text:
\section{Introduction}

Dirac fermions have been becoming one of most significant subjects in contemporary condensed matter
physics~\cite{Novoselov2005Nature,Neto2009RMP,Kane2007PRL,Roy2009PRB,Moore2010Nature,Hasan2010RMP,
Qi2011RMP,Sheng2012Book,Bernevig2013Book} in the last decade and attract a vast of both experimental~\cite{Novoselov2005Nature,Neto2009RMP,Hasan2010RMP,Qi2011RMP,
Sheng2012Book,Bernevig2013Book} and theoretical efforts~\cite{Neto2009RMP,Kane2007PRL,Roy2009PRB,
Moore2010Nature,Hasan2010RMP,Qi2011RMP,Sheng2012Book,Bernevig2013Book}. These Dirac systems harbor
a variety of types, such as two-dimensional (2D) graphene~\cite{Neto2009RMP}, Weyl~\cite{Neto2009RMP,Burkov2011PRL,Yang2011PRB,Savrasov2011PRB,
Huang2015PRX,Weng2015PRX,Hasan2015Science,Hasan2015NPhys,Ding2015NPhys},
Dirac~\cite{WangFang2012PRB,Young2012PRL,Steinberg2014PRL,Hussain2014NMat,
LiuChen2014Science,Ong2015Science} semimetals and etc., which usually possess
several discrete Dirac nodal points with gapless low-energy excitations
and display a linear dispersion in two or three directions~\cite{Neto2009RMP,Hasan2010RMP,Qi2011RMP,
Huang2015PRX,Hasan2015Science,Ding2015NPhys}. Recently, some groups elucidate~\cite{Hasegawa2006PRB,Pardo2009PRL,Katayama2006JPSJ,Dietl2008PRL,Delplace2010PRB,Wu2014Expre}
that there exist another kind of analogous material, dubbed as the 2D semi-Dirac (SD) material,
which is of remarkable interest 2D Dirac-like material as its dispersion is parabolic in one
direction and linear in the other, such as the $\mathrm{VO_2-TiO_2}$ multilayer systems
(nanoheterostructures)~\cite{Pardo2009PRL}, quasi-two dimensional organic conductor
$\alpha-(\mathrm{BEDT-TTF)_2I_3}$ salt under uniaxial pressure~\cite{Katayama2006JPSJ},
tight-binding honeycomb lattices for the presence of a magnetic field~\cite{Dietl2008PRL},
and photonic systems consisting of a square array of elliptical dielectric cylinders~\cite{Wu2014Expre}.
These unique and unusual properties of the low-energy excitations~\cite{Banerjee2009PRL,Delplace2010PRB,Saha2016PRB}
may result in distinct behaviors from the general Dirac fermions in the low-energy regime~\cite{Neto2009RMP,Hasan2010RMP,Qi2011RMP,Huang2015PRX, Hasan2015Science,Ding2015NPhys}.
Recently, K. Saha~\cite{Saha2016PRB} study this 2D SD system and indeed found that
some unconventional properties can be engineered. For instance, an electromagnetic field
can induce certain topological phase transition and the Chern number can be changed qualitatively
during this phase transition in the presence of light. Moreover, Uchoa and Seo~\cite{Uchoa1704.08780}
have pointed out that the superconductor quantum critical point can be actualized by
electric field and strain effects.

It is worth pointing out that the fermion-fermion interactions have been insufficiently
taken into account in previous studies. In this circumstance, the information of correlated
physical properties that are closely pertinent to these locally four-fermion interactions, in particular
the stability of ground states, the distance of Dirac nodal points in the semimetal states,
the transport properties and so on~\cite{Sachdev1999Book,Altland2002PR,
Lee2006RMP,Neto2009RMP,Fradkin2010ARCMP,Hasan2010RMP,Sarma2011RMP,Qi2011RMP,
Kotov2012RMP}, may be partially neglected or cannot be fully captured especially in the
low-energy regime. Incorporating more potential ingredients of physical degrees
of freedom in low-energy regime
would be of significant help to improve the description of their low-energy behaviors~\cite{Sachdev1999Book,
Altland2002PR,Lee2006RMP,Neto2009RMP,Fradkin2010ARCMP,Sarma2011RMP,
Kotov2012RMP}.  It is therefore imperative to examine what the role is played by the contribution from
fermion-fermion interactions to these low-energy phenomena of physical
properties in the 2D SD materials.

Within this work, we concentrate on four primary sorts of
short-range four-fermion interactions as designated in Eq.~(\ref{Eq_S_eff}). Additionally,
impurities are well-known to be present in the real systems and usually can play an essential
role in modern condensed matter physics~\cite{Ramakrishnan1985RMP,Nersesyan1995NPB,Mirlin2008RMP},
which can give rise to a plenty of prominent phenomena in low-energy regime~\cite{Ramakrishnan1985RMP,
Nersesyan1995NPB,Mirlin2008RMP,Efremov2011PRB,Efremov2013NJP,Korshunov2014PRB,Fiete2016PRB,
Nandkishore2013PRB,Potirniche2014PRB,Nandkishore2017PRB,Roy2016PRB-2,Roy1604.01390,Roy1610.08973,Roy2016SR}.
In Fermi systems, there conventionally own three sensible types of impurities~\cite{Nersesyan1995NPB,Stauber2005PRB}
that usually be named as random chemical potential, random mass, or random gauge potential~\cite{Nersesyan1995NPB,Stauber2005PRB,Wang2011PRB,Wang2013PRB} depending
upon their distinct couplings with fermions presented in Eq.~(\ref{Eq_S_f-d}). It has been proved that
these impurities can drive a multitude of interesting and unusual behaviors of physical
properties in these fermionic systems as widely shown in previous studies~\cite{Sachdev1999Book,Altland2002PR,Novoselov2005Nature,Aleiner2006PRL,Aleiner2006PRL-2,
Lee2006RMP,Neto2009RMP,Fradkin2010ARCMP,Hasan2010RMP,Sarma2011RMP,Wang2011PRB,Wang2013PRB,
Kotov2012RMP,Lee1702.02742}. In order to collect more physical information and fully understand
unconventionally physical properties in the low-energy regime, we suggeste to
turn on the four-fermion interactions and assume the presence of certain
amount of impurities simultaneously. At this stage, an intriguing question is therefore naturally raised
whether the basic results in noninteracting case with clean limit can be revised or even qualitatively
changed by the interplay between these four-fermion interactions and impurities.
Unambiguously answering this question would be of great profit for us to
further understand and uncover the unique properties of 2D SD materials and
even instructive to explore new Dirac-like materials.

In this paper, we are going to treat all the four types of short-range
four-fermion interactions and three kinds of impurities on the same footing via adopting the
momentum-shell renormalization group (RG) approach~\cite{Wilson1975RMP,Polchinski9210046,Shankar1994RMP}.
In this respect, the effects of four-fermion interactions, impurities, and
their interplays can be fully and unbiasedly incorporated into our consideration.
After taking into account all one-loop corrections from the competition between four-fermion
interaction and impurity, the coupled flow equations
of all associated interaction parameters for both the pure four-fermion interactions and presence of
impurity are derived after practicing the standard RG
analysis~\cite{Wilson1975RMP,Polchinski9210046,Shankar1994RMP,Wang2011PRB}.
To proceed, we employ these coupled flow equations that determine the evolutions of interaction
parameters replying on the energy scales to investigate whether and how the
low-energy behaviors of 2D SD systems can be affected or revised compared to
their clean and noninteracting counterparts, and additionally potential phenomena
would be trigged.

After performing both theoretical and numerical analysis, we find that several interesting
results have been extracted from these evolutions of all the correlated interaction
parameters. First, all of four-fermion parameters, to one-loop level,
are irrelevant in the RG language~\cite{Shankar1994RMP,Huh2008PRB,Wang2011PRB}.
This means the contribution from four-fermion
interactions becomes less and less significant and finally vanishes at the
lowest-energy limit~\cite{Shankar1994RMP}. As a result, these energy-dependent
interaction parameters at clean limit cannot qualitatively change the low-energy
behaviors of the 2D SD system~\cite{Shankar1994RMP,Huh2008PRB,Wang2011PRB}.
Consequently, both the trivial insulating and nontrivial semimetal states
are very stable against the four-fermion interactions. However, in the case of presence
of both four-fermion interaction and impurity, we find that the fates of these interaction
parameters can be modified qualitatively under certain initial conditions due to the strong
between fermion-fermion interactions and impurities. To be concrete,
these irrelevant interaction parameters can be transferred to irrelevant relevant
couplings after taking into account one-loop corrections
from interplay between fermion-fermion interactions and impurity~\cite{Wilson1975RMP,Polchinski9210046,
Shankar1994RMP,Wang2011PRB}, which lead to the divergence of interaction
parameters and instabilities at the critical energy scale. This conventionally suggests
that some phase transition~\cite{Fradkin2009PRL,Vafek2012PRB,Vafek2014PRB,
Wang2017QBCP,Altland2006Book,Vojta2003RPP,Metzner2000PRL,Metzner2000PRB,Wang2014PRB,Chubukov2010PRB,
Chubukov2012ARCMP,Chubukov2016PRX}, in our system certain insulator-semimetal phase transition expected,
would be generated under certain conditions although the states are qualitatively
stable against solely four-fermion interactions. In addition, we are informed that
the distance of two Dirac nodal points in the semimetal phase is sensitive
to the interplay and competition between the four-fermion interaction and impurity.
Specifically, we find that the four-fermion interactions and impurity respectively
decrease and increase the distance of Dirac nodal points.  One may expect that the
revisions of distance of Dirac nodal point would affect the interplay between the gapless
excitations in the low-energy regime. Moreover, several non-Fermi liquid
behaviors that deviate from the properties of conventional Fermi liquids theory~\cite{Altland2002PR},
for instance the quasiparticle residue $Z_f$ and the density of states (DOS) of the quasiparticle,
are obviously displayed at the lowest-energy limit caused by the intimate interplay and competition between
four-fermion interaction and random gauge potential or random mass.

The rest of this paper is organized as follows. In Sec.~\ref{Sec_model}, we bring out
the microscopic model and derive our effective quantum field theory. The forthcoming is
Sec.~\ref{Sec_RG_clean} that we provide the RG transformations for momenta, energies
and fields. All the one-loop corrections to the coupling parameters at clean limit
are computed in this section. In Sec.~\ref{Sec_RG_imp}, we subsequently move to derive
the coupled flow equations of all related parameters by means of the standard RG analysis
in the presence of both four-fermion interaction and impurity. In Sec.~\ref{Sec_int_delta},
we examine the stability of the ground states of 2D SD systems against the four-fermion
interactions at clean limit. Sec.~\ref{Sec_int-imp} and Sec.~\ref{Sec_NFL} are
accompanied by studying the low-energy behaviors affected by the interplay between
four-fermion interaction and impurity. Finally, we present a short summary in Sec.~\ref{Sec_summary}.

\section{Model and effective theory}\label{Sec_model}

\subsection{Noninteracting model}

The noninteracting Hamiltonian describing low-energy electronic bands of a SD
material can be generally proposed as~\cite{Banerjee2009PRL,Delplace2010PRB,Saha2016PRB}
\begin{eqnarray}
\mathcal{H}_0(\mathbf{k})=\mathbf{d}(\mathbf{k})\cdot\sigma,\label{Eq_H_0}
\end{eqnarray}
with $\mathbf{d}(\mathbf{k})=(\alpha k^2_x-\delta,vk_y,0)$ and $\mathbf{k}=(k_x,k_y,0)$.
Here, the parameters $\alpha$, $v$, and $\delta$ describe the inverse of quasiparticle
mass along $x$, the Dirac velocity along $y$, and the energy gap, respectively. In the rest of
this paper, we restrict the focus on the 2D systems. In this respect, we would easily
arrive at the the energy eigenvalues from this noninteracting
Hamiltonian (\ref{Eq_H_0})~\cite{Saha2016PRB}
\begin{eqnarray}
E^{\pm}(\mathbf{k})=\pm\sqrt{(\alpha k^2_x-\delta)^2+v^2k^2_y},\label{Eq_alpha}
\end{eqnarray}
with the notations $\pm$ respectively corresponding to the conduction and valence bands.
There are in all three potential ground states~\cite{Banerjee2009PRL,Delplace2010PRB,
Huang2015PRB,Saha2016PRB}: (i) $\delta=0$, the linear dispersion for $k_y$ direction
and parabolical for $k_x$ direction with a gapless spectrum; (ii) $\delta<0$, a trivial
insulating phase with a nonzero energy gap; and (iii) $\delta>0$, a 2D SD semimetal state
with two gapless Dirac nodal points at $(\pm\sqrt{\frac{\delta}{\alpha}},0)$.

We subsequently can address the noninteracting effective action
after performing several conventional transformations,
\begin{eqnarray}
S_0&=&\int\frac{d\omega}{(2\pi)}\int\frac{d^2\mathbf{k}}{(2\pi)^2}\Psi^\dagger(i\omega,\mathbf{k})
\Bigl[-i\omega+(\alpha k^2_x-\delta)\sigma_1\nonumber\\ \nonumber\\
&&+vk_y\sigma_2\Bigr]\Psi(i\omega,\mathbf{k}),\label{Eq_S_0}
\end{eqnarray}
with the spinor $\Psi(i\omega,\mathbf{k})$ characterizing the low-energy excitations from the Dirac nodes.
The $\sigma_j,j=1,2,3$ corresponds to the Pauli matrices. In order to study the low-energy behaviors,
we need to include the fermionic interactions besides this free term.

\begin{figure}
\centering
\includegraphics[width=3.0in]{Fig1_self_energy.eps}
\vspace{0.05cm}
\caption{One-loop corrections to the fermion propagator at clean limit (the dashed
line indicates the four-fermion interaction).}\label{Fig_fermion_propagator_correction}
\end{figure}

\subsection{Effective theory}

To proceed, we introduce the four-fermion interactions to construct our effective action.
After bringing out the four allowed kinds of short-range four-fermion interactions incorporated
into the non-interacting action (\ref{Eq_S_0})~\cite{Fradkin2009PRL,Vafek2012PRB,Vafek2014PRB,Wang2017QBCP},
we henceforth are left with the updated effective action
\begin{widetext}
\begin{eqnarray}
S_{\mathrm{eff}}&=&\int\frac{d\omega}{2\pi}\int\frac{d^2\mathbf{k}}{(2\pi)^2}\Psi^\dagger(i\omega,\mathbf{k})
[-i\omega+(\alpha k^2_x+\delta)\sigma_1+vk_y\sigma_2]\Psi(i\omega,\mathbf{k})+
\sum^3_{a=0}u_a\int\frac{d\omega_1d\omega_2d\omega_3}{(2\pi)^3}\int
\frac{d^2\mathbf{k}_1d^2\mathbf{k}_2d^2\mathbf{k}_3}{(2\pi)^{6}}\nonumber\\
&&\times\Psi^\dagger(\omega_1,\mathbf{k}_1)\sigma_a\Psi(\omega_2,\mathbf{k}_2)
\Psi^\dagger(\omega_3,\mathbf{k}_3)\sigma_a
\Psi(\omega_1+\omega_2-\omega_3,\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3).\label{Eq_S_eff}
\end{eqnarray}
\end{widetext}
Here, the $\sigma_j,j=1,2,3$ again delineates the Pauli matrices and  $\sigma_0=I_{2\times2}$ the unit
matrix. The $u_a$, $a=0,1,2,3$ collects all sorts of given four-fermion interactions. This effective action
allows us to directly extract the free fermionic propagator,
\begin{eqnarray}
G_0(\omega,\mathbf{k})
&=&\frac{1}{-i\omega+\alpha k^2_x\sigma_1+vk_y\sigma_2}.\label{Eq_propagators}
\end{eqnarray}

To simplify our study and facilitate the evaluations, we can regard the $\delta$-term as
a quadratic interaction and its evolution upon lowering the energy scale will be attained in
the impending sections~\cite{Shankar1994RMP}. Based on these, we can expect the corresponding state
which the system locates at initially. In such circumstances, we are allowed to investigate how the
four-fermion interactions affect the low-energy behaviors of this 2D SD materials
via performing the momentum-shell RG analysis~\cite{Wilson1975RMP,Polchinski9210046,Shankar1994RMP}
of our effective theory~(\ref{Eq_S_eff}).

\section{One-loop RG analysis}\label{Sec_RG_clean}

\subsection{RG rescaling transformations}

According to the standard procedures of RG framework~\cite{Shankar1994RMP,Kim2008PRB,Huh2008PRB,
She2010PRB,She2015PRB,Roy2016PRB,Wang2011PRB,Wang2013PRB,Wang2013NJP,Vafek2012PRB,Wang2017QBCP,
Vafek2014PRB,Wang2014PRD,Wang2015PRB,Wang2017PRB}, we need to integrate out the fields
in the momentum shell $b\Lambda<k<\Lambda$ with $b<1$ to derive the evolutions of interaction
parameters, where $\Lambda$ represents the energy scale and the variable parameter $b$
can be written as $b=e^{-l}$ with a running energy scale $l>0$~\cite{Shankar1994RMP,
Kim2008PRB,Huh2008PRB,She2010PRB,She2015PRB,Wang2011PRB}.

Under the RG consideration, the parameters $\alpha$ and $v$ maybe flow upon lowering
the energy scale after collecting the one-loop interaction corrections. We will show later that the flows of
these two parameters would be revised and they enter into the coupled flow equations
of all interaction parameters after collecting the contribution from four-fermion interactions.
At this stage, all of interaction parameters are not independent but mutually and intimately
associated with others. Hence, the interacting couplings can also play a direct or indirect
role in the evolutions of the parameters $\alpha$ and $v$, which are essential to determine
the low-energy fate of 2D SD materials.

Before going to the one-loop RG calculations, we firstly dwell on the RG rescaling transformations.
In the spirt of momentum-shell RG theory~\cite{Shankar1994RMP,Huh2008PRB,She2010PRB,She2015PRB,Wang2011PRB},
we can choose the free action $S_0\sim-i\omega+\alpha k^2_x\sigma_1+vk_y\sigma_2$ as
the freely invariant fixed point that is invariant under the
RG transformation. As a result, we instantly address
the re-scaling transformations of momenta, energy and fermionic fields, namely,~\cite{Shankar1994RMP,Huh2008PRB,She2010PRB,She2015PRB,Wang2011PRB},
\begin{eqnarray}
k_{x}&=&k'_{x}e^{-\frac{1}{2}l},\label{Eq_RG_scales_k}\\
k_{y}&=&k'_{y}e^{-l},\label{Eq_RG_scales_k}\\
\omega&=&\omega'e^{-l},\label{Eq_RG_scales_omega}\\
\Psi(i\omega,\mathbf{k})&=&\Psi'(i\omega',\mathbf{k}')
e^{\frac{1}{2}\int_{0}^{l}dl\left(\frac{7}{2}-\eta_f\right)},\label{Eq_RG_scales_psi}
\end{eqnarray}
where the parameter $\eta_f$ delineates the anomalous fermion dimension~\cite{Shankar1994RMP,Huh2008PRB},
which would be generated by the four-fermion interaction
or impurities owning to their one-loop corrections.

\begin{figure}
\centering
\includegraphics[width=3.399in]{Fig2_interaction.eps}
\vspace{-0.3cm}
\caption{One-loop corrections to the four-fermion interacting couplings in clean limit (the dashed
line indicates the four-fermion interaction). All
subfigures (i)-(v) capture the contributions from purely four-fermion interactions distinguished by
different topological properties.}\label{Fig_fermion_interaction_correction}
\end{figure}

To proceed, we would like to present several clarifications on the RG re-scaling transformation
of energy $\omega$. It is indeed not unambiguous for us to choose its RG transformation
in that our microscopic model~(\ref{Eq_S_0}) owns a unconventional dispersion, namely,
linear in one direction and quadratical in the other, which is qualitatively distinct from two limit cases,
i.e. linear or quadratical in both $x$ and $y$ directions. Fortunately, it is worth highlighting that the
focus of present work is primarily on how the interaction parameters evolve upon lowering the energy
scale, in particular in the low-energy regime. This consequently suggests that the low-energy
degrees of freedom play a more significant role than their higher-energy counterparts.
In addition, learning from the noninteracting theory~(\ref{Eq_S_0}), the linear term $vk_y$ is
more dominant compared to the quadratical term $\alpha k^2_x$ while $k$ is adequate small
in the low-energy regime. These both indicate that the $z=1$ part (here $z$ depicts the dynamical
critical exponent) should be more important or take a leading responsibility for potentially
unique properties in the low-energy regime. Due to the exotic feature of dispersion
in the 2D SD material, one can expect the re-scaling transformation of $\omega$ (\ref{Eq_RG_scales_omega})
can properly capture the key ingredients of the low-energy physics. Therefore, we, in current work,
adopt the RG transformation of $\omega$, namely $\omega=\omega'e^{-l}$.

\subsection{Coupled flow equations}\label{Sec_RGeqs_clean}

In order to facilitate the performance of the standard momentum-shell RG analysis~\cite{Shankar1994RMP},
it is convenient to rescale the momenta and energy by $\Lambda_0$ that is linked to the lattice
constant, i.e. $k\rightarrow k/\Lambda_0$ and  $\omega\rightarrow\omega=\omega/\Lambda_0$,
and redefine the energy scale as $\Lambda=\Lambda_0/b$ with $b=e^{-l}$ and $l>0$ denoting
the changes of energy scales~\cite{Shankar1994RMP,Huh2008PRB,She2010PRB,Wang2011PRB}.
The free fermion propagator would receive one-loop corrections caused by the four-fermion
interaction as provided in Fig.~\ref{Fig_fermion_propagator_correction},
which can be explicitly expressed as
\begin{eqnarray}
\Sigma^{i}
&=&-\frac{u_1\mathcal{C}_1}{8\pi^2}\sigma_1l,\label{Eq_Sigma_i_clean}\\
\Sigma^{ii}&=&0,
\end{eqnarray}
with designating
\begin{eqnarray}
\mathcal{C}_1\equiv\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2\alpha\cos\theta}
{\sqrt{\alpha^2\cos^2\theta+v^2\sin^2\theta}}.\label{Eq_C_1}
\end{eqnarray}
These one-loop corrections to the fermion propagator directly results in
\begin{eqnarray}
\eta_f=0,\label{Eq_eta_clean}
\end{eqnarray}
In this sense, we are straightforwardly informed that $v$ does not evolve via lowering
the energy scale~\cite{Shankar1994RMP,Roy2016PRB}, namely
\begin{eqnarray}
\frac{dv}{dl}&=&0,\label{Eq_RGeqs_v_clean}
\end{eqnarray}
and
\begin{eqnarray}
\frac{d\alpha}{dl}&=&0.\label{Eq_RGeqs_alpha_clean}
\end{eqnarray}
By virtue of the RG transformations (\ref{Eq_RG_scales_k})-(\ref{Eq_RG_scales_psi}) and
collecting the one-loop corrections (\ref{Eq_Sigma_i_clean}) and (\ref{Eq_RGeqs_v_clean}), we subsequently
arrive at the flow equation of parameter $\delta$ after fulfilling the standard momentum-shell RG analysis~\cite{Shankar1994RMP,Huh2008PRB,She2010PRB,Wang2011PRB},
\begin{eqnarray}
\frac{d\delta}{dl}&=&\left(1-\frac{u_1\mathcal{C}_1}{8\pi^2}\right)\delta.
\end{eqnarray}

\begin{figure}
\centering
\includegraphics[width=3.35in]{Fig3_self_energy_imp.eps}
\vspace{-0.105cm}
\caption{One-loop corrections to the fermion propagator (self-energy) in the presence of impurities
(The dashed and wave lines respectively indicate the four-fermion interaction and
the impurity).}\label{Fig_fermion_propagator_correction_2}
\end{figure}

Next, we move to address the evolutions of the fermion-fermion interaction couplings.
All one-loop corrections to the
four sorts of four-fermion interactions are depicted in Fig.~\ref{Fig_fermion_interaction_correction}
and the detailed results are presented in Appendix~\ref{Appendix_four-fermion-interaction}.
For instance, there are in all five one-loop diagrams contributing to the fermion interacting
coupling $u_0$ as shown in Fig.~\ref{Fig_fermion_interaction_correction},
whose contributions are displayed in Eqs.~(\ref{Eq_S_i_ii_u_0})-(\ref{Eq_S_v_u_0}) of Appendix~\ref{Appendix_four-fermion-interaction}.
Thereafter, summarizing all of these related one-loop contributions~\cite{Shankar1994RMP,Huh2008PRB,
She2010PRB,She2015PRB,Wang2011PRB,Wang2013NJP,Wang2014PRD,Wang2015PRB,Wang2017PRB} forthrightly gives
rise to
\begin{eqnarray}
&&\delta S_{u_0}\nonumber\\
&=&\int\frac{d\omega_1d\omega_2d\omega_3}{(2\pi)^3}\int^{b}
\frac{d^2\mathbf{k}_1d^2\mathbf{k}_2d^2\mathbf{k}_3}{(2\pi)^6}
\Psi^\dagger(\omega_1,\mathbf{k}_1)\sigma_0\Psi(\omega_2,\mathbf{k}_2)\nonumber\\
&&\times\Psi^\dagger(\omega_3,\mathbf{k}_3)\sigma_0\Psi(\omega_1+\omega_2
-\omega_3,\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3)u_0\nonumber\\ \nonumber\\
&&\times \left[1-\frac{(u_2+u_3)(\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)
+u_1(3\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)}{8\pi^2}l\right]\nonumber\\ \nonumber\\
&\approx&\int\frac{d\omega_1d\omega_2d\omega_3}{(2\pi)^3}\int^{b}
\frac{d^2\mathbf{k}_1d^2\mathbf{k}_2d^2\mathbf{k}_3}{(2\pi)^6}
\Psi^\dagger(\omega_1,\mathbf{k}_1)\sigma_0\Psi(\omega_2,\mathbf{k}_2)\nonumber\\ \nonumber\\
&&\times\Psi^\dagger(\omega_3,\mathbf{k}_3)\sigma_0\Psi(\omega_1+\omega_2
-\omega_3,\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3)\nonumber\\ \nonumber\\
&&\times u_0e^{\left[-\frac{(u_2+u_3)(\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)
+u_1(3\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)}{8\pi^2}l\right]}\nonumber\\ \nonumber\\
&=&\int\frac{d\omega_1d\omega_2d\omega_3}{(2\pi)^3}\int^1
\frac{d^2\mathbf{k}_1d^2\mathbf{k}_2d^2\mathbf{k}_3}{(2\pi)^6}\Psi^\dagger(\omega_1,\mathbf{k}_1)
\sigma_0\Psi(\omega_2,\mathbf{k}_2)\nonumber\\ \nonumber\\
&&\times\Psi^\dagger(\omega_3,\mathbf{k}_3)\sigma_0\Psi(\omega_1+\omega_2
-\omega_3,\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}_3)\nonumber\\
&&\times u_0e^{\left[-1-\frac{(u_2+u_3)(\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)
+u_1(3\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)}{8\pi^2}\right]l},
\end{eqnarray}
which afterward results in the energy-dependent running equation of $u_0$ in the sprit
of momentum-shell RG theory~\cite{Shankar1994RMP,Huh2008PRB,She2010PRB,Wang2011PRB}
\begin{eqnarray}
\frac{du_0}{dl}
&=&\frac{u_0}{8\pi^2}\Bigl[(u_2+u_3)(\mathcal{C}_0-\mathcal{C}_2
-\mathcal{C}_3)\nonumber\\
&&+u_1(\mathcal{C}_0-3\mathcal{C}_2-\mathcal{C}_3)-4\pi^2\Bigr].
\end{eqnarray}

The RG flow equations of other coupling parameters can be deduced by paralleling above steps with employing
the corresponding one-loop corrections provided in Appendix~\ref{Appendix_four-fermion-interaction}.
After long but straightforwardly algebraic procedures, we eventually arrive at the coupled flow equations
of all related interaction parameters at clean limit~\cite{Shankar1994RMP,Huh2008PRB,She2010PRB,Wang2011PRB},
\begin{widetext}
\begin{eqnarray}
\frac{dv}{dl}&=&0,\label{Eq_RGeqs_alpha_clean}\\
\frac{d\alpha}{dl}&=&0,\label{Eq_RGeqs_alpha_clean}\\
\nonumber\\
\frac{d\delta}{dl}&=&\left(1-\frac{u_1\mathcal{C}_1}{8\pi^2}\right)\delta,\label{Eq_RGeqs_delta_clean}\\
\nonumber\\
\frac{du_0}{dl}&=&\frac{u_0(u_2+u_3)(\mathcal{C}_0-\mathcal{C}_2-\mathcal{C}_3)
+u_0u_1(\mathcal{C}_0-3\mathcal{C}_2-\mathcal{C}_3)-4\pi^2u_0}{8\pi^2},\label{Eq_RGeqs_u0_clean}\\
\nonumber\\
\frac{du_1}{dl}&=&\frac{u_1(u_0-u_2-u_3)(\mathcal{C}_0-\mathcal{C}_2)
+u_1(u_0-3u_2-u_3)\mathcal{C}_3-\mathcal{C}_2
(u^2_0+u^2_1+u^2_2+u^2_3)-4\pi^2u_1}{8\pi^2},\label{Eq_RGeqs_u1_clean}\\
\nonumber\\
\frac{du_2}{dl}&=&\frac{u_2(u_0-u_1-u_3)(\mathcal{C}_0+\mathcal{C}_2
-\mathcal{C}_3)-4\pi^2u_2}{8\pi^2},\label{Eq_RGeqs_u2_clean}\\
\nonumber\\
\frac{du_3}{dl}&=&\frac{u_3(u_0-u_1-u_2)(\mathcal{C}_0+\mathcal{C}_2
+\mathcal{C}_3)-4\pi^2u_3}{8\pi^2},\label{Eq_RGeqs_u3_clean}
\end{eqnarray}
\end{widetext}
where the three new coefficients $\mathcal{C}_i$, $i=0,2,3$ are representatively
defined by
\begin{eqnarray}
\mathcal{C}_0&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta
\frac{2}{\sqrt{\alpha^2\cos^2\theta+v^2\sin^2\theta}},\label{Eq_C_0}\\
\mathcal{C}_2&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2\alpha^2\cos^2\theta}
{(\alpha^2\cos^2\theta+v^2\sin^2\theta)^{\frac{3}{2}}},\label{Eq_C_2}\\
\mathcal{C}_3&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2v^2\sin^2\theta}
{(\alpha^2\cos^2\theta+v^2\sin^2\theta)^{\frac{3}{2}}},\label{Eq_C_3}
\end{eqnarray}
with $\mathcal{C}_1$ being brought out in Eq.~(\ref{Eq_C_1}).

These coupled evolutions of
interaction parameters (\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}) indicate
that the four-fermion interacting couplings $u_a$, $a=0,1,2,3,4$ are pertinently
coupled and mutually influence each other via varying the energy scales. Their intimate
interplay may play a crucial role in determining the low-energy behaviors of physical
quantities, which will be carefully investigated and addressed in next section.

\section{One-loop RG analysis in the presence of impurity}\label{Sec_RG_imp}

In the real fermion systems, impurity is well
known to contribute significant effects to the low-energy behaviors of physical quantities~\cite{Ramakrishnan1985RMP,
Nersesyan1995NPB,Mirlin2008RMP} and consequently lead to a wealth of unconventional
phenomena in these Fermi systems~\cite{Ramakrishnan1985RMP,Nersesyan1995NPB,Mirlin2008RMP,
Efremov2011PRB,Efremov2013NJP,Korshunov2014PRB,Fiete2016PRB,
Nandkishore2013PRB,Potirniche2014PRB,Nandkishore2017PRB,Roy2016PRB-2,
Roy1604.01390,Roy1610.08973,Roy2016SR}. It is therefore of remarkable interest and
necessary to investigate how the impurity works before moving to examine how the
low-energy behaviors physical quantities are affected by the coupled flow equations (\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}) generated by the presence
of four-fermion interactions. Before going further, we would like to
suppose from now on that the semi-Dirac fermions are still extended in the presence
of weak impurity although there remains some debate on whether 2D Dirac/
semi-Dirac fermions extended or localized in the presence of impurity~\cite{Sarma2011RMP,Mirlin2008RMP}.
To proceed, we within this section endeavor to incorporate three important types of impurities
into our effective action (\ref{Eq_S_eff}), which are distinguished by their unique
couplings with fermions~\cite{Nersesyan1995NPB,Stauber2005PRB}, and named random
chemical potential, random mass, and random gauge potential respectively~\cite{Nersesyan1995NPB,Stauber2005PRB,Wang2011PRB,Wang2013PRB}.
Starting from the new effective theory incorporating the impurity, we extract
the revised version of coupled RG evolutions after collecting the interplay
between fermion and impurity.

\subsection{Fermion-impurity interaction}

The interplay between fermion and impurity can be conventionally described by the following expression~\cite{Ramakrishnan1985RMP,Nersesyan1995NPB,Stauber2005PRB,Mirlin2008RMP,Wang2011PRB}
\begin{eqnarray}
S_{\mathrm{fd}}=v_D\int d^2\mathbf{x}\Psi^\dagger(\mathbf{x})
\gamma\Psi(\mathbf{x})D(\mathbf{x}),\label{Eq_S_f-d}
\end{eqnarray}
where the Pauli matrix $\gamma$ represents different types of impurities and
$\gamma=\sigma_0$, $\gamma=\sigma_2$, and $\gamma=\sigma_{1,3}$ respectively
correspond to the random chemical potential, random mass and random gauge potential.
The coupling parameter $v_D$ denotes the impurity strength of a single impurity. The impurity field
$D(\mathbf{x})$ would be restricted by a quenched, Gauss-white potential under the conditions~\cite{Nersesyan1995NPB,Stauber2005PRB,Wang2011PRB,Altland2006Book,Coleman2015Book}
\begin{eqnarray}
\langle D(\mathbf{x})\rangle=0,\hspace{0.5cm}\langle D(\mathbf{x})D(\mathbf{x'})\rangle
=\Delta\delta^2(\mathbf{x}-\mathbf{x'}),\label{Eq_S_d-d}
\end{eqnarray}
here the $\Delta$ is assumed to measure the concentration of the impurity and can be taken as a constant
controlled by the experiment. After performing a Fourier transformation, we are left with the corresponding
action in momentum space~\cite{Nersesyan1995NPB,Stauber2005PRB,Wang2011PRB},
\begin{eqnarray}
S_{\mathrm{fd}}=v_D\int d^2\mathbf{k}d^2\mathbf{k'}d\omega\Psi^\dagger(\mathbf{k},\omega)
\gamma\Psi(\mathbf{k'},\omega)D(\mathbf{k-k'}).\label{Eq_S_f-d2}
\end{eqnarray}

\begin{figure}
\centering
\includegraphics[width=4.699in]{Fig4a_int_delta_M1.eps}\vspace{-2.16cm}
\includegraphics[width=4.699in]{Fig4b_int_delta_1.eps}
\vspace{-2.23cm}
\caption{(Color online) Flows of interaction parameters due to the corrections
from the four-fermion interactions. The initial value of $\delta$ is taken to
be negative (a) or positive (b) and others equally.}\label{Fig_int_delta_M1}
\end{figure}

Based on these information, we eventually achieve the effective theory for the presence of impurity
by combing the effective action in Sec.~\ref{Sec_RG_clean} at clean limit and the fermion-impurity
interaction (\ref{Eq_S_f-d2}),
\begin{eqnarray}
S'_{\mathrm{eff}}&=&S_{\mathrm{eff}}+S_{\mathrm{fd}}.
\end{eqnarray}
Paralleling the one-loop analysis for fermion propagator in Sec.~\ref{Sec_RGeqs_clean} and computing
the one-loop fermion-impurity diagrams in Fig.~\ref{Fig_fermion_propagator_correction_2} for the presence
of random chemical potential, we find that the fermion propagator would gain a nontrivial revision
from the fermion-impurity interaction, namely
\begin{eqnarray}
\Sigma
&=&\frac{\Delta v^2_D\mathcal{D}_0}{8\pi^2}(i\omega)l.\label{Eq_Self_energy_0}
\end{eqnarray}
The other two sorts of fermion-impurity interactions both give rise to nonzero
corrections by practicing similar procedures,
\begin{eqnarray}
\Sigma
&=&-\frac{\Delta v^2_D\mathcal{D}_0}{8\pi^2}(i\omega)l.\label{Eq_Self_energy_123}
\end{eqnarray}

In such circumstance, we next can utilize these nontrivial corrections
to produce certain nonzero anomalous fermion dimensions, which are list as
\begin{eqnarray}
\eta^{\mathrm{chem}}_f&=&-\frac{\Delta v^2_D\mathcal{D}_0}{8\pi^2}=-\eta^{\mathrm{gaug}}_f
=-\eta^{\mathrm{mass}}_f.\label{Eq_eta}
\end{eqnarray}
where the indexes $\mathrm{chem},\mathrm{gaug},\mathrm{mass}$ denote the random chemical potential,
random gauge potential and random mass, representatively. These nonzero anomalous dimensions
will be intensely instructive to derive the coupled flow equations of interaction parameters
in next subsection, which are sharp contrast to
a trivial value at clean limit in Eq.~(\ref{Eq_eta_clean})~\cite{Nersesyan1995NPB,Huh2008PRB,Wang2011PRB}.
In addition, the impurity strength can participate in the coupled RG evolutions as a new member and
affect the revised evolutions of all correlated interaction parameters.

\subsection{Coupled flow equations in the presence of impurity}

After carrying out the analogous steps in section~\ref{Sec_RG_clean} with
employing the one-loop corrections in Appendix~\ref{Appendix_four-fermion-interaction_dis},
we consequently summarize the coupled flow equations of all related parameters
in the presence of random chemical potential,
\begin{widetext}
\begin{eqnarray}
\left.\begin{array}{ll}
\frac{dv}{dl}=\frac{\Delta v^2_D\mathcal{D}_0}{8\pi^2}v,\vspace{0.53cm}\\
\vspace{0.53cm}
\frac{d\alpha}{dl}=\frac{\Delta v^2_D\mathcal{D}_0}{8\pi^2}\alpha,\\
\vspace{0.53cm}
\frac{d\delta}{dl}=\left(1-\frac{u_1\mathcal{C}_1}{8\pi^2}+\frac{\Delta v^2_D\mathcal{D}_0}{8\pi^2}\right)\delta,\\
\vspace{0.53cm}
\frac{du_0}{dl}=\frac{2(\Delta v^2_D\mathcal{D}_0-2\pi^2)u_0
-u_0(u_2+u_3)(\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)
-u_0u_1(3\mathcal{C}_2+\mathcal{C}_3-\mathcal{C}_0)
-4\Delta v^2_D(\mathcal{D}_2+\mathcal{D}_3)}{8\pi^2},\\
\vspace{0.53cm}
\frac{du_1}{dl}=\frac{2(\Delta v^2_D\mathcal{D}_0-2\pi^2)u_1-u_1(u_2+u_3-u_0)(\mathcal{C}_0-\mathcal{C}_2)
-u_1(3u_2+u_3-u_0)\mathcal{C}_3-\mathcal{C}_2
(u^2_0+u^2_1+u^2_2+u^2_3)+2\Delta v^2_D(\mathcal{D}_2-\mathcal{D}_3)}{8\pi^2},\\
\vspace{0.53cm}
\frac{du_2}{dl}=\frac{2(\Delta v^2_D\mathcal{D}_0-2\pi^2)u_2-u_2(u_1+u_3-u_0)(\mathcal{C}_0+\mathcal{C}_2
-\mathcal{C}_3)+2\Delta v^2_D(\mathcal{D}_3-\mathcal{D}_2)}{8\pi^2},\\
\vspace{0.53cm}
\frac{du_3}{dl}=\frac{2(\Delta v^2_D\mathcal{D}_0-2\pi^2)u_3-u_3(u_1+u_2-u_0)(\mathcal{C}_0+\mathcal{C}_2
+\mathcal{C}_3)+6\Delta v^2_D(\mathcal{D}_2+\mathcal{D}_3)}{8\pi^2},\\
\vspace{0.53cm}
\frac{d v_D}{dl}=\frac{\Delta v^2_D\mathcal{D}_0+2[\Delta v^2_D-(u_0+u_1+u_2+u_3)]
(\mathcal{D}_2+\mathcal{D}_3)}{8\pi^2}v_D.
\end{array}\right. \label{Eq_RGeqs_int_dis_gamma0}
\end{eqnarray}
\end{widetext}
In order to present more compact evolutions of the interaction
parameters (\ref{Eq_RGeqs_int_dis_gamma0}),
we here have nominated four new coefficients that are
designated as
\begin{eqnarray}
\mathcal{D}_0&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2}{\alpha^2 \cos^2\theta+v^2\sin^2\theta},\label{Eq_D_0}\\
\mathcal{D}_1&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2\alpha\cos\theta}
{\alpha^2\cos^2\theta+v^2\sin^2\theta},\label{Eq_D_1}\\
\mathcal{D}_2&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2\alpha^2\cos^2\theta}
{(\alpha^2\cos^2\theta+v^2\sin^2\theta)^2},\label{Eq_D_2}\\
\mathcal{D}_3&=&\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}d\theta\frac{2v^2\sin^2\theta}
{(\alpha^2\cos^2\theta+v^2\sin^2\theta)^2}.\label{Eq_D_3}
\end{eqnarray}
The coupled flow equations for the presence of other two types of impurities
can derived analogously and are provided in the Appendix~\ref{Appendix_RGeqs_dis}.
In such situation, we can study the effects of these coupled flow equations that collect
contribution from both four-fermion interaction and impurity on the
low-energy behaviors of the semi-Dirac materials in the following section.

\section{Stability of the ground states against the fermionic interaction}\label{Sec_int_delta}

We have already presented the coupled RG flow equations of correlated coupling parameters
in Sec.~\ref{Sec_RG_clean} for the presence of fermion-fermion interactions. In the light
of these running Eqs.~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}), we are
informed that all these interaction parameters participate in the coupled evolutions
and mutually affect each other upon lowering energy scale. We subsequently concentrate
on the fate of parameter $\delta$ in low-energy regime under taking into account
one-loop interaction corrections.

Clearly, the parameter $\delta$ becomes energy-dependent and is coupled
with the evolutions of other parameters via reading the information provided in Eqs.~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}). To be specific,
the parameter $\delta$ flows via decreasing the energy scale and
it manifestly replies upon the coupling parameter $u_1$ and
additionally collects the contribution from the parameters $u_0$, $u_2$, and $u_3$
owning to their closely coupled RG equations. As told in Ref.~\cite{Saha2016PRB},
the parameter $\delta$ is extremely crucial to pin down
the ground state of the 2D SD system: (i) $\delta=0$, the spectrum is gapless with linear
dispersion along $k_y$; (ii) $\delta<0$, it is a gapped system with a trivial insulating phase; and
(iii) $\delta>0$, there exists two gapless Dirac nodal points at $(\pm\sqrt{\delta/\alpha},0)$
and this implies certain nontrivially topological state sets in. It is indeed that the system would
choose one of these states and be stable while we begin with the noninteracting
action~(\ref{Eq_H_0})~\cite{Saha2016PRB}.

However, we would like to stress that, the evolution of parameter $\delta$ with lowering the energy scale,
in apparently contrast to action (\ref{Eq_H_0}), is reconstructed profoundly while the four-fermion
interactions are switched on as presented in Eqs.~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}).
In addition,  all interaction parameters are not independent but
closely coupled with each other. In this respect, it is intensely
instructive to ask whether the low-energy properties of parameter $\delta$
can be qualitatively modified by incorporating into the one-loop corrections
due to the four-fermion interactions, namely whether these three ground states
are stable against these short-range four-fermion interactions?

In order to response to these questions, we have to numerically analyze the coupled RG
equations~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}). To this end, we
first take the initial value of parameter $\delta$ to be negative, i.e.,
$\delta(l=0)<0$ and rescale all the parameters by dividing their initial values, which are
supposed an equal starting constant for unbiased consideration.  After carrying out the
numerical calculations of coupled Eqs.~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean})
with these initial conditions, we obtain interesting results as illustrated in Fig.~\ref{Fig_int_delta_M1}(a). This result directly suggests that the semimetal state with $\delta<0$ is insensitive to the four-fermion interaction and stable in the low-energy regime. Similarly, we representatively assume the parameter
$\delta$ to be positive initially and deliver the results after paralleling the
previous steps as designated in Fig.~\ref{Fig_int_delta_M1}(b). The qualitative results for the
case $\delta(l=0)=0$ are analogous and not shown here. According to the tendency of evolutions
upon decreasing the energy scale in Fig.~\ref{Fig_int_delta_M1}, we are unambiguously told that
the sign of parameter $\delta$ cannot be changed qualitatively by the fermion-fermion interactions.
This thereafter indicates both the trivial insulating state and nontrivial topological state are considerable
stable against the four-fermion interactions~\cite{Saha2016PRB}.

We stop here to present several comments on these
numerical results. At tree level, we easily obtain that all of the quartic interaction parameters
are irrelevant in the RG language after implementing the RG analysis of effective action, i.e.,  Eq.~(\ref{Eq_S_eff})~\cite{Shankar1994RMP,Huh2008PRB,Wang2011PRB}. In addition,
the numerical calculations of one-loop coupled flow equations countenance
this point as demonstrated in Fig.~\ref{Fig_int_delta_M1}. In the spirt of momentum-shell RG
theory~\cite{Shankar1994RMP}, this means that the fermion-fermion interaction parameters
are irrelevant even to one-loop level and as a result the contribution from four-fermion
interactions becomes less and less significant and finally vanishes at the
lowest-energy limit~\cite{Ramakrishnan1985RMP,Nersesyan1995NPB,Mirlin2008RMP}.
Consequently, these parameters cannot qualitatively destroy
the ground sates of the 2D SD material.
Since the low-energy states of 2D SD system are insensitive
to these short-range four-fermion interaction, it is imperative to investigate
the effects of impurities, which are well known to be always
present in the real systems, and in particular interplay between
four-fermion interaction and impurity on these states.

\section{Low-energy behaviors affected by the interaction and impurity}\label{Sec_int-imp}

Impurities are well-known one of most significant facets in producing a multitude of prominent phenomena of
Fermi systems~\cite{Ramakrishnan1985RMP,Nersesyan1995NPB,Mirlin2008RMP}. Within this section,
we are going to study and answer whether our previously basic results with switching on the four-fermion
interactions at clean limit in the 2D SD system are robust under certain number of
impurities and their competition with fermion-fermion interaction.
In Fermi systems, there are three typical types of impurities, which are dubbed by
random chemical potential, random mass, and random gauge potential~\cite{Nersesyan1995NPB,Stauber2005PRB,
Wang2011PRB}. Without loss of generality, all these three sorts of impurities will be equally
investigated. After taking into account the contribution for the presence of both four-fermion
interaction and impurity, the coupled RG flow equations of interaction parameters are
modified from Eq.~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean}) to
Eq.~\ref{Eq_RGeqs_int_dis_gamma0}, Eq.~(\ref{Eq_RGeqs_int_dis_gamma13}),
and Eq.~(\ref{Eq_RGeqs_int_dis_gamma2}).
Under such circumstances, we can expect the remarkably revised behavior
of parameter $\delta$ in the low-energy regime.

\subsection{Interaction-impurity induced phase transition}

To proceed, we endeavor to numerically calculate
Eq.~(\ref{Eq_RGeqs_int_dis_gamma0}), Eq.~(\ref{Eq_RGeqs_int_dis_gamma13}),
and Eq.~(\ref{Eq_RGeqs_int_dis_gamma2}) to check the energy-dependent properties
against the impurity. At the outset, we consider the presence of random chemical potential
with the coupled flow equations~(\ref{Eq_RGeqs_int_dis_gamma0}) and the other
two cases will be followed.

In order to facilitate our analysis, we parallel the steps in Sec.~\ref{Sec_int_delta}
and initially let $\delta(l=0)=0$ and other parameters are assumed to own an equal starting value.
Before moving further, we elucidate the information on the strength of impurity.
For physical consideration, we introduce the impurity scattering rate to qualify the
initial value of impurity strength as~\cite{Coleman2015Book}
\begin{eqnarray}
\tau^{-1}\sim\frac{\Delta v^2_D}{\alpha},
\end{eqnarray}
where $\alpha$, $\Delta$ and $v_D$ are nominated in  Eqs.~(\ref{Eq_alpha}), (\ref{Eq_S_f-d})
and (\ref{Eq_S_d-d}), representatively. In the sprit of the perturbative theory,
we need to restrict the initial value of $\tau^{-1}(l=0)\equiv\tau^{-1}_0\leq\Lambda_0$.
After carrying out the similarly numerical performance, we find that the parameter
$\delta$ exhibits analogously compared to the pure-interaction case when the starting value
of impurity scattering rate is considerable small, namely $\tau^{-1}_0$ is much smaller than $\Lambda_0$.
In this case, the ground state of system is stable and the corresponding numerical results
are not shown here. On the contrary, while the initial value of impurity scattering rate is large,
for instance $\tau^{-1}_0>0.1\Lambda_0$, it is of particular interest that the sign of parameter
$\delta$ with a negative starting value can be changed upon lowering the energy scale as displayed in Fig.~\ref{Fig_int_imp_sign}.

\begin{figure}
\centering
\includegraphics[width=4.699in]{Fig5_int_imp_sign.eps}
\vspace{-2.16cm}
\caption{(Color online) Flows of parameters due to the corrections from the interplay between
four-fermion interactions and random chemical potential. The sign of parameter $\delta$ is changed
upon lowering the energy scale.}\label{Fig_int_imp_sign}
\end{figure}

This apparently signals that the system
undergoes a phase transition~\cite{Fradkin2009PRL,Vafek2012PRB,Vafek2014PRB,
Wang2017QBCP,Altland2006Book,Vojta2003RPP,Metzner2000PRL,Metzner2000PRB,Wang2014PRB,Chubukov2010PRB,
Chubukov2012ARCMP,Chubukov2016PRX}.  As shown in Fig.~\ref{Fig_int_imp_sign},
it is interesting to point out that the fermionic coupling parameters are divergent at
certain critical energy (dubbed by $l=l_c$). Guided by the spirt of momentum-shell RG~\cite{Wilson1975RMP,Polchinski9210046,Shankar1994RMP}
and phase transition theory~\cite{Sachdev1999Book,Altland2006Book,Vojta2003RPP}, we are informed,
compared to the clean-limit case with only switching the four-fermion interactions in Sec.~\ref{Sec_int_delta}, that the irrelevant fermion-fermion interaction parameters
can be transferred to irrelevant relevant interaction couplings in the low-energy
regime after collecting one-loop corrections due to the close interplay between
four-fermion interactions and impurity. Generally,
these irrelevant relevant parameters are responsible for the divergences of interaction parameters.
A multitude of previous studies~\cite{Fradkin2009PRL,Vafek2012PRB,Vafek2014PRB,
Wang2017QBCP,Altland2006Book,Vojta2003RPP,Metzner2000PRL,Metzner2000PRB,Wang2014PRB,Chubukov2010PRB,
Chubukov2012ARCMP,Chubukov2016PRX} reveal that these divergent coupling parameters
are the evident signals of instability and phase transitions. Consequently, we infer that the
intimate interplay between four-fermion interaction and impurity together triggers some phase transition
in the low-energy regime. At this stage, a good candidate for our 2D SD material occurs,
namely, a trivial insulator experiencing certain phase transition to become a nontrivial Dirac
semimetal owning to the strong interplay between random chemical potential and fermion-fermion interactions.

Subsequently, we move to the cases for presence of random gauge potential and random mass
via considering Eq.~(\ref{Eq_RGeqs_int_dis_gamma13}) and Eq.~(\ref{Eq_RGeqs_int_dis_gamma2})
in Appendix~\ref{Appendix_RGeqs_dis}. Paralleling the analysis for the random chemical potential
indicates these two sorts of impurities share the qualitative results with the random chemical
potential: while the impurity scattering rate is small, the ground states of 2D SD systems are
strongly stable against the interplay between fermion-fermion interaction and impurities;
however, the potential phase transition from a trivial insulator to a gapless Dirac nodal
system would be generated by the random gauge potential or random mass if the initial
value of scattering rate of impurity is adequate large, which is as the same order
as the value in Fig.~\ref{Fig_int_imp_sign}. The primary difference from the random
chemical potential is that critical energy scales $E_c$ at which certain phase transition
is generated are distributed into three distinct values. After performing both analogously
numerical and theoretical studies, we are left with the orders of critical energy scales as
\begin{eqnarray}
E^{\mathrm{chem}}_c<E^{\mathrm{mass}}_c<E^{\mathrm{gaug}}_c.
\end{eqnarray}
where $\mathrm{chem},\mathrm{mass},\mathrm{gaug}$ describes
the random chemical potential, random mass and random gauge potential, representatively.
As the numerical results are similar to the presence of random chemical potential,
we do not provide them here.

Based on these analysis, we conclude that the ground states of 2D SD materials
are qualitatively stable against all three types of impurities at a small impurity scattering
rate, but certain phase transition can be generated under specific
condition as exhibited in Fig.~\ref{Fig_int_imp_sign} while the initial value of
impurity scattering rate exceeds certain critical value to make the fermion-fermion
interaction parameters irrelevantly relevant and the interplay between four-fermion
interaction and impurity remarkably significant.

\begin{figure}
\centering
\includegraphics[width=4.699in]{Fig6_int_imp_sign_FP.eps}\vspace{-5.6cm}\\
\hspace{-1.3cm}\includegraphics[width=1.6in]{Fig6-inset_int_imp_sign_FP_sub.eps}
\vspace{1.0cm}
\caption{(Color online) Relatively fixed points of flows of parameters due
to the corrections from the interplay between four-fermion interactions
and random chemical potential. The initial values are followed by the
Fig.~\ref{Fig_int_imp_sign} and the values of parameters are measured by
$v_D$. Inset: evolution of parameter $\delta$ in the sign-change regime.}\label{Fig_int_imp_sign_FP}
\end{figure}

\subsection{Relatively fixed point and phase transition}

Before closing the study of the impurity-induced phase transition,
we would like to provide our comments on this particular
phase transition, which is conventionally accompanied by the sign change of parameter $\delta$
shown in the last subsection.

To this end, we first need to seek the critical fixed point in the parameter space
that is lined to the critical energy scale represented by $l_c$ (or $E_c$) at
which the interaction couplings are divergent and instabilities take place~\cite{Shankar1994RMP,Vafek2012PRB,Vafek2014PRB,Metzner2000PRL,Metzner2000PRB,
Chubukov2010PRB,Chubukov2016PRX}. Learning from Fig.~\ref{Fig_int_imp_sign} informs us
that the interaction coupling parameters (or their absolute values) are increased
quickly and finally divergent at certain critical energy
scale. In order to make our analysis remain weak coupling, one can measure all of interaction couplings
with one of them (no sign change) and try to obtain the relatively fixed point of their
rations at this critical energy scale~\cite{Shankar1994RMP,Vafek2012PRB,Vafek2014PRB}.
Motivated by this idea and after performing analogously numerical analysis, we produce the relative
evolutions of interaction parameters and acquire our relatively
fixed point, namely, $[(v,\alpha,u_0,u_1,u_2,u_3,u_4,v_D)/v_D]^*
\approx(0,0,-5.0\times10^{-4},9.0\times10^{-5},2.4\times10^{-4},1.2\times10^{-3},1)$
by following the initial values of coupled flows in Fig.~\ref{Fig_int_imp_sign}.
In particular, we find that $[\delta/v_D]^*$ is substantially large and nearly divergent as
presented in Fig.~\ref{Fig_int_imp_sign_FP} in 2D SD system. Combing the evolutions of interaction
parameters in Fig.~\ref{Fig_int_imp_sign} and Fig.~\ref{Fig_int_imp_sign_FP},
it singles out that the divergence of parameter $\delta$ at the critical energy scale
should be intimately associated with the leading instability and simultaneously
accompanied by a phase transition. Given the perturbative restriction of our momentum-shell
RG framework~\cite{Shankar1994RMP}, we admit that our approach may gradually break
down and even be invalid at $l\geq l_c$ as the $\delta(l_c+\delta l)$ goes towards the strong coupling.
However, it is well proved that this analysis can conventionally provide constructive information
for leading instability and potential phase transition of the fermionic systems via approaching
the critical energy scale~\cite{Fradkin2009PRL, Vafek2012PRB,Vafek2014PRB,Wang2017QBCP,Altland2006Book,
Vojta2003RPP,Metzner2000PRL,Metzner2000PRB,Wang2014PRB,Chubukov2010PRB,Chubukov2012ARCMP,
Chubukov2016PRX}. Under all these consideration, we are encouraged to infer that some phase
transition must be trigged by the strong interplay between fermion-fermion interaction and
impurity in the vicinity of the critical energy scale denoted by $l_c$.

In order to judge which instability is dominant or which phase transition should be taken place,
we in principle need to calculate all of susceptibilities of potential instabilities allowed by
the symmetries of related system around its relatively fixed point and pick up the leading
one to be tied to the specific phase transition~\cite{Vafek2012PRB,Vafek2014PRB,Metzner2000PRL,
Metzner2000PRB,Chubukov2010PRB,Chubukov2016PRX}.

However, one usually, without loss of generality, can also
directly read the primary type of phase transition from the evolutions of interaction parameters~\cite{Chubukov2010PRB,Chubukov2012ARCMP}. Generally, the activated phase
transition is linked to the largest parameter at
the critical energy scale~\cite{Chubukov2010PRB,Chubukov2012ARCMP}.
As presented in Fig.~\ref{Fig_int_imp_sign} and Fig.~\ref{Fig_int_imp_sign_FP}, the parameter
$\delta$ features the largest value at the critical energy scale. Additionally,
the leading parameter $\delta$ is forced to go towards (positively) strong coupling
at $l=l_c$. Hence one can expect it would be positive at the low-energy regime.
Moreover, it clearly exhibits the sign change of the parameter $\delta$ upon
approaching the critical energy scale as displayed in the inset figure of
Fig.~\ref{Fig_int_imp_sign_FP}. In present study, we only try to put our
focus on how the four-fermion interaction and impurity qualitatively influence
the stabilities of the ground states of 2D SD systems. Therefore for the sake
of simplicity we suggest that the 2D SD system experiences a phase transition from
a trivial insulator to a nontrivial Dirac semimetal according to the spirt from
Refs.~\cite{Chubukov2010PRB, Chubukov2012ARCMP} caused by the interplay between
random chemical potential and four-fermion interaction.

\subsection{Sign change and phase transition}

To further clarify the implication of sign change,
we, within this subsection, provide further discussions on the relationship between
the sign change of parameter $\delta$ and phase transition. As studied in
above two subsections, the $\delta=0$ corresponds to an unstable fixed point that is
suggested to be linked to the phase transition. It is worth pointing out that the key ingredient
producing the sign change of $\delta$ is the divergent quantum fluctuation
at the phase transition point.

To be specific, we would like to emphasize, for physical consideration, the sign change
of $\delta$ is not solely taken place but closely and intimately associated with
the phase transition point at which the instability sets in accompanied concomitantly with the divergent
susceptibilities at the critical energy scale. By approaching the
quantum critical point (of the phase transition), the quantum fluctuations become stronger and
stronger, and finally divergent at that point. These unusual behaviors enter into
the coupled flow equations via the one-loop corrections and then greatly influence the
value of right-hand side of flow equations~(\ref{Eq_RGeqs_int_dis_gamma0}) in the
vicinity of phase transition point, such as the parameter $\delta$'s equation. As a result, the
sign change and divergence of parameter $\delta$ are triggered simultaneously under certain initial
conditions of four-fermion parameters and impurity strength.

For the mathematical consideration, we, for convenience, dub the right hand of flow equation
of $\delta$ as the slope of the $\delta$ equation nominated by $S_\delta$ and assume the unstable
fixed point takes place at $l_c$. If one starts from a $\delta_0<0$ with $S_{\delta_0}>0$, then
$\delta$ gradually approaches the fixed point via lowering the energy scale. On the contrary,
$\delta$ goes to the strong coupling once we begin from a $\delta_0>0$ still with $S_{\delta_0}>0$.
This explicitly implies the single-direction tendency of a parameter can be realized and closely tied
to the sign of $S_\delta$. To be concretely, if the sign of $S_\delta$ is the same while an infinitesimal
perturbation is tuned, $l_c\rightarrow l\pm\delta l$,
the sign change of unstable fixed point can be realized~\cite{Herbut2007Book}.
Guided by above discussion, we can straightforwardly numerically examine the $S_\delta$ away from
$l_c$ in current work (however the theoretical proof is hardly done for these complicatedly coupled
flow equations (\ref{Eq_RGeqs_int_dis_gamma0})). Before going further, we would like
to highlight that the "initial state" should be understood as a relative notation.
For instance, supposing the divergent behaviors set in at $l_c$,
we can define the "initial states" with $l^{\pm}_0\equiv l_c\pm\delta l$.
To be specific, the evolution can follow the steps: (i) initially,
$\delta$ starts from a negative value at $l^+_0$, and its absolute value
decreases in that the numerical calculation of coupled RG equations indicates $S_\delta(l^+_0)>0$;
(ii) while $l\rightarrow l_c$ it goes nearly zero, but the zero point is an unstable
fixed point impacted by the strong quantum fluctuation; (iii) then adding an infinitesimal
increase with $l^-_0$, the slope can change qualitatively by the divergent quantum
fluctuation still with $S_\delta(l^-_0>0)$ informed by the numerical evaluation,
which breaks the balance of the zero point fixed point and the parameter is  directly forced to
the strong coupling.

To reiterate, the intimate relationship between the evolutions of
coupled flow equations and strong quantum fluctuation yields to the singular behavior
of the $\delta$'s RG equation, and eventually results in the the sign change and divergence
of $\delta$ at phase transition point under certain initial conditions.

\begin{figure}
\centering
\includegraphics[width=4.699in]{Fig7_distance_nodal.eps}
\vspace{-2.3cm}
\caption{(Color online) Evolution of the distance ($\lambda/\lambda_0$)
between two nodal points in the $\delta>0$ states in the presence of pure
interaction ($\mathrm{int}$) and interaction+impruity ($\mathrm{int+imp}$)
before approaching the critical energy scale.}\label{Fig_int_imp_distance}
\end{figure}

\subsection{Distance of Dirac nodal points}

In addition to the phase transition investigated in former parts of Sec.~\ref{Sec_int-imp},
we here display another interesting behavior triggered by the interplay between four-fermion
interactions and impurity.

As addressed in Sec.~\ref{Sec_model}, the 2D SD material possesses two gapless Dirac nodal points
for $\delta>0$, namely at $(\pm\sqrt{\frac{\delta}{\alpha}},0)$.  Despite the gapless states with
$\delta>0$ being considerably stable against the short-range four-fermion interaction or impurity (assuming
the initial value of impurity scattering rate is not too large), the distance between these
two Dirac nodal points may be varied by the four-fermion interaction or impurity.
To proceed, their distance can be expressed as~\cite{Saha2016PRB}
\begin{eqnarray}
\lambda=2\sqrt{\frac{\delta}{\alpha}}.
\end{eqnarray}
This straightforwardly indicates that the distance $\lambda$ flows upon lowering
the energy scale as the correlated parameters both $\delta$ and $\alpha$ evolve and
are coupled complicatedly with other interaction parameters along the coupled RG flow equations~(\ref{Eq_RGeqs_alpha_clean})-(\ref{Eq_RGeqs_u3_clean})
or (\ref{Eq_RGeqs_int_dis_gamma0}), (\ref{Eq_RGeqs_int_dis_gamma13})
and (\ref{Eq_RGeqs_int_dis_gamma2}). We therefore are forced to
numerically analyze these coupled RG running equations to investigate the energy-dependent
properties of the distance of Dirac nodal points $\lambda$. Before moving further, we here would like to
stress that the following study is restricted to the energy scales which are larger than the
critical energy scale represented by $l_c$.

We subsequently take the starting value of $\delta$
positive to ensure the starting state harbors two Dirac nodal points and other parameters to
possess the similar values to Fig.~\ref{Fig_int_delta_M1}'s. The corresponding numerical results of
the energy-dependent distance for both cases are delineated in Fig.~\ref{Fig_int_imp_distance}.
We deduce that the four-fermion interaction can reduce the distance of Dirac nodal points after
capturing the basic information from Fig.~\ref{Fig_int_imp_distance}. On the contrary,
the presence all three sorts of impurities would increase the
distance $\lambda$ (as the difference among distinct types of impurities is negligible, we
only leave one line for impurity in Fig.~\ref{Fig_int_imp_distance}). As a consequence, we
conclude that the distance of Dirac nodal points in the semimetal states of 2D SD system
is sensitive to the four-fermion interaction or impurity, which representatively try to
decrease or increase the distance $\lambda$ as depicted in Fig.~\ref{Fig_int_imp_distance}.
One may expect that the increase of distance of Dirac nodal point would be somehow harmful to
the possibility of interplay between the gapless excitations in the low-energy regime.
Hence, this result is qualitatively consistent with the conclusion in Sec.~\ref{Sec_int_delta}
and Sec.~\ref{Sec_int-imp} that the ground state of 2D SD system is stable against
the four-fermion interaction and sensitive to the interplay between the
four-fermion interaction and impurity.

\section{Non-Fermi liquid behaviors}\label{Sec_NFL}

Based on our RG analysis in last several sections, the impurity indeed play a crucial role in
pinning down the physical quantities in the low-energy regime. We finally investigate
how the physical quantities behavior at the lowest-energy limit. In this section, we
focus on the quasiparticle residue $Z_f$ and DOS of the quasiparticle. As the random
chemical potential is a relevant parameter, our perturbative RG method
may break down at the lowest-energy limit and thus cannot clearly present an answer
to the fate of Landau quasiparticles~\cite{{Altland2002PR}}. Therefore, we here only
focus on the random gauge potential and random mass.

At the outset, we study the residue $Z_f$, which can be introduced via
\begin{eqnarray}
Z_f=\frac{1}{1-\frac{\partial \mathrm{Re} \Sigma^{\mathrm{R}}(\omega)}{\partial \omega}},
\end{eqnarray}
where $\mathrm{Re}\Sigma^{\mathrm{R}}$ represents the real part of retarded fermion self-energy.
It is worth pointing out that whether the notation of quasiparticle is well- or unwell- defined
(Fermi liquid or non-Fermi liquid) is closely associated with this renormalization factor $Z_{f}$.
On one side, a finite $Z_f$ corresponds to the Fermi liquid. On the other, the non-Fermi liquid (behavior) is usually accompanied by the residue $Z_f\rightarrow0$.
Subsequently, we can derive the evolution of $Z_f$ depending on the energy scale after performing
momentum-shell RG analysis and collecting the one-loop corrections to anomalous dimensions Eq.~(\ref{Eq_Self_energy_0})
and Eq.~(\ref{Eq_eta}), namely~\cite{Ludwig1994PRB,Xu2008PRB,WLZ2016NJP,Wang2017PRB2}
\begin{eqnarray}
\frac{d Z_f}{dl}=-\eta^i_f Z_f,
\end{eqnarray}
where $i=\mathrm{gaug},\mathrm{mass}$ as shown in Eq.~(\ref{Eq_eta}) representatively correspond
to the random gauge potential and random mass. By carrying the numerical analysis,
we find that $\lim_{l\rightarrow\infty}Z_f=0$ for both the random gauge potential and random mass,
this straightforwardly signals that the Dirac fermions are no longer well-defined
Landau quasiparticles.  As a result, non-Fermi liquid behaviors have been activated.

Additionally, we move to the DOS of the quasiparticle, whose flow $\rho$ under RG analysis can be
expressed as ~\cite{Ludwig1994PRB,WLZ2016NJP} for the 2D Dirac semimetal,
\begin{eqnarray}
\frac{d \ln\rho(\omega)}{d\ln(\omega)}=\frac{1+\eta^i_f}{1-\eta^i_f},\label{Eq_rho}
\end{eqnarray}
here again $i=\mathrm{gaug},\mathrm{mass}$ denotes distinct types of disorders.
This implies the DOS $\rho(\omega)\sim\omega$ for the clean and noninteracting 2D Dirac semimetal.
In order to examine the low-energy behaviors of the DOS in the presence of impurities,
we perform the numerical analysis after combining Eq.~(\ref{Eq_rho}) and coupled flow
equations for the presence of impurities, namely, Eq.~(\ref{Eq_RGeqs_int_dis_gamma13})
and Eq.~(\ref{Eq_RGeqs_int_dis_gamma2}) and subsequently are informed that the DOS at the lowest-energy
regime deviates from the linear behavior $\rho(\omega)\sim\omega$
due to the intimate interplay between four-fermion interactions
and random gauge potential or random mass. This is therefore also a signature
of non-Fermi liquid behavior.

\section{Summary}\label{Sec_summary}

In summary, we have carefully investigated the effects of four-fermion
interactions, impurities, and the interplay between fermionic interactions and
impurities on the ground states of 2D SD materials. In order to capture more
information of physical ingredients in the low-energy regime, all allowed types of
short-range four-fermion interactions by symmetries and three kinds of impurities,
namely random chemical potential, random mass, and random gauge potential~\cite{Nersesyan1995NPB,Stauber2005PRB,Wang2011PRB}, are
included and treated on the same footing by means of utilizing the momentum-shell RG theory~\cite{Wilson1975RMP,Polchinski9210046,Shankar1994RMP}.
The coupled flow equations for all related interaction parameters are derived after carrying
out the standard RG analysis~\cite{Wilson1975RMP,Polchinski9210046,Shankar1994RMP}
for both the pure interaction and presence of impurity, which are
employed to study the low-energy behaviors of 2D SD systems affected by the
interplay between the fermionic interaction and impurity. After performing both
analytic and numerical consideration of these coupled flow equations,
whether and how theses fermion-fermion interactions and impurity
impact the low-energy properties of the 2D SD materials are
fully studied and presented.

To be specific, we first consider the clean limit case with only switching on four sorts of
four-fermion interactions. After analyzing the evolutions of the
related interaction parameters upon lowering the energy scales, we clearly
display that all of four-fermion parameters
are irrelevant in the RG language~\cite{Shankar1994RMP,Huh2008PRB,Wang2011PRB}
to one-loop level. This indicates that the contribution from four-fermion
interactions becomes less and less significant and finally vanishes at the
lowest-energy limit~\cite{Shankar1994RMP}. Based on these theoretical and
numerical studies, we therefore conclude that both the trivial insulating
state and gapless semimetal state are considerably stable
against the short-range fermion-fermion interactions. However, while the impurity is
taken into account simultaneously, the close interplay between four-fermion interaction and impurity
can play a crucial role in determining the ground state of 2D SD system.
Concretely, these irrelevantly fermionic parameters can be transferred to
irrelevant relevant after collecting the corrections from interplay between interaction and impurity.
Additionally, if the initial value of impurity scattering exceeds certain critical
value, we find that the quartic interaction couplings are divergent and sign of parameter
$\delta$ can be changed concomitantly at a critical energy scale.
This conventionally suggests that some phase transition would
be triggered~\cite{Fradkin2009PRL,Vafek2012PRB,Vafek2014PRB,
Wang2017QBCP,Altland2006Book,Vojta2003RPP,Metzner2000PRL,Metzner2000PRB,Wang2014PRB,Chubukov2010PRB,
Chubukov2012ARCMP,Chubukov2016PRX}. For our 2D SD material, we expect
that the system experiences a phase transition from a trivial
insulator to a nontrivial Dirac, which is activated together by the interplay between four-fermion
interactions and impurities in the low-energy regime under specific conditions although the states are
qualitatively stable against all three types of impurities~\cite{Fradkin2009PRL,Vafek2012PRB,Vafek2014PRB,
Wang2017QBCP,Altland2006Book,Vojta2003RPP,Metzner2000PRL,Metzner2000PRB,Wang2014PRB,Chubukov2010PRB,
Chubukov2012ARCMP,Chubukov2016PRX}.

In addition, we find that the distance of two Dirac nodal points in
this 2D SD material is sensitive to the four-fermion interactions or impurities. On one side,
all three sorts of impurities are helpful to increase the distance of Dirac nodal points.
On the other, the four-fermion interaction is preferable to decreases it.
This interesting result is expected to be in line with the basic conclusion in Sec.~\ref{Sec_int_delta}
and Sec.~\ref{Sec_int-imp} that the ground state of 2D SD system is stable against
the four-fermion interaction and sensitive to the interplay between the
four-fermion interaction and impurity. Furthermore, several
deviations from Fermi liquid behaviors~\cite{Altland2002PR}, such as the quasiparticle residue $Z_f$
and the DOS, are displayed at the lowest-energy limit caused by the interplay between
the four-fermion interactions and random gauge potential or random mass.
These exotic properties are manifest signals of non-Fermi liquid behaviors.

Before closing this section we would like to present brief comments on
the long-range fermion-fermion interaction. The long-range interactions may be possibly
formally rewritten as~\cite{Boettcher2016PRB,Hirata2016NComm}
\begin{eqnarray}
S_{\mathrm{int}}&=&
\sum^3_{a=0}u_a\int\frac{d\omega_1d\omega_2}{(2\pi)^2}\int
\frac{d^2\mathbf{k}_1d^2\mathbf{k}_2}{(2\pi)^{4}}
\Psi^\dagger(\omega_1,\mathbf{k}_1)\sigma_a\nonumber\\
&&\times\Psi(\omega_1,\mathbf{k}_1)
\Psi^\dagger(\omega_2,\mathbf{k}_2)\sigma_a
\Psi(\omega_2,\mathbf{k}_2),
\end{eqnarray}
which would give rise to totally different coupled evolutions of related parameters
in that one independent momentum and frequency would be vanished. Accordingly,
this interaction is expected to trigger more interesting behaviors.
However, one central problem is that the long-range
interactions are hardly present due to all kinds of screenings.
In particular, we take into account the effects of impurity scatterings, which
are unavoidably allowed in real systems and are very detrimental
for the long-range interactions. Under this circumstance,
it seems inappropriate to study the long-range interaction in this work, which
we leave for a future study hypothesizing the restricted conditions of presence of
long-range interaction being satisfied.

Finally, we expect that impeding experiments will be achieved to detect and verify whether there exists
the analogous phase transition from the gapped to gapless phases in the 2D SD materials, the variance
for distance of Dirac nodal points in the gapless phase, and also possibly non-Fermi
liquid behaviors at the low-energy regime. This may be profitable
for us to further understand and uncover the unique properties of 2D SD materials and even instructive to
motivate the new materials by virtue of many experimental methods~\cite{LCJS2009PRL,Beenakker2009PRL,
Rosenberg2010PRL,Hasan2011Science,Bahramy2012NC,Viyuela2012PRB,Bardyn2012PRL,Garate2003PRL,
Oka2009PRB,Lindner2011NP,Gedik2013Science,CBHR2016PRL}.

\section*{ACKNOWLEDGEMENTS}

J.W. is supported by the National Natural Science Foundation of China
under Grant No. 11504360.

redpajama_set_name: RedPajamaArXiv


Plainview-Old Bethpage Chamber of Commerce - May 15, 2019: Nassau Council of Chambers 40th Birthday Celebration!
May 15, 2019: Nassau Council of Chambers 40th Birthday Celebration!
"It will be a networking extravaganza!" promises Francesca Carlow, president of the Nassau Council of Chambers of Commerce. Members of all the county's local chambers are invited to help cheer on the Council as they mark this milestone.
It will be held at the Atrium at RXR Plaza in Uniondale from 6:00pm to 8:00pm.
This event is complimentary for chamber members, but RSVP is required.


Why Can't My Church Get A Loan?
and said to me, "Deal!".
The members of a church will likely have many questions when they first initiate the loan process. How much does the church need to borrow? Which potential lenders actually make loans to churches? What documentation will be required during the loan process? What will be the interest rate? How much will be the monthly payments? Yet, before the answers to any of these questions can be of any real significance, the first thing the church members will want to know is, can we qualify for the loan?
Choose BDM Mortgage Services for Quick Funding of California Churches with minimal paperwork & repayment schedule option. Check out the 8 reasons you should choose us!
Ever wonder how to fund your next church project? We show off a little diagram to help make the workflow easier.


text:
Florida Democrats on offensive in health care fight
Posted By Brandon Larrabee, News Service of Florida on Wed, Jun 28, 2017 at 11:56 AM
Photo by Monivette Cordeiro
Even as the U.S. Senate delayed a planned procedural vote Tuesday on a controversial Republican plan to overhaul the Affordable Care Act, the proposal was roiling Florida's political landscape ahead of the 2018 elections.
Democrats running for the Senate and in the state's governor's race hammered away at the GOP, suggesting that they saw a chance to go on the offensive over an issue that has dogged them for years. Republicans running in the marquee contests, meanwhile, seemed to be doing everything they could to take a definitive stand on the legislation.
At the heart of the controversy is a bill that, according to the nonpartisan Congressional Budget Office, would lead to 22 million fewer Americans having health insurance in 2026 than would be the case under current law.
Senate Majority Leader Mitch McConnell, R-Ky., unveiled the bill last week as part of the GOP's longstanding promise to "repeal and replace" the Affordable Care Act, commonly known as Obamacare. McConnell and other GOP leaders had hoped to advance the bill with a key vote Wednesday, but pulled the plug after it became clear the move would fail.
By then, Democrats were already blasting away at Republican candidates in Florida over the measure, underscoring changes in Medicaid spending and the reductions in tax credits for some low-income workers. The attacks suggested Democrats now see Obamacare, which had generally hurt the party since 2010 but has recently increased in popularity, as a net positive.
Former Congresswoman Gwen Graham, a Democrat who's running for governor, labeled the Republican bill "heartless" as she delivered more than 4,000 petitions against "Trumpcare," named for President Donald Trump, to the Florida Capitol office of U.S. Sen. Marco Rubio.
Graham also blasted term-limited Republican Gov. Rick Scott for not agreeing to Medicaid expansion under Obamacare, and expressed hope that the GOP bill wouldn't pass.
"But if that does happen, then we're going to need an even better governor here in Florida," she said. "The one we've had ... should not be able to sleep at night."
Tallahassee Mayor Andrew Gillum, one of Graham's opponents for the Democratic nomination, called for an amendment to the Florida Constitution that would recognize health care as "a fundamental right of all Floridians."
"There is a public trust for the government to care for its citizens, and our state can no longer be ambiguous about that moral obligation," Gillum said in a statement issued by his campaign. "When healthcare is under attack in Washington, we're going to lean into the challenge of healthcare in the Sunshine State and live our values."
Highlighting the potential differences that exist even within the party, Graham was equivocal Tuesday when asked whether she would support such an amendment, stressing her support for a public option in the state.
"I think health care is a right," she said. "But I want to make sure the way we go about it is doable."
Winter Park businessman Chris King, the third declared Democratic candidate for governor, didn't comment publicly on the bill Tuesday, but has in the past criticized a previous version passed by the House and attacked Scott and Agriculture Commissioner Adam Putnam for not taking a stance on the bill.
Putnam is the only major, declared Republican candidate for governor in 2018. A spokeswoman said Tuesday that Putnam's campaign had no comment about his position on the bill.
American Bridge, a Democratic campaign organization, said Monday that Putnam was "complicit" in the federal legislation.
"If Adam Putnam wants to be governor, he should come out and propose a plan of his own instead of hiding behind this disaster of a bill that gives tax breaks to the wealthiest few at the expense of Floridians' health," said Lizzy Price, a spokeswoman for the group.
Scott spent Tuesday in Washington, D.C., lobbying federal officials on the legislation while dancing around questions about his position during television appearances.
"I think it's very important to repeal and replace Obamacare," Scott told business channel CNBC when asked whether he would vote for the bill. "I'm up here as a governor. I don't have a vote. I'm up here as a governor trying to make sure the bill is good for Florida families. I know the bill is a work in progress."
That didn't stop U.S. Sen. Bill Nelson, a Democrat likely to face a re-election challenge from Scott next year, from trying to tie his would-be opponent to the bill.
"Rick Scott is supporting and urging Republican senators to vote for a bill that makes huge cuts to Medicaid, takes coverage away from 22 million people and allows insurance companies to hike rates for older Americans," Nelson said in a statement issued by his office. "If he really cared about the people of Florida, he'd be doing the exact opposite of what he's doing now."
Tags: florida democrats, health care florida, Image
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Putnam raises more cash in governor's race than Democratic opponents combined
Florida lawmakers got wealthier this past year
Gov. Rick Scott's worth is almost $150 million
Orlando activists stage die-in at Marco Rubio's office over Trumpcare
On the Senate's no-good, very bad health care bill
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text:
'80s, Awesome Lists, Decades, Playlist Reviews
Curated By: Pitchfork
Published By: Pitchfork
Post By: Stephanie Garr
What's This Playlist All About? "The most trusted voice in music" works its way through a decade we all can't seem to get enough of with this disclaimer: "Longtime readers may remember that, in 2002, we made a list of The Top 100 Albums of the 1980s. That list was shorter, sure, but it also represented a limited editorial stance we have worked hard to move past; its lack of diversity, both in album selections and contributing critics, does not represent the voice Pitchfork has become. For this new list, we gathered votes from more than 50 full-time staffers and regularly contributing writers to open up our discussion."
What You Get: The usual suspects crowd the top of the list (Prince, MJ, Madonna, Beastie Boys, Public Enemy, New Order), but dig into the heart of it and you may find some real hidden gems. You'll find the throbbing, funky post-punk of Bronx band ESG; the brilliant sampling of hip-hop greats EPMD; the Satanic doggedness of death metal gods Morbid Angel; the infectious South African rhythms of The Indestructible Beat of Soweto compilation; and the intricate computer patchworks of electronic pioneer Laurie Spiegel. Let's just say the whole 575-song mix certainly has the diversity promised.
Greatest Discovery: At No. 130 is Scientist's Scientist Rids the World of the Evil Curse of the Vampires, excellently described by reviewer Eddie "Stats" Houghton as "one of the greatest dub albums ever, transforming the swing of dancehall's catchiest tunes into their spookiest, most expansive selves. Historically, this record is a precursor to trip-hop and dubstep, but even encountered as an isolated sonic experience, the tracks are revelatory, uniquely suffused with an eerie joy."
Do We Really Need Another '80s Playlist? This list is surprisingly fresh, and it may even be worth starting from the bottom, as you'll likely discover some unexpected treasures you've never heard before. In other words, yes, another '80s playlist will do just fine. There's still plenty to discover from the decade that just won't die—thankfully.


text:
homemarket Newsstocks News
Here's why Tata MF is positive on pharma but not so much on discretionary consumption space
market | Apr 1, 2020 12:01 PM IST
By Latha Venkatesh | Anuj Singhal | Sonia Shenoy | Apr 1, 2020 12:01 PM IST (Published)
A combination of lower valuations, least impact from COVID-19, some bit of regulatory leniency or relaxations makes the pharma sector look good at this point of time, said Rahul Singh, CIO-equities of Tata Mutual Fund.
Rahul Singh, CIO-equities of Tata Mutual Fund is of the view that market is reflecting the overall uncertainty that people are feeling even at a personal level. People are hoarding all kinds of things - funds have seen redemptions, there is liquidation because everyone seems to want to hoard cash.
"So at this point of time, what is very important to analyse or realize is that economies generally bounce back after periods like these. The difference between the previous slowdowns and this slowdown is that this one is a self-imposed, forced one and we have no way of judging how long this is going to continue, whether it is six-eight weeks or whether it is going to be six-nine months. So, in that scenario, some bit of defensive positioning -- being overweight in consumers, to some extent utilities and also pharmaceuticals, which is a new sector that is looking interesting from a valuation perspective is the maximum you can do," he advised in an interview with CNBC-TV18.
At the same time, he said there are good quality companies in various other sectors, which are becoming reasonably attractive and strong businesses. So, we are keeping an eye on them and adding to those as and when the value emerges in those sectors and stocks, he said.
"This is the most rational way to go about rather than worrying about whether we have formed a bottom or not because that is something which is a function of so many different kinds of newsflow in the US, in Europe, in India that it becomes quite challenging to always try and time the bottom," he added.
"Discretionary consumption is something which is likely to be hit probably longer than anything else. There is a theory around how consumption habits could change. I don't entirely believe that an event like this in a country like India will structurally change the consumption habits or change the consumer behavior to an extent that we are in a structural slowdown in consumption. However, given the hit on the incomes and the sense of security, which has come down at the individual level, discretionary consumption is probably the most badly positioned to withstand this kind of a slowdown," said Singh.
"Within discretionary there are pockets which could come out of this stronger like the digital delivery of products and services, digital food delivery. So you cannot paint the whole discretionary space with one wide brush but that is one segment which we would be watching very carefully in terms of any potential stress, if this situation continues for longer," he said.
According to him, market would not mind another 21-30 days of lockdown if we are very sure that we are going to come out of this coronavirus spread 100 percent. The uncertainty is around the fact that what if the lockdown does not succeed and what if it succeeds partially then how long does it continue and will there be second and third wave of this and that is where discretionary consumption starts to look a little weak," he added.
When asked about infrastructure and autos, he said they were anyways going through a slowdown over the last two years. So the valuations had started to become reasonable. The margin expectations which were very high had come down, so we were at a good point of time in terms of trying to say that some of these stocks had formed a bottom.
"This leg of slowdown is an additional layer of slowdown, which has been added on to these sectors. Therefore I don't see these stocks or sectors any differently from any other sector which is going through a similar slowdown," he added.
In terms of pharmaceutical sector, he said, "Incrementally, we are more positive on that space than we have ever been in the last two years because there are two-three things at work." From a disruption point of view, once China has been up and running, the disruption from COVID-19 itself has been the minimum in terms of the impact. There are stocks where the valuation of the international businesses is now less than one time sales, so from a valuation point of view they are look good,." he said.
There is an opportunity in the pharma space for some of these companies to come out of the FDA issues, if the strictness with which FDA was looking at some of the issues gets relaxed.
"In addition, the companies have been working towards a pipeline, some of those pipelines have been stuck in various FDA issues but over the last two-three years a lot of companies have been correcting their balancesheet, building up their pipeline for complex generics and some of that pipeline will start to see a light of the day in FY22," he said adding that a combination of valuation, least impact from COVID-19, some bit of regulatory leniency or relaxation makes this sector pretty good to look at it at this point of time.
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Reduce Tent-sion: Go Camping in Northumberland
Johanna Flashman October 29, 2020 Leave a comment
Back in September, I went on a weekend camping trip in Northumberland National Park and it was beautiful. Not only that, I was tent testing, so it was all in the name of work.
To be honest, I knew very little about this area before going on this trip. I knew Google Maps had a big green splotch that was "Northumberland National Park" and it was vaguely close to Newcastle Upon Tyne (Newcastle). I also knew Cam, my partner, was going to the area on a trip with some friends so I could get a ride to have my own adventure. Let the (COVID-safe) planning begin!
First line of business: find someone to come with me. I could go solo, but the convenient thing about Northumberland is it's right in between the North York Moors and Edinburgh meaning a friend from there could swing down and join me. Reunion and companionship!
My friend also unknowingly agreed to be my model for the trip as well. (Photo: Johanna Flashman)
Where to Camp in Northumberland
Second line of business: where to stay. Considering I was testing out this tent (it's a phenomenal bikepacking tent from NEMO that's for 2021), I was going to be camping, so that narrowed down the search.
Compared to the US, in England there aren't park-run campsites and you can't just go backpacking and do dispersed camping in a national forest area. In Scotland, they have a "right to roam" law where you can pretty much camp anywhere as long as you're respectful and don't keep the tent set up during the day. But in England, you have to find a designated, private campsite to pitch your tent. Pitch, please.
So I looked for a private campsite that let me pitch a tent and was close to the park. There are several different campsites in the area, but I chose Winshields Farm Campsite. It's straight forward, conveniently located, and the owner Malcolm was really friendly and easy to work with – we're working out some website copywriting now!
Winshields Farm Campsite even has this super cute sign pointing to it from the Hadrian's Wall path. (Photo: Johanna Flashman)
A few other places I considered, in case you're looking were:
Hadrian's Wall Camping and Caravan Site (£12-20/night)
Brockalee Farm Caravan Site (Contact for prices)
Herding Hill Farm Camping and Glamping Site (£20-34/night)
Greencarts Campsite and Bunkhouse (£8-11/night/person)
Stonehaugh Campsite (£10-13.50/night)
Fallowfield Dene (Contact for prices)
What to Do in Northumberland
What specifically got me going about Northumberland were two long-distance walking routes (Hadrian's Wall Path and Pennine Way), the international dark sky park, and climbing. Both nights we were there it was rainy and cloudy, so I can't speak to the dark skies but we did see both the Pennine Way and Hadrian's Wall. I didn't do any climbing this trip, but I did scout out some of the rock and definitely want to get on it at some point.
Hadrian's Wall Path
Hadrian's Wall used to be a whole lot taller, but it's still there, which is pretty incredible. (Photo: Johanna Flashman)
Hadrian's Wall Path follows the UNESCO World Heritage Site, Hadrian's Wall, which was a coast-to-coast wall that the Roman's built to guard the northern frontier. There are tower and fort remains and informational museums along the way.
We only walked a very short portion of the path, but the whole thing goes on for 84 miles from Newcastle upon Tyne to Cumbria. People will walk the entire thing over an average six days, but you can also take your time or speed through it. (The Fastest Known Time is 16 hours 25 minutes.)
Pennine Way
The turn-off for the Pennine Way. (Photo: Johanna Flashman)
To be honest, we didn't really walk much on the Pennine Way, but it goes with the Hadrian's Wall Path for a bit and we saw the turn-off for it.
According to National Trails, the Pennine Way is the "oldest and most iconic national trail." It's 268 miles of walking from the Edale in the Peak District to Kirk Yetholm, just north of the England/Scotland border. The whole thing typically takes between 16 and 19 days, though the fastest known time is two days, 13 hours, and 35 minutes.
International Dark Sky Park
Northumberland has low levels of light pollution making it the most "pristine dark skies" in England. In 2013, the area became England's first International Dark Sky Park and has been trying to conserve it ever since. Since we didn't get any clear skies while we were there, I can't speak to this personally, but the stars are supposed to be spectacular. I hope to go back and see them at some point!
We didn't do any rock climbing on this trip, but that didn't stop me from dragging my friend with me to scramble along the rocks in front of a climbing area. The rocks look amazing and it has some top class, beginner-friendly trad routes I can't wait to try in the future!
Why yes, I do really love rocks. (Photo: Johanna Flashman)
Check out UK Climbing for more information on some of the crags I'm talking about. Jezebel is the one I'm going for first.
Have you had a good experience in Northumberland? Share it in the comments, I'd love to hear about it!
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History is filled with African American men who have been leaders and role models. This list includes great actors, athletes and performers like Jackie Robinson, Sidney Poitier and Nat King Cole. It also includes politicians and activists such as Martin Luther King, Medgar Evers, and, of course, President Barack Obama. But it also includes unsung heroes such as fathers, brothers, uncles and cousins, people who have served as role models and protectors. That includes coaches, ministers, teachers, principals, police officers, doctors and lawyers.
The purpose of this site is to honor African American men who are making significant contributions. The criteria are based on talent, intelligence, dignity and grace. It is not just intended for stars or public figures, though their work should be recognized, beyond their looks. We want to highlight some unsung heroes who deserve recognition, but rarely get it; and still they keep giving of themselves. Go to our Nominate page to make a recommendation. Go to our Vote page to vote on this month's nominees.