\prime(e,f,g)}=$
$\displaystyle\Big{(}\frac{S_{d}}{(2\pi)^{d}v_{f}^{2}\Lambda^{2-d}}\Big{)}l\Big{[}-2\Delta_{M}^{\prime}\big{(}\Delta_{S}+\Delta_{M}-2\Delta_{V}\big{)}+4\Delta_{S}^{\prime}\Delta_{V}^{\prime}\Big{]}$
(53) $\displaystyle\delta\Delta_{V}^{\prime(e,f,g)}=$
$\displaystyle\Big{(}\frac{S_{d}}{(2\pi)^{d}v_{f}^{2}\Lambda^{2-d}}\Big{)}l\Big{[}2\Delta_{M}^{\prime}\Delta_{S}^{\prime}\Big{]}$
(54)

According to above results, label and rescale the couplings as follows:

$\displaystyle G_{0}=\frac{4}{a(1+a)^{2}},\quad
G_{1}=\frac{4(1+2a)}{da(1+a)^{2}},\quad
G_{2}=\frac{4(2+a)}{da(1+a)^{2}},\quad\frac{g^{2}\Lambda^{-\epsilon}S_{d}}{2(2\pi)^{d}v_{f}^{3}}\rightarrow
g^{2},\quad\frac{\Delta_{\Gamma}\Lambda^{1-\epsilon}S_{d}}{(2\pi)^{d}v_{f}^{2}}\rightarrow\Delta_{\Gamma}.$
(55)

Then, the results obtained for Fig. 2(a)-Fig. 2(h) can be simplified, leading
to the one-loop quantum corrections of the action. These one-loop results
produce the RG equations

$\displaystyle\frac{dv_{f}}{dl}$
$\displaystyle=v_{f}\Big{[}g^{2}(G_{1}-G_{0})-\sum_{\Gamma}\Delta_{\Gamma}-\sum_{\Gamma}\Delta_{\Gamma}^{\prime}\Big{]},$
(56) $\displaystyle\frac{da}{dl}$
$\displaystyle=g^{2}\Big{(}\frac{1-a^{2}}{2a}+a(G_{0}-G_{1})\Big{)}+a\sum_{\Gamma}\Delta_{\Gamma}+a\sum_{\Gamma}\Delta_{\Gamma}^{\prime},$
(57) $\displaystyle\frac{dg^{2}}{dl}$ $\displaystyle=\epsilon
g^{2}-g^{4}\big{(}1-G_{0}+3G_{1}\big{)}+\big{(}4\Delta_{S}-\sum_{\Gamma}\Delta_{\Gamma}+3\sum_{\Gamma}{\Delta}_{\Gamma}^{\prime}\big{)}g^{2},$
(58) $\displaystyle\frac{d\Delta_{S}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{S}+2\Delta_{S}\big{(}\Delta_{S}+\Delta_{M}+2\Delta_{V}\big{)}+4\Delta_{M}\Delta_{V}+(G_{0}-3G_{2}-2G_{1})\Delta_{S}g^{2},$
(59) $\displaystyle\frac{d\Delta_{M}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{M}-2\Delta_{M}\big{(}\Delta_{S}+\Delta_{M}-2\Delta_{V}\big{)}+4\Delta_{S}\Delta_{V}-(G_{0}+3G_{2}+2G_{1})\Delta_{M}g^{2},$
(60) $\displaystyle\frac{d\Delta_{V}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{V}+2\Delta_{M}\Delta_{S}-(G_{0}+2G_{1})\Delta_{V}g^{2},$
(61) $\displaystyle\frac{d\Delta_{S}^{\prime}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{S}^{\prime}+2\Delta_{S}^{\prime}\big{(}\Delta_{S}+\Delta_{M}+2\Delta_{V}\big{)}+4\Delta_{M}^{\prime}\Delta_{V}^{\prime}+(G_{0}-3G_{2}-2G_{1})\Delta_{S}^{\prime}g^{2},$
(62) $\displaystyle\frac{d\Delta_{M}^{\prime}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{M}^{\prime}-2\Delta_{M}^{\prime}\big{(}\Delta_{S}+\Delta_{M}-2\Delta_{V}\big{)}+4\Delta_{S}^{\prime}\Delta_{V}^{\prime}-(G_{0}+3G_{2}+2G_{1})\Delta_{M}^{\prime}g^{2},$
(63) $\displaystyle\frac{d\Delta_{V}^{\prime}}{dl}$
$\displaystyle=\big{(}\epsilon-1\big{)}\Delta_{V}^{\prime}+2\Delta_{M}^{\prime}\Delta_{S}^{\prime}-(G_{0}+2G_{1})\Delta_{V}^{\prime}g^{2},$
(64)

Since the initial values of the parameters are
$\Delta_{\Gamma}^{0}=\Delta_{\Gamma}^{\prime 0}$, $\Delta_{\Gamma}$ and
$\Delta_{\Gamma}^{\prime}$ will flow in the same way. We can simplify the
equations by taken $\Delta_{\Gamma}^{\prime}=\Delta_{\Gamma}$ and produce the
Eqs. (8) - (13).

For the case of $(3+1)D$, due to the change of disorder typies, the one-loop
corrections of disorder couplings need to recalculate, the extension is direct
for which we do not show details here.

## Appendix C: general disorder case

In this section, we study the case where all the disorder potential is
appeared. We focus on the regime near the Lorentz symmetric fixed point,
$a=1+\delta a$ with small $\delta a\ll 1$. The RG equations in 2D now become,

$\displaystyle\frac{dv_{f}}{dl}$ $\displaystyle=$ $\displaystyle
v_{f}\Big{[}g^{2}\frac{2}{3}\delta a-2\sum_{\Gamma}\Delta_{\mu}\Big{]},$ (65)
$\displaystyle\frac{d\delta a}{dl}$ $\displaystyle=$
$\displaystyle\Big{(}-\frac{5}{3}g^{2}+2\sum_{\Gamma}\Delta_{\Gamma}\Big{)}\delta
a,$ (66) $\displaystyle\frac{dg^{2}}{dl}$ $\displaystyle=$
$\displaystyle\epsilon g^{2}-g^{4}\big{(}3-2\delta
a\big{)}+\big{(}4\Delta_{S}+2\sum_{\Gamma}\Delta_{\Gamma}\big{)}g^{2},$ (67)
$\displaystyle\frac{d\Delta_{S}}{dl}$ $\displaystyle=$
$\displaystyle\big{(}\epsilon-1\big{)}\Delta_{S}+2\Delta_{S}\big{(}\Delta_{S}+\Delta_{M}+2\Delta_{V}\big{)}+4\Delta_{M}\Delta_{V}+(-4+\frac{17}{3}\delta
a)\Delta_{S}g^{2},$ (68) $\displaystyle\frac{d\Delta_{M}}{dl}$
$\displaystyle=$
$\displaystyle\big{(}\epsilon-1\big{)}\Delta_{M}-2\Delta_{M}\big{(}\Delta_{S}+\Delta_{M}-2\Delta_{V}\big{)}+4\Delta_{S}\Delta_{V}-(6-\frac{29}{3}\delta
a)\Delta_{M}g^{2},$ (69) $\displaystyle\frac{d\Delta_{V}}{dl}$
$\displaystyle=$
$\displaystyle\big{(}\epsilon-1\big{)}\Delta_{V}+2\Delta_{M}\Delta_{S}-(3-\frac{14}{3}\delta
a)\Delta_{V}g^{2},$ (70)

There is a fixed point given by
$(v_{f},a^{*},g^{2\ast},\Delta_{S}^{\ast},\Delta_{M}^{\ast},\Delta_{V}^{\ast})=(v_{f},1,\frac{\epsilon}{3},0,0,0)$.
For small disorder case, the stability matrix at this fixed point is

$\displaystyle\left(\begin{array}[]{cccccc}0&\frac{2}{9}\epsilon
v_{f}&0&-2v_{f}&-2v_{f}&-2v_{f}\\\ 0&-\frac{5}{9}\epsilon&0&2&2&2\\\
0&\frac{2}{9}\epsilon^{2}&-\epsilon&2\epsilon&\frac{2}{3}\epsilon&\frac{2}{3}\epsilon\\\
0&0&0&-1-\frac{\epsilon}{3}&0&0\\\ 0&0&0&0&-1-\epsilon&0\\\
0&0&0&0&0&-1\end{array}\right).$ (77)

The eigenvalues of the stability matrix are all negative except for one
marginal direction at $v_{f}$. We can conclude that the coupling between
Yukawa potential and disorder will suppress weak random disorder potential.
Similarly argument can also be applied to three dimensional case.

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