# Characteristics of Dark Current from an S-band RF Gun with Exchangeable
Photocathode System

Frank Jackson<EMAIL_ADDRESS>Louise Cowie  Marta Furmaniak
Phillipe Goudket  Boris Militsyn  Julian McKenzie  Tim Noakes  UKRI/STFC
Daresbury Laboratory

###### Abstract

Field emission or dark current from RF electron guns is problematic for
accelerators for several reasons, and can be more difficult to predict,
measure and understand, compared to the main beam. The work presented here is
a study of the dark current emitted from the CLARA 3GHz 2.5 cell electron gun
with exchangeable photocathode, which is a type of gun used for single pass
high quality low emittance beams for uses such as Free Electron Lasers. The
main purpose of these studies was to identify sources of dark current emission
within the gun, by examining the dark current distribution and patterns on
scintillating screens. It is shown that the features of the patterns can be
reproduced qualitatively, and to some extent quantitatively, by finite element
simulations. Thus, certain surfaces within the gun and cathode apparatus which
contribute most significantly to problematic dark current can be identified,
and further development and preparation of the guns and their cathodes can be
informed to minimise dark current.

††preprint: APS/123-QED

##  I Introduction

Field emission of electrons (or ‘dark current’) in RF guns of the type used to
produce high quality beam such as for Free Electron Lasers (FELs) have the
potential to cause many different problems related to machine protection. In
FELs for example, gun dark current transported to FEL undulators may cause
undulator damage [1].

CLARA (Compact Linear Accelerator for Research and Applications) at Daresbury
Laboratory was originally designed as a FEL test facility [2], and includes a
high gradient RF electron gun very similar to that used in existing X-FELs.
Dark current from the CLARA gun has been observed since first commissioning
and has been of concern as the machine has developed. For example, the
background level caused by gun dark current has been recorded in some non-FEL
experiments and applications (electron diffraction studies, very high electron
energy for radiotherapy investigations) [3] [4]. Most recently the level of
dark current, following a gun upgrade, has increased significantly to a level
where the first beam diagnostic screen can be dominated by dark current, and
concerns remain about its effect on cathode performance, damage to diagnostic
devices like charged couple devices, discolouration of beamline viewports, and
even potentially effects on the optical components of the photoinjector laser
transport line.

The understanding and mitigation of dark current from the CLARA gun, and other
similar guns, presents several difficulties. The field emission may be
generated from several different parts of the inner surface of the gun, and
the field emission particles have a wide ranging kinematic properties compared
to the core electron beam, in particular a large spread of energies. The
physical and chemical properties of the photocathode and other gun parts,
particularly in the cathode area, have a strong influence on the dark current.
However the effects are not well understood, and difficult to measure to a
high degree of precision. Disentangling the effects on the macroscopic
features of dark current, typically observed on scintillating beam diagnostics
screens, is a complex problem. There have been several investigations of these
features in different electron guns worldwide, some similar to the CLARA gun.
For the Linac Coherent Light Source (LCLS) gun an analytical model has
provided a representation of the dark current screen patterns [5]. The APEX
gun at Lawrence Berkeley National Laboratory (LBNL) has investigated the dark
current screen appearance and associated it to emission points in the cathode
area [6]. Detailed simulations have been used in several guns to model dark
current emission and transport to screens: at the Argonne Wakefield
Accelerator facility (AWA) the macroscopic distribution of surface microscopic
field emitters was deduced from screen measurements using these simulations
and achieved a high degree of agreement with surface scanning electron
microscopy measurements [7]; a similar simulation has been used to model dark
current transport at POLFEL [8]; recently a detailed study of dark current
screen patterns at the PITZ gun has successfully reproduced the observed
features in simulation and associated them surface locations of the cathode
[9]. Modelling of the RF electromagnetic (EM) field and consideration of
microscopic emission points, particularly on the surfaces of the cathode area
of the guns, has been a key element of the above studies.

For the CLARA gun, particle-in-cell (PIC) simulations have been developed
using CST Studio [10], to understand the appearance of dark current screen
observations, particularly following the gun upgrade. In the following paper
we describe the simulation of dark current generation and transport in the
CLARA gun and injector, and a comparison of its features to screen
observations. It is shown that components of the cathode area can be readily
identified as emission sources using these simulations. The results can be
used to inform future gun development to mitigate dark current.

##  II THE CLARA INJECTOR AND GUN

A recent description of the CLARA accelerator is given in [11]. The CLARA gun
and photoinjector has been in operation at Daresbury Laboratory since 2013.
The gun is a 2.5 cell copper cavity designed to reach a maximum field of 100
MV/m and produce a beam of several MeV energy in single bunches operating at
10 Hz repetition rate. In the initial installation the gun copper backplate
served as a bulk photocathode and the electron beam was generated with a laser
of wavelength 266 um. In 2019 the gun was upgraded by replacing the backplate
with a cathode exchange mechanism, and since then hybrid copper cathodes of
different properties have been commissioned [12].

Figs. 1 and 2 show the upgraded CLARA gun and mechanical design of the gun
region. The gun incorporates a main solenoid magnet to allow beam focussing,
and ‘bucking’ solenoid to cancel the solenoid field on the surface of the
cathode. The design of the cathode exchange mechanism was influenced by a
project to develop an alternative high repetition rate gun for CLARA (up to
400 Hz). This development sought to optimise performance of the gun, through
control of electromagnetic field by shaping of inner gun surfaces including
those of the cathode areas. A key element of this design was the elliptical
design of cavity irises and the profile of the cathode insert (plug). The
cathode plug profile has a flat central section of 6 mm diameter and an
elliptical edge (rim). The gun electric field is enhanced at the cathode edges
by the plug opening. In the design the ellipticity was optimised to maximise
the ratio of the peak field in the centre of the cathode to that on the edges.
An elliptical shape falling back to 0.6 mm from the cathode face was found to
be the optimum plug shape, achieving a ratio of around 0.9. In other words,
even when optimised, the peak field on the cathode edges was still slightly
larger then the field in the centre (as can be seen in Fig 3).

Figure 1:  The upgraded 2.5 cell CLARA gun region. The cathode plug can be
inserted into the back of the gun (the load-lock mechanism is not included in
the diagram). The focussing solenoids surrounding the gun (main and bucking)
are displayed. Also shown are the waveguide and coupler and beampipe at gun
exit. Figure 2:  Close up view of the CLARA gun cathode region, consisisting
of the retractable cathode insert(plug) and opening in the gun backplate in
which it sits.

The CLARA machine and injector beamline is described elsewhere [11]; the
injector line consists of the gun and a beampipe approximately 1m long
containing a lightbox through which the photoinjector laser is directed onto
the cathode, a wall current monitor to measure beam charge, and a
scintillation (YAG) screen on which the beam and the dark current emerging
from the gun is observed. Immediately after the screen is the first main
accelerating cavity where electrons gain their full energy. The beampipe is
nominally 35 mm radius, but the coupler and light box assembly contain
narrower apertures through which the beam and dark current pass before they
reach the first screen.

##  III CLARA DARK CURRENT SIMULATION OUTLINE

Finite element method (FEM) simulations of the dark current generation and
transport in the CLARA injector have been developed using CST. Both the
electric field in the gun and the dark current are simulated within CST, based
on FEM methods. The solenoid magnetic field is simulated in an external code
OPERA [13] and imported into CST. This method provides a full 3-D model of the
situation. It was first used for simulations of dark current in the CLARA gun
with its original bulk photocathode and was able to provide a level of
agreement with experimental observations [14]. When the CLARA gun was upgraded
the CST simulations were modified to reflect the changes, and the complete
model used here includes the gun with new cathode geometry, the coupler
geometry, and the beam pipe extending through the light box and its narrower
aperture, up to the first beam diagnostic screen.

The gun EM field is simulated using the FEM field solver of CST, using the gun
and coupler section of the model. Fig. 3 shows the solved electric field of
the gun in the x-z plane. To simulate dark current production and transport
the PIC simulation of CST is used, which incorporates the EM field generated
from the previous step, and the combined gun solenoid and bucking solenoid
field modelled by the OPERA package. The dark current generated from the gun
surfaces uses the Fowler Nordheim (FN) model of field emission, describing the
current density of surface field emission, depending exponentially on the
applied electric field $E$

$J=aE^{2}\exp(-b/E)$ (1)

where $a$ and $b$ depend on the properties of surface and material (work
function and surface roughness). In the simulation field emission points are
uniformly distributed over the gun surfaces and the FN model determines the
emitted current density at each point using the simulated electric field
there.

The emitted particles are then tracked through the gun and solenoid fields to
the position of the first screen. There are several features related to the
discretisation of the FEM nature of this model. A tetrahedral mesh is used
solve the EM field, which better describes curved surfaces compared to a
hexahededral mesh; however the PIC simulation uses a hexahedral mesh to track
the particles through the field. The particle generation uses triangulation to
model emission points over the 3-D surfaces of the gun. The parameters of each
discretisation, and compatibility between them, may affect both the fine and
coarse results of the simulation. In the simulations of the simpler pre-
upgrade gun [14], convergence of the amount of charge emitted for each surface
was achieved by choosing a sufficiently dense set of mesh parameters. The more
complex geometries of the upgraded gun proved did not allow convergence to be
achieved. The goal of the studies was to examine the appearance of the dark
current on the screen, and so convergence of the total emitted and transported
charge was not necessary, as long as the parameterisation was sufficent to
achieve physically reasonable results (i.e. demonstrating sufficient symmetry
and uniformity). Thus the meshing parameters and the field emission parameters
(FN parameters $a$ and $b$) where adjusted to achieve this.

Figure 3:  CLARA gun E-field strength in x-z plane. Top: expanded view of gun,
colour denotes the field strength (red strongest, blue weakest). Bottom:
E-field strength along cathode backplate surface, with cathode region profile
superimposed.

The PIC mesh had different density in different regions of the gun, depending
on the geometric complexity of each region. Fig 4 shows a closer detail of the
meshing density in the area of the cathode where the most complex geometry in
the model arises. The smallest mesh size in the transverse and longitudinal
dimensions is 0.2 mm.

Figure 4:  The PIC meshing density in the region of the photo-cathode.

##  IV RESULTS AND DISCUSSION

Following development of the CST model of the upgraded CLARA gun, several
experimental observations of dark current were made during the commissioning
of the gun, and an attempt was made to match these in the CST simulation.
Several different photocathodes of different materials and preparations were
used in the commissioning which are detailed in [12].

Previous general characteristics of the dark current, such as 6D phase space
distributions, have previously been reported [14], and similar general
characteristics were seen here. This included the finding that, for nominal
operating parameters and assuming uniform emission properties over all
surfaces, only the backplate and cathode regions contributed to dark current
escaping the gun. In the upgraded gun simulations is was found that the
emission from cathode region (including backplate rim) dominated the beamline
dark current so greatly that the backplate emission was omitted from the final
simulations. Other characteristics included a large energy spread of the dark
current from each surface, with significant dark current at the nominal energy
of the beam. There was not enough difference in the energy distributions from
each surface to suggest the contributions from the flat cathode region and the
rims could be separated and identified by dipole spectroscopy.

The main experimental observations presented the appearance of dark current on
a scintillating screen near the gun, for different solenoid strengths. During
operation of the first photocathode used in the commissioning of the CLARA gun
(called cathode ‘#13’ see [12] ), the screen image was taken at 3 different
solenoid/bucking-solenoid configurations, showing distinctively different
coarse and fine features of the dark current. The solenoid field
configurations included 1) the main solenoid on but the bucking solenoid off
2) the bucking solenoid set to a strength which cancels the main solenoid
field on the cathode surface which is the nominal setting used for beam
focussing 3) the bucking solenoid and main solenoid on but at strength where
the field does not cancel on the cathode surface.

The measured and simulated screen images are shown in Fig. 5. For the
simulated images, the colour map (middle row) used is the charge per 2D bin,
to compare with the intensity colour map of the screen images.

Figure 5:  Dark current at the first CLARA diagnostic screen for 3 different
gun solenoid configurations (see text), for measured and simulated data. Top
row: measured. Middle row: simulation including emission from cathode flat
surfaces and cathode and backplates rim (colour scale is the particle
density). Bottom row: Simulation including emission from cathode rim (red) and
backplate rim (purple).

The notional gun field (at its nodal maxima) during these measurements was
estimated to be 67 MV/m (using the measured RF power to the gun, a previously
measured beam momentum vs RF power calibration, and a simplified simulation
with a 1-D model of the field). This may omit some realistic features and
errors in the actual gun field. Thus in the CST simulations the overall
scaling of the gun field was set to a range (approximately 20 %) around 67
MV/m, and the results which gave the closest match to observed screen images
are shown here. In a similar way the FN parameters for each surface, which
influence the emitted dark current as described above, were adjusted in an ad-
hoc manner to find a closer agreement with the data (a consistent set of FN
parameters was chosen for all simulation results compared to data here).

It was found that contributions from the rims in the cathode region (see Fig.
2.) gave rise to the distinctive ‘ring’ features most closely matching the
features of observations, which is clarified by the colour coded comparison of
the separated contributions from only the rims (Fig 5 bottom row). The
relative brightness of the ring features may indicate more intense emission
from the rim surfaces compared to the flat central cathode surface which may
be expected due to electric field enhancement. The surface roughness on the
rims would be expected to be greater than the flat cathode surface (which is
diamond polished) and microscopic field enhancement on the (greater) geometric
surface imperfections of the rims would generate more field emission. A more
complete formulation of the FN model [15] defines surface imperfections by a
field enhancement factor $\beta$ and gives the field emission current
(integrated over a RF cycle) as $\sim\beta^{2.5}e^{1/\beta}$. While there is
some uncertainty in using this model to extract quantitative values of $\beta$
for RF guns [15], the exponential form indicates strong dependence of the
emission on surface quality, which supports the reasoning above. While the
(non-enhanced) electric field strength varies over the cathode region, it
reaches maxima on the rims (see Fig 3.) which again supports the increased
contribution to dark current of the rims.

The results indicate that relatively bright central features (middle column of
Fig. 5) are not necessarily due to field emission from the centre of the
cathode. They may instead be field emission from the rims which are then
focused to the centre of the screen by the solenoid. Some fine features seen
in the screen images may be a result of non-uniformly distributed
imperfections on the surfaces, which are not reproduced by the uniformly
distributed emission points of the simulations.

There were other further observations made throughout the subsequent gun
commissioning and operation of different photocathodes which support these
findings. For example during the commissioning of a Mo/Cu hybrid photocathode
(named ‘#7’) it was retracted for heat cleaning and reinserted; later another
hybrid cathode (named ‘#16’) with different surface preparation (diamond-
turned) was inserted. The appearance of dark current on the screen during
these changes was observed and compared. This is shown in Fig 6 where
approximately the same gun electric field and solenoid strength was used in
all the images (note these approximately matched, but were slightly different
to, those used in the middle column of Fig 6, with the solenoid field
cancelled on the cathode surface). The images taken before and after
retraction and reinsertion of the cathode highlight the dark current features
from emitters on the cathode compared to those on the surrounding backplate
and rim. Similarly the inserting of a different cathode shows a clear
difference in some features and similarity on others, again allowing the
identification of different sources and confirming the ability of the
simulations to predict these features.

Figure 6:  Dark current screen images at different stages of CLARA gun
conditioning, with different cathode status. Hybrid cathode (‘#7’) first and
second insertion (left and middle), and diamond turned cathode (‘#16’)
(right). In the top row the colour scale is adjusted to emphasise the
comparison of the features; in the bottom row the color scale is identical for
each figure. The similarity of the outer features in the three images indicate
emission from the backplate rim, while the difference in the inner features
indicate emission from the cathode.

An interesting observation is the difference in brightness of the dark current
on the screen for different cathodes and/or different insertions of the same
cathode. This may be due to the effect of very small ($\mu$m) differences in
positioning of the cathode and/or the length of the different cathodes. The
geometrical change this makes to the gun cavity may change the EM field
strength on the cathode surfaces (as seen in [9]), and thus change the
intensity of the dark current emission. A wall current monitor (WCM) located
approximately 20 cm upstream of the screen was used to measure the dark
current, which confirms the reduction in dark current (around 15 %) seen after
retraction and reinsertion of the cathode seen in Fig 6.

##  V CONCLUSIONS

Despite the lack of quantitative convergence, the results above demonstrate
the usefulness of simulations in the ability to characterize the effect of
each emitting source on the screen pattern. In the screen images of the dark
current the emission from the backplate and from the cathode insertion has
been identified. This is clear due to the comparison of different experimental
conditions (retraction and reinsertion of cathodes), and is also supported by
the simulations, where different emission sources can be isolated, and shows
some level of agreement with observation. For the CLARA gun, these
observations lead to the following conclusions.

Field emission from the curved section of the backplate is significant and,
for certain parameters, may dominate the dark current transported through the
first part of the beam-line. For the nominal beam setting of the solenoids it
appears as an outer bright ring on the screen.

Field emission from the cathode (flat surface and and the curved rim) leads to
features observed in the central part of the screen, when the solenoid
strength is set approximately to its nominal beam focussing setting. The
central bright spot may not be due to ‘field emitters’ in the centre of the
cathode, since simulations indicate field emission from the edge of the
cathode may be focused to the centre of the screen.

There are several issues and uncertainties with the simulations that would
require further work to evaluate further. There are various ways in which the
electric and magnetic field in the simulations may differ from reality. At the
emission surfaces where the particle energy is low, this may significantly
affect the dynamics of the field emission. In simulation, meshing artefacts
may affect the gun electromagnetic field shape, even for perfectly described
geometries. Other effects which might be thought small may also be important;
for example mutual coil excitation included in the modelling of combined
solenoid and bucking solenoid field were observed to make a significant
different to the simulated screen patterns. We do not have detailed in-situ
3-D measurements of the magnetic and electric fields, which may include other
such small effects, to compare against the modelled fields.

##  VI ACKNOWLEDGEMENTS

We wish to thank members of ASTeC (Accelerator Science and Technology Centre)
and Technology Departments of STFC (Science and Technology Facilities Council)
for providing technical and operational support of the CLARA accelerator,
which made these experiments possible. We wish to particularly thank I. Gessey
and P. Tipping for development of the CST field emission simulations of the
CLARA gun.

##  VII REFERENCES

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# Superconducting phase transition in planar fermionic models with Dirac cone
tilting

Y. M. P. Gomes<EMAIL_ADDRESS>Departamento de Física Teórica,
Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil
Rudnei O. Ramos<EMAIL_ADDRESS>Departamento de Física Teórica, Universidade
do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil  Physics
Department, McGill University, Montreal, Quebec, H3A 2T8, Canada

###### Abstract

The chiral and superconducting gaps are studied in the context of a planar
fermion model with four-fermion interactions. The effect of the tilt of the
Dirac cone on both gaps is shown and discussed. Our results point to two
different behaviors exhibited by planar fermionic systems. We show that there
is a threshold value $\tilde{t}^{*}$ for the effective tilt parameter such
that when $|{\bf\tilde{t}}|<\tilde{t}^{*}$, the superconducting phase persists
for negative values of the superconducting coupling constant. For positive
values of the superconducting coupling constant, the induction of a
superconducting gap by a chemical potential exists and which is similar to the
one seen in graphene-like systems. For $|{\bf\tilde{t}}|>\tilde{t}^{*}$ and a
negative superconducting coupling constant, the superconducting phase can be
present, but it is restricted to a smaller area in the phase portrait. Our
analysis also shows that when $|{\bf\tilde{t}}|>\tilde{t}^{*}$ and for
positive values for the superconducting coupling constant, the induction of a
superconducting gap in the presence of a chemical potential is ruled out. In
this case, the increase of the chemical potential works in favor of the
manifestation of a metallic phase.

##  I Introduction

The possibility of the superconducting phase in the Weyl fermion system is one
of the popular topics in condensed matter physics. The discovery of a tilted
Weyl dispersion in realistic materials, for example, type-II Weyl semimetals,
has accelerated the related research on this topic. The detailed verification
of the phase diagram for these types of systems is of relevance for
researchers in the field. Here we approach this important problem from the
point of view of quantum field theory techniques. Since the seminal work of
Gross and Neveu intro1 , where the authors use quantum field theory (QFT)
tools to describe two-dimensional massless fermions with quartic interactions,
much attention was expended to apply QFT techniques in low-dimensional systems
and with special attention to condensed matter problems. One of the most
interesting examples of the applications of QFT in condensate matter is the
study of graphene intro2 . In this almost planar system, the electrons obey
linearly dispersing relations and the fermionic excitations are well described
by a relativistic Dirac equation in (2+1)-dimensions.

The Lorentz symmetry is respected by the electrons in graphene due to its
relativistic characteristics, but this feature is an exception compared to the
majority of materials in condensed matter. Some of the condensed matter
systems, where the dispersion in the proximity of band touching points can be
generically linear and resemble the Weyl equation, do not respect Lorentz
symmetry Grassano2020 ; zhang2021 ; kost2022 ; Tamashevich2022 . Even though
there are quasiparticles in the aforementioned systems that behave like Weyl
fermions weyl1 , these systems are described by Weyl-like Hamiltonians and,
thus, these quasiparticles are by construction massless and more stable
against gap formation in comparison to Dirac ones weyl2 .

Our proposal in this paper is to study how the properties associated with Weyl
fermions influence the formations of chiral and superconducting gaps in planar
systems. Superconductivity was studied in three-dimensional Weyl semi-metals
of both types I and II with a particular effect of the tilting of the Dirac
cone Rosenstein2018 . From the experimental side, despite the challenges,
there have been studies in heterostructures consisting of thin films of half-
metal and spin-singlet superconductor hao2017 . The theoretical study of
superconducting instabilities in Dirac and Luttinger fermions has also been
recently analyzed szabo2021 . Here we will extend the usual Weyl Hamiltonian
used in the description of Weyl semimetals (WSM) goerbig1 ; goerbig2  by
introducing two forms of four-fermion interactions that will allow for a
chiral phase and a superconducting phase. We also analyze the properties of
this system under the effects of a finite chemical potential, which in
practice models the doping process. This will allow us to study the allowed
phase transitions in a $(2+1)$\- dimensional Gross-Neveu (GN)-type model,
which describes the competition between the chiral symmetry breaking and
superconductivity. These two phenomena will dispute the true ground state of
the system through the intensity of the coupling constants and as a function
of the chemical potential. Let us also recall that chiral symmetry and its
breaking can be seen as a way to describe the metal-insulator phase transition
in these planar systems. Thus, the study of chiral symmetry breaking in planar
systems by GN-like four-fermion interactions has become a useful tool for
qualitative analysis of the two-dimensional system and has already been used
successfully in many different contexts Caldas:2008zz ; Caldas:2009zz ; gn1 ;
gn2 ; gn3 ; Ramos:2013aia ; Klimenko:2012tk ; Klimenko:2013gua ; Ebert:2015hva
; Ebert:2016ygm ; Zhukovsky:2017hzo ; Zerf:2017zqi ; Fernandez:2021dfk ;
Drut:2007zx ; Juricic:2009px ; Herbut:2009vu ; Rostami:2020set ;
Khunjua:2021hhb ; Khunjua:2021fus .

In this paper, we also want the address the question of the production of a
superconducting phase in the model and how the tilting of the Dirac cone
affects it. The phenomenon of electron pairing in the vast majority of
superconductors follows the Bardeen–Cooper–Schrieffer (BCS) theory of
superconductivity. The BCS theory describes the condensation of electrons into
pairs with anti-parallel spins in a singlet state with an $s$-wave symmetry.
The $s$-wave channel will be the superconducting channel that will be
addressed in this paper. Several works have already indicated that
superconductivity appears in planar systems, such as twisted bilayer graphene
superc1 , normal trilayer graphene superc2 , and twisted trilayer graphene as
well superc3 ; superc4 . However, here we address the the effects caused by
the tilt of the Dirac cone on the combined chiral and superconducting phases
and how it might influence, in particular, the superconducting gap.

The tilting, the coupling constants for the chiral and pairing interactions in
the superconducting channel, and the chemical potential provide four
independent parameters. From the coupling constants, we can present a phase
diagram in the case where the tilt factor and the chemical potential take
values that are of practical interest. In particular, we find that the
existence of the superconducting phase strongly depends on whether the tilt
factor is larger or smaller than a threshold value, $\tilde{t}^{*}$, which we
explicitly estimate both analytically and numerically, besides of depending as
well on the sign of the pairing interaction in the superconducting channel.

The remainder of this paper is organized as follows. In Sec. II, we briefly
discuss the main properties of two-dimensional Dirac and Weyl semimetal
systems. In Sec. III, we present the extension of the model that describes the
four-fermion interactions for the excitonic and superconducting channels. The
effective thermodynamic potential for the system is derived through the mean-
field and one-loop semi-classical approximation level. The effects of he
anisotropy, tilting of the Dirac cone, and chemical potential are taking into
account in this derivation. In Sec. IV, we show and discuss the effect of the
chemical potential $\mu$ on the effective thermodynamic potential and we
present the chiral and superconductivity gap equations of the system. In Sec.
V, we discuss the phase transition of the system as a function of the chemical
potential. In Sec. VI, our conclusions and remarks are presented, along also
with the discussion of the possible implications of our results to some
current experimental planar materials of interest. Two appendices are also
included where some technical details are presented. Throughout this paper, we
will be considering the natural units where $\hbar=k_{B}=c=1$.

##  II Two dimensional Weyl semimetals

In this section, we present the main details of the representation of the low
energy electronic excitations in the two-dimensional Weyl semimetals. Within
the tight-binding approximation calculated for the honeycomb-like lattices,
the low energy dynamics of the two-dimensional system of Weyl fermions can be
described by the Hamiltonian goerbig1 ; goerbig2

$H_{t}({\bf p})=v_{F}\left[({\bf t}\cdot{\bf
p})\tau^{0}+(\xi_{x}p_{x})\tau^{x}+(\xi_{y}p_{y})\tau^{y}\right],$ (1)

where $v_{F}$ is the Fermi velocity, ${\bf t}$ is called the tilt vector and
that describes the Dirac cone tilt, ${\bf\xi}=(\xi_{x},\xi_{y})$ is the vector
that describes the anisotropy of the material, $\tau^{0}=\mathbb{1}$ is the
$2\times 2$ identity matrix and $\tau^{x,y}$ are the Pauli matrices. In the
limit ${\bf t}\rightarrow 0$ and $\xi_{x}=\xi_{y}=1$, we recover the
Hamiltonian of the isotropic graphene. The tilt vector ${\bf t}$ is related to
the separation between the Dirac cones in the Weyl semi-metal. A consequence
of the non-null tilt term in Eq. (1) is that the Dirac points, denoted by $D$
and $D^{\prime}$, no longer coincide with the Brillouin corners K and
$K^{\prime}$ (see, e.g., Ref. goerbig1 ). In particular, type-I Weyl semi-
metals are characterized by $|{\bf t}|<1$, while type-II ones are
characterized by $|{\bf t}|>1$. From the Hamiltonian given by Eq. (1), one
finds that the spectrum is given by

$E_{\lambda}({\bf p})=v_{F}\Big{[}{\bf t}\cdot{\bf
p}+\lambda\sqrt{(\xi_{x}p_{x})^{2}+(\xi_{y}p_{y})^{2}}\Big{]},$ (2)

where $\lambda=\pm 1$ represent the conduction and valence bands,
respectively. Note that to be able to associate $\lambda=+1$ with a positive
and $\lambda=-1$ with a negative energy state, it is required that goerbig1 ;
goerbig2

$\sqrt{\left(\frac{t_{x}}{\xi_{x}}\right)^{2}+\left(\frac{t_{y}}{\xi_{y}}\right)^{2}}=|{\bf\tilde{t}}|<1,$
(3)

where $|{\bf\tilde{t}}|$ is called the effective tilt parameter.

The Hamiltonian given by Eq. (1) commutes with the chirality operator defined
as

$\mathcal{C}=\frac{(\xi_{x}p_{x})\tau_{x}+(\xi_{y}p_{y})\tau_{y}}{\sqrt{(\xi_{x}p_{x})^{2}+(\xi_{y}p_{y})^{2}}},$
(4)

with the eigenvalues given by $\alpha=\pm 1$. Taking into account all the
degeneracies of the system, the free Weyl fermion can be described with a
four-component spinor and a Dirac-like Lagrangian density can be written as
follows (see also, e.g., Ref. Gomes:2021nem ):

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N}i\bar{\psi}_{k}M^{\mu\nu}\gamma_{\mu}\partial_{\nu}\psi_{k},$
(5)

where $\psi$ is a four-component Dirac fermion. The $\gamma$-matrices are
written as

$\gamma^{\mu}=\tau^{\mu}\otimes\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (6)

with $\mu=0,1,2$, $\tau^{\mu}=(\tau_{z},i\tau_{x},i\tau_{y})$,
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$ and $\tau_{z}$ is the third Pauli
matrix. The $\gamma$-matrices obey the identity
$\gamma^{\mu}\gamma^{\nu}=\eta^{\mu\nu}+i\epsilon^{\mu\nu\lambda}\gamma_{3}\gamma_{\lambda}$,
where $\gamma_{3}=\begin{pmatrix}\mathbb{1}&0\\\ 0&-\mathbb{1}\end{pmatrix}$
and $\eta^{\mu\nu}=diag(+,-,-)$. Thus, it is straightforward to prove that the
$\gamma$-matrices obey the algebra
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$. The matrix $M$ in Eq. (5) is
explicitly given by

$M=\begin{pmatrix}1&-v_{F}t_{x}&-v_{F}t_{y}\\\ 0&-v_{F}\xi_{x}&0\\\
0&0&-v_{F}\xi_{y}\end{pmatrix}.$ (7)

We can see $M$ as representing an analogous of an effective metric. One also
notices that $M^{\mu\nu}$ contains the parameters that explicitly break the
Lorentz symmetry, which is a consequence of the tilting of the Dirac cone. It
is easy to show that the Lagrangian density given by Eq. (5) has a discrete
chiral symmetry given by $\psi\rightarrow\gamma_{5}\psi$ and
$\bar{\psi}\rightarrow-\bar{\psi}\gamma_{5}$, with

$i\gamma_{5}=\begin{pmatrix}0&\mathbb{1}\\\ -\mathbb{1}&0\end{pmatrix}.$ (8)

Throughout the next sections, one follows the Ref. Gomes:2021nem  and choose
the mass term that breaks the chiral symmetry as $\bar{\psi}\psi$.

##  III Chiral and difermion interactions

To write an effective Lagrangian density that can describe the
(2+1)-dimensional Weyl semimetal with both chiral symmetry breaking (excitonic
pairing) and superconductivity (Cooper pairing), two forms of four-fermion
interactions can be introduced Klimenko:2012tk . One of them is a four fermion
interaction for the scalar fermion-antifermion and the other one is for the
scalar difermion channel. The complete model can then be written as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N}\bar{\psi}_{k}(iM^{\mu\nu}\gamma_{\mu}\partial_{\nu}+\gamma^{0}\mu)\psi_{k}+\frac{G_{1}v_{F}}{2N}\Bigg{(}\sum_{k=1}^{N}\bar{\psi}_{k}\psi_{k}\Bigg{)}^{2}$
(9)
$\displaystyle+\frac{G_{2}v_{F}}{2N}\sum_{k=1}^{N}(\psi_{k}^{T}C\psi_{k})\sum_{j=1}^{N}(\bar{\psi}_{j}C\bar{\psi}_{j}^{T}),$

where $C=i\gamma^{2}$ is the charge conjugation matrix and $G_{1}$ and $G_{2}$
are the coupling constants for the chiral and difermion channels. The coupling
constants $G_{1}$ and $G_{2}$ are negative for an attractive interaction,
while they are positive for a repulsive interaction. The attractive/repulsive
nature of the couplings will be decisive for the phase transition patterns
analyzed in the subsequent sections. The effective action of the model can be
expressed as

$\displaystyle\exp(iS_{eff})$ $\displaystyle=$ $\displaystyle\int
D\bar{\psi}D\psi D\Delta D\Delta^{*}D\sigma$ (10) $\displaystyle\times$
$\displaystyle\exp\left\\{\int
d^{3}x\left[\frac{N}{2G_{1}v_{F}}\sigma^{2}+\frac{N}{2G_{2}v_{F}}\Delta^{*}\Delta\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.\sum_{k=1}^{N}\bar{\psi}_{k}(iM^{\mu\nu}\gamma_{\mu}\partial_{\nu}+\gamma^{0}\mu+\sigma)\psi_{k}\right.\right.$
$\displaystyle+$
$\displaystyle\left.\left.\frac{\Delta^{*}}{2}\psi_{k}^{T}C\psi_{k}+\frac{\Delta}{2}\bar{\psi}_{k}C\bar{\psi}_{k}^{T}\right]\right\\},$

where $\sigma=\frac{G_{1}v_{F}}{N}\sum_{j=1}^{N}\bar{\psi}_{j}\psi_{j}$,
$\Delta=\frac{G_{2}v_{F}}{N}\sum_{j=1}^{N}\psi_{j}^{T}C\psi_{j}$ and
$\Delta^{*}=\frac{G_{2}v_{F}}{N}\sum_{j=1}^{N}\bar{\psi}_{j}C\bar{\psi}_{j}^{T}$.
We can explicitly integrate over the fermion field (for the technical details,
see Appendix A) and the effective action can be rewritten as
$S_{eff}(\sigma,\Delta,\Delta^{*})=N\int
d^{3}x\Omega(\sigma,\Delta,\Delta^{*})$, where $\Omega$ is the effective
thermodynamics potential,

$\displaystyle\Omega(\sigma,\Delta,\Delta^{*})$ $\displaystyle=$
$\displaystyle\frac{1}{2G_{1}v_{F}}\sigma^{2}+\frac{1}{2G_{2}v_{F}}\Delta^{*}\Delta$
(11) $\displaystyle+$
$\displaystyle\sum_{i=1}^{2}\int\frac{d^{3}p}{(2\pi)^{3}}ln\lambda_{i}(p),$

with $\lambda_{i}$ denoting the eigenvalues of $B=CDC^{-1}D^{T}-|\Delta|^{2}$,
with $D=M^{\mu\nu}\gamma_{\mu}\partial_{\nu}+\gamma^{0}\mu-\sigma$, which are
given by

$\displaystyle\lambda_{1,2}$ $\displaystyle=$
$\displaystyle\sigma^{2}+\left[p_{0}-v_{F}({\bf t}\cdot{\bf
p})\right]^{2}-v_{F}^{2}{\bf\tilde{p}}^{2}-\mu^{2}-|\Delta|^{2}$
$\displaystyle\pm 2\sqrt{\sigma^{2}\left\\{\left[p_{0}-v_{F}({\bf t}\cdot{\bf
p})\right]^{2}-v_{F}^{2}{\bf\tilde{p}}^{2}\right\\}+v_{F}^{2}\mu^{2}{\bf\tilde{p}}^{2}}.$

with ${\bf\tilde{p}}=(\xi_{x}p_{x},\xi_{y}p_{y})$. Using the identity

$\displaystyle\int_{-\infty}^{\infty}dp_{0}\ln(p_{0}-A)=i\pi|A|~{},$ (13)

we find that

$\displaystyle\sum_{i=1}^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\ln\lambda_{i}(p)$
$\displaystyle=$
$\displaystyle-\int\frac{d^{2}p}{(2\pi)^{2}}(|\Sigma^{+}|+|\Sigma^{-}|),$

where

$\displaystyle\Sigma^{\pm}=v_{F}({\bf t}\cdot{\bf
p})+\sqrt{\tilde{E}^{2}+\mu^{2}+|\Delta|^{2}\pm
2\sqrt{\sigma^{2}|\Delta|^{2}+\mu^{2}\tilde{E}^{2}}},$

and $\tilde{E}^{2}=v_{F}^{2}{\bf\tilde{p}}^{2}+\sigma^{2}$. Finally, for
constant configurations $\sigma_{0}=\langle\sigma\rangle$ and
$\Delta_{0}=\langle\Delta\rangle=\langle\Delta^{*}\rangle$, we find

$\displaystyle\Omega(\sigma_{0},\Delta_{0},\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2G_{1}v_{F}}\sigma_{0}^{2}+\frac{1}{2G_{2}v_{F}}\Delta_{0}^{2}$
(16) $\displaystyle-$
$\displaystyle\int\frac{d^{2}p}{(2\pi)^{2}}\left(|\Sigma_{0}^{+}|+|\Sigma_{0}^{-}|\right),$

where $\Sigma^{\pm}_{0}=\Sigma^{\pm}(\sigma=\sigma_{0},\Delta=\Delta_{0})$.
Note that the momentum integral in Eq. (16) is divergent in the ultraviolet
limit and, thus, the effective potential given by Eq. (16) needs to be
renormalized. The renormalization of Eq. (16) is described below.

###  III.1 Renormalization

Taking $\mu=0$ in Eq. (16), we will have that $\Sigma^{\pm}_{0}={\bf
t}\cdot{\bf p}+\sqrt{{\bf\tilde{p}}^{2}+(\sigma_{0}\pm\Delta_{0})^{2}}$ and,
therefore,

$\displaystyle\Omega(\sigma_{0},\Delta_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2G_{1}v_{F}}\sigma_{0}^{2}+\frac{1}{2G_{2}v_{F}}\Delta_{0}^{2}$
$\displaystyle-$ $\displaystyle\int\frac{d^{2}p}{(2\pi)^{2}}\left|v_{F}({\bf
t}\cdot{\bf
p})+\sqrt{v_{F}^{2}{\bf\tilde{p}}^{2}+(\sigma_{0}+\Delta_{0})^{2}}\right|$
$\displaystyle-$ $\displaystyle\int\frac{d^{2}p}{(2\pi)^{2}}\left|v_{F}({\bf
t}\cdot{\bf
p})+\sqrt{v_{F}^{2}{\bf\tilde{p}}^{2}+(\sigma_{0}-\Delta_{0})^{2}}\right|.$

The linear term ${\bf t}\cdot{\bf p}$ in Eq. (LABEL:unrenor) vanishes in the
integration over the angular variable111We use the identity
$\int_{0}^{2\pi}d\theta|a\cos\theta+b|=2\pi b\Theta(b-a)$, where $\Theta(x)$
is the Heaviside function, for $a>0$ and $b>0$., but the integral in Eq.
(LABEL:unrenor) is still divergent. Thus, applying the re-scaling
$\xi_{x,y}p_{x,y}\rightarrow p_{x,y}$ and integrating with the introduction of
a momentum cutoff $\Lambda$, one defines the renormalization conditions,

$\displaystyle\frac{1}{g_{1}(m)}$ $\displaystyle=$ $\displaystyle
v_{F}\frac{d^{2}\Omega(\sigma_{0},\Delta_{0})}{d\sigma_{0}^{2}}\Big{|}_{\sigma_{0}=m,\Delta_{0}=0}$
(18) $\displaystyle=$ $\displaystyle\frac{1}{G_{1}}+\frac{2m}{\pi
v_{F}\xi_{x}\xi_{y}}-\frac{\Lambda}{\pi v_{F}\xi_{x}\xi_{y}},$

and

$\displaystyle\frac{1}{g_{2}(m^{\prime})}$ $\displaystyle=$ $\displaystyle
v_{F}\frac{d^{2}\Omega(\sigma_{0},\Delta_{0})}{d\Delta_{0}^{2}}\Big{|}_{\sigma_{0}=0,\Delta_{0}=m^{\prime}}$
(19) $\displaystyle=$ $\displaystyle\frac{1}{G_{2}}+\frac{2m^{\prime}}{\pi
v_{F}\xi_{x}\xi_{y}}-\frac{\Lambda}{\pi v_{F}\xi_{x}\xi_{y}},$

where $m$ and $m^{\prime}$ are regularization scales. Going further, defining
the renormalized couplings $g_{1}$ and $g_{2}$ as

$\displaystyle\frac{1}{g_{1}}=\frac{1}{g_{1}(m)}-\frac{2m}{\pi
v_{F}\xi_{x}\xi_{y}},$ (20)

and

$\displaystyle\frac{1}{g_{2}}=\frac{1}{g_{2}(m^{\prime})}-\frac{2m^{\prime}}{\pi
v_{F}\xi_{x}\xi_{y}},$ (21)

the renormalized effective thermodynamic potential finally can be expressed as

$\displaystyle\Omega^{\text{ren}}(\sigma_{0},\Delta_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2g_{1}v_{F}}\sigma_{0}^{2}+\frac{1}{2g_{2}v_{F}}\Delta_{0}^{2}$
(22) $\displaystyle+\frac{(\sigma_{0}+\Delta_{0})^{3}}{6\pi
v_{F}^{2}\xi_{x}\xi_{y}}+\frac{|\sigma_{0}-\Delta_{0}|^{3}}{6\pi
v_{F}^{2}\xi_{x}\xi_{y}}~{}.$

###  III.2 Phase diagram of the system at $\mu=0$

Let us first specialize in the analysis of the effective thermodynamic
potential and its properties in the case of a null chemical potential. In this
perspective, we analyze the two sectors, the chiral and the superconductor
ones, individually. This will allows us to extract the main characteristics of
the model. After this analysis, we can then compare the results and show where
each phase will be mandatory in the system.

The minima of Eq. (22) are given in terms of the gaps $\bar{\sigma}_{c}$ and
$\bar{\Delta}_{c}$, which are defined as $\bar{\sigma}_{c}=\pi
v_{F}\xi_{x}\xi_{y}/|g_{1}|$ and $\bar{\Delta}_{c}=\pi
v_{F}\xi_{x}\xi_{y}/|g_{2}|$. By analyzing the thermodynamic potential given
by Eq. (22), in the absence of tilting, it can be established that the system
can be characterized by three phases, according to the values of $\sigma$,
$\Delta$, and the coupling constants. We follow the same classification used
in Ref. Klimenko:2013gua  which studied the nontilted system. Phase I: this is
the symmetric phase, where both vacuum expectations values for the chiral and
superconducting phases are zero, $\sigma_{0}=\Delta_{0}=0$, and which can take
place when for $g_{1}>0$ and $g_{2}>0$. Phase II: in this phase
$\sigma_{0}=\bar{\sigma}_{c}\neq 0$ and $\Delta_{0}=0$ and it can happen when
$g_{1}<0$. Phase III: in this phase $\sigma_{0}=0$ and
$\Delta_{0}=\bar{\Delta}_{c}\neq 0$ and it can happen when $g_{2}<0$. When
$g_{1}$ and $g_{2}$ are simultaneously negative, the system is characterized
as phase II if $|g_{1}|>|g_{2}|$ and as phase III for $|g_{1}|<|g_{2}|$. In
the next two sections, we will analyze how the effects of both tilting and
chemical potential affect these different phases allowed by the model.

##  IV Tilting effects on the superconducting gap

Let us now turn on the effects of the tilting of the Dirac cone on the
different three phases allowed by the model and described at the end of Sec.
III. It is useful to first focus on the pure chiral phase (when
$\Delta_{0}=0$), where we here briefly reproduce some of the results obtained
in Ref. Gomes:2021nem . After that, we will analyze the case of the
superconducting gap in details.

###  IV.1 The pure chiral phase ($\Delta_{0}=0$)

By considering the pure chiral phase, i.e., by considering $\Delta_{0}=0$, one
notices that

$(\Sigma^{\pm}_{0})|_{\Delta_{0}=0}=\mathcal{E}^{\pm}_{\sigma}=v_{F}({\bf
t}\cdot{\bf p})+\big{|}\tilde{E}_{0}\pm\mu\big{|},$ (23)

where $\tilde{E}_{0}=\sqrt{v_{F}^{2}{\bf\tilde{p}}^{2}+\sigma_{0}^{2}}$.
Assuming $\mu>0$, one finds in this case that the effective thermodynamic
potential (22) becomes

$\displaystyle\Omega^{ren}(\sigma_{0},0,\mu)=\frac{\sigma_{0}^{2}}{2g_{1}v_{F}}+\frac{\sigma_{0}^{3}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}$
$\displaystyle-\int\frac{d^{2}p}{(2\pi)^{2}}\Bigg{(}|\mathcal{E}^{+}_{\sigma}|+|\mathcal{E}^{-}_{\sigma}|-2E_{\sigma}\Big{)}$
$\displaystyle=\frac{\sigma_{0}^{2}}{2g_{1}v_{F}}+\frac{\sigma_{0}^{3}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}-\int\frac{d^{2}p}{(2\pi)^{2}}\Big{(}\mu-\tilde{E}_{\sigma}+|\mu-\tilde{E}_{\sigma}|\Big{)}$
$\displaystyle=\frac{\sigma_{0}^{2}}{2g_{1}v_{F}}+\frac{\sigma_{0}^{3}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}-2\int\frac{d^{2}p}{(2\pi)^{2}}(\mu-\tilde{E}_{\sigma})\Theta(\mu-\tilde{E}_{\sigma}),$

with $\tilde{E}_{\sigma}=v_{F}({\bf t}\cdot{\bf
p})+\sqrt{v_{F}^{2}{\bf\tilde{p}}^{2}+\sigma_{0}^{2}}$ and we have used the
identity $x+|x|=2x\Theta(x)$. From Eq. (LABEL:potchiral), one can derive the
gap equation,

$1+\frac{{\rm
sign}(g_{1})\sigma_{0}}{\bar{\sigma}_{c}}+2g_{1}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{\Theta(\mu-\tilde{E}_{\sigma})}{\sqrt{v_{F}^{2}{\bf\tilde{p}}^{2}+\sigma_{0}^{2}}}=0.$
(25)

It follows from Eq. (25) that for $g_{1}>0$ the chiral symmetry is maintained
for any $\mu>0$. We can also see that the effect of the effective tilt
parameter $|{\bf\tilde{t}}|$ in Eq. (25) is to enhance the effect of the
chemical potential and, hence, to lower the point of chiral symmetry
restoration. In particular, for $g_{1}<0$ the chiral symmetry breaks for
$\mu<\mu_{c}$ and is restored for $\mu>\mu_{c}$, where the critical chemical
potential $\mu_{c}$ is found to be given by Gomes:2021nem

$\mu_{c}=\sqrt{1-|{\bf\tilde{t}}|^{2}}\bar{\sigma}_{c}.$ (26)

Therefore, we can say that the presence of the non-vanishing tilt parameter
tends to facilitate the chiral symmetry restoration. One also finds from Eq.
(25) that the chiral order parameter, which is the solution of (25), jumps
discontinuously from $\sigma_{0}=\bar{\sigma}_{c}$ to $\sigma_{0}=0$ as we
change the chemical potential from $\mu<\mu_{c}$ to $\mu>\mu_{c}$. This is a
first-order transition that exists for both the non-tilted case
$|{\bf\tilde{t}}|=0$ and the tilted case $|{\bf\tilde{t}}|\neq 0$.

The charge density $n$ is defined as

$n=-N\frac{\partial\Omega^{ren}(\sigma_{0},0,\mu)}{\partial\mu}\Bigg{|}_{\sigma_{0}=\langle\sigma_{0}\rangle}.$
(27)

The exact expression for $n$ can be readily calculated from
$\Omega^{ren}(\sigma_{0},0,\mu)$ and it reads

$\displaystyle n(g_{1}>0)=\frac{N\mu^{2}}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}(1-|{\bf\tilde{t}}|^{2})^{3/2}},$ (28)

when $g_{1}>0$ and

$n(g_{1}<0)=\frac{N(\mu^{2}-\mu_{c}^{2})}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}(1-|{\bf\tilde{t}}|^{2})^{3/2}}\Theta\left(\mu^{2}-\mu_{c}^{2}\right),$
(29)

when $g_{1}<0$.

In the next section one turns to the analysis of the superconducting phase.

###  IV.2 The pure superconducting phase ($\sigma_{0}=0$)

In the case of a pure superconducting phase, i.e., considering now
$\sigma_{0}=0$, and using the identity

$(\Sigma^{\pm}_{0})|_{\sigma_{0}=0}=\mathcal{E}^{\pm}_{\Delta}=v_{F}({\bf
t}\cdot{\bf p})+\sqrt{(v_{F}|{\bf\tilde{p}}|\pm\mu)^{2}+\Delta_{0}^{2}},$ (30)

the effective thermodynamic potential (22) can be written as

$\displaystyle\Omega^{ren}(0,\Delta_{0},\mu)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{0}^{2}}{2g_{2}v_{F}}+\frac{\Delta_{0}^{3}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}-\int\frac{d^{2}p}{(2\pi)^{2}}\left\\{|\mathcal{E}^{+}_{\Delta}|+|\mathcal{E}^{-}_{\Delta}|-2\left[v_{F}({\bf
t}\cdot{\bf
p})+\sqrt{v_{F}^{2}|{\bf\tilde{p}}|^{2}+\Delta_{0}^{2}}\right]\right\\}.$

Performing the momentum integrals in Eq. (LABEL:veffsup), one finds

$\displaystyle\Omega^{ren}(0,\Delta_{0},\mu)=\frac{\Delta_{0}^{2}}{2g_{2}v_{F}}+\frac{(\mu^{2}+\Delta_{0}^{2})^{3/2}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}$ (32)
$\displaystyle-\frac{\mu^{2}\sqrt{\mu^{2}+\Delta_{0}^{2}}}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}-\frac{\mu\Delta_{0}^{2}}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\ln\left(\frac{\mu+\sqrt{\mu^{2}+\Delta_{0}^{2}}}{\Delta_{0}}\right)$
$\displaystyle+\frac{1}{2\pi\xi_{x}\xi_{y}v_{F}^{2}}I(\Delta_{0},\mu),$

where the function $I(\Delta_{0},\mu)$ is derived explicitly in the Appendix B
and given by Eq. (55). The effective thermodynamic potential given by Eq. (32)
is shown in Fig. 1(a) for $g_{2}<0$, while for $g_{2}>0$ it is shown in Fig.
1(b) for $g_{2}>0$, where we have considered some representative values of the
effective tilt parameter and for the chemical potential. The emergence of a
superconducting gap $\Delta\neq 0$ due to the combined effect of the tilt and
chemical potential is noted. Let us analyze in more details the contribution
of the tilt parameter for the superconducting gap. New features generated by
the tilt of the Dirac cone will influence the superconducting gap for both the
$g_{2}>0$ and $g_{2}<0$ scenarios and are explained below.

(a) $g_{2}<0$ (b) $g_{2}>0$

Figure 1: Effective thermodynamic potential for (a) $g_{2}<0$ and (b)
$g_{2}>0$, in units of $\Omega_{0}=N\bar{\Delta}_{c}^{3}/(\pi
v_{F}^{2}\xi_{x}\xi_{y})$ as a function of $\Delta/\bar{\Delta}_{c}$.

From the effective thermodynamic potential one derives the gap equation,

$\displaystyle{\rm
sign}(g_{2})+\sqrt{x^{2}+y^{2}}-y\ln\left(\frac{y+\sqrt{x^{2}+y^{2}}}{x}\right)$
$\displaystyle+\frac{1}{2x}\frac{\partial I(x,y)}{\partial x}=0,$ (33)

where $x=\Delta_{0}/\bar{\Delta}_{c}$ , $y=\mu/\bar{\Delta}_{c}$. The
superconducting gap that is induced by the chemical potential and the tilt
parameter is shown in Figs. 2(a) and 2(b), for the cases of $g_{2}<0$ and for
$g_{2}>0$, respectively.

(a) $g_{2}<0$ (b) $g_{2}>0$

Figure 2: Superconducting gap for (a) $g_{2}<0$ and (b) $g_{2}>0$ induced by
the chemical potential and the tilt parameter in units of $\bar{\Delta}_{c}$
for some representative values of $|{\bf\tilde{t}}|$.

In Fig. 2, the numerical results for the superconducting gap are shown as a
function of the chemical potential and some representative values for the
effective tilt parameter $|\tilde{\bf t}|$.

Figure 3: The normalized chemical potential $\mu^{*}_{<}$ as a function of
$|{\bf\tilde{t}}|$. The thin vertical dashed line represents the threshold
value $|{\bf\tilde{t}}|=\tilde{t}^{*}$.

In the case $g_{2}<0$, which is shown in Fig. 2(a), one can see that the tilt
increases $\Delta_{c}$ for a given $\mu>\mu^{*}_{<}$. In the particular value
$\mu^{*}_{<}=\mu^{*}_{<}(|{\bf\tilde{t}}|)$ is where the tilt parameter starts
to contribute to the superconducting gap. The behavior of $\mu^{*}_{<}$ as a
function of $|{\bf\tilde{t}}|$ is shown in Fig. 3. We find that there is a
threshold value for the effective tilt parameter, $\tilde{t}^{*}$, such that
when $|{\bf\tilde{t}}|<\tilde{t}^{*}$, the superconducting gap is given by
$\Delta_{0}=\Delta_{{\bf t}=0}$ for any $\mu$. However, for values of
$|{\bf\tilde{t}}|>\tilde{t}^{*}$ and when $\mu>\mu^{*}_{<}$, the
superconducting gap is given by $\Delta_{0}=\Delta_{{\bf t}}$. On the other
hand, in the case of $\mu<\mu^{*}_{<}$, the superconducting gap is given
instead again by $\Delta_{0}=\Delta_{{\bf t}=0}$. Let us obtain an explicit
estimation for this particular value for the effective tilt parameter,
$\tilde{t}^{*}$. For $\mu>\mu^{*}_{<}$, the superconducting gap takes the
exact form $\Delta_{c}(\mu>\mu^{*}_{<})=\Delta_{\bf t}$, where

$\Delta_{{\bf t}}=\frac{|{\bf\tilde{t}}|\mu}{\sqrt{1-|{\bf\tilde{t}}|^{2}}}.$
(34)

Moreover, in order to extract the asymptotic behavior of the superconducting
gap shown in Fig. 2, one first notes that in the gap equation for the non-
tilted case ($\tilde{\bf t}=0$), the last term in Eq. (33) vanishes. Hence,
for $\tilde{\bf t}=0$,

$\displaystyle{\rm
sign}(g_{2})+\sqrt{x^{2}+y^{2}}-y\ln\left(\frac{y+\sqrt{x^{2}+y^{2}}}{x}\right)=0.$

Now, it is reasonable to assume that in the large $y$ limit the normalized gap
solution $x$ becomes a linear function of the normalized chemical potential
$y$, i.e., $x=\lambda y+c$, with $c$ a constant. Hence, considering the
asymptotic limit $y,x\ll c$ and multiplying Eq. (LABEL:app31) by $1/y$, one
obtain that $\lambda$ satisfies

$\sqrt{\lambda^{2}+1}-\ln\left(\frac{\sqrt{\lambda^{2}+1}+1}{\lambda}\right)\approx
0.$ (35)

The above equation has one positive solution given by $\lambda\simeq 0.66$.
The threshold value $\tilde{t}^{*}$ for which the effective tilt parameter
begins to drive the superconducting gap is determined when the superconducting
gap, given by Eq. (34), becomes parallel to the asymptotic linear behavior of
the tilt-less gap equation, i.e., we must have $\Delta_{\bf t}=\lambda\mu$.
This leads to the relation

$\frac{|\tilde{\bf t}|}{\sqrt{1-|\tilde{\bf t}|^{2}}}=\lambda~{}~{}.$ (36)

The solution of the above equation gives us the result $\tilde{t}^{*}\simeq
0.55$ when using the solution for $\lambda$ obtained from Eq. (36). This
result agrees quite well with the numerical results expected from Figs. 2 and
3.

When $g_{2}<0$, for any value of the tilt parameter
$|{\bf\tilde{t}}|<\tilde{t}^{*}$ the effect of the tilt parameter in the
superconducting gap vanishes for any $\mu$, and the superconducting gap of the
system obeys the solid gray curve shown in Fig. 2(a). We can also analyze the
situation for the case of $g_{2}>0$. Analyzing now the case for $g_{2}>0$, we
are able to uncover another structure for the superconducting gap. As can be
seen in Fig. 2(b), in this case we have two different situations. When
$|{\bf\tilde{t}}|<\tilde{t}^{*}$, the tilt parameter only contributes for the
chemical potential up to the values $\mu_{>}^{*}$, $\mu<\mu^{*}_{>}$, e.g., as
in the case seen by the blue curve in Fig. 2(b). This particular value
$\mu^{*}_{>}=\mu^{*}_{>}(|{\bf\tilde{t}}|)$ sets a lower limit where the tilt
parameter stops contributing to the superconducting gap. The behavior of
$\mu^{*}_{>}$ is shown in Fig. 4. When $|{\bf\tilde{t}}|>\tilde{t}^{*}$, the
superconducting gap will be exactly
$\Delta_{c}(|{\bf\tilde{t}}|>\tilde{t}^{*})=\Delta_{{\bf t}}$. As seen in Fig.
4, we now have that when $g_{2}>0$, for values of $\mu>\mu^{*}_{>}$ the
superconducting gap is given by $\Delta_{0}=\Delta_{{\bf t}=0}$ and for
$\mu<\mu^{*}_{>}$ the superconducting gap is given by $\Delta_{0}=\Delta_{{\bf
t}}$. For $|{\bf\tilde{t}}|>\tilde{t}^{*}$ the superconducting gap is given by
$\Delta_{0}=\Delta_{{\bf t}}$ for any $\mu$.

Figure 4: Plot of the normalized chemical potential $\mu^{*}_{>}$. The thin
vertical dashed line represents the threshold value
$|{\bf\tilde{t}}|=\tilde{t}^{*}$.

Finally, we can explicitly compute the charge density,

$n=-N\frac{\partial\Omega^{ren}(0,\Delta_{0},\mu)}{\partial\mu}\Bigg{|}_{\Delta_{0}=\langle\Delta_{0}\rangle},$
(37)

which it can be expressed through an exact expression and given by

$\displaystyle n$ $\displaystyle=$ $\displaystyle\frac{N}{4\pi
v_{F}^{2}\xi_{x}\xi_{y}}\left[\mu\sqrt{\mu^{2}+\Delta_{0}^{2}}\right.$ (38)
$\displaystyle+$
$\displaystyle\left.\Delta^{2}_{0}\ln\left(\frac{\mu+\sqrt{\mu^{2}+\Delta_{0}^{2}}}{\Delta_{0}}\right)\right]\Bigr{|}_{\Delta_{0}=\langle\Delta_{0}\rangle}$
$\displaystyle+$ $\displaystyle N\frac{\partial
I(\Delta_{0},\mu)}{\partial\mu}\Bigr{|}_{\Delta_{0}=\langle\Delta_{0}\rangle},$

where $\langle\Delta_{0}\rangle$ is the solution of Eq. (33), which can be
found numerically for both the $g_{2}>0$ and $g_{2}<0$ cases. From the
inequality ${\rm
Re}\sqrt{|{\bf\tilde{t}}|^{2}\mu^{2}-(1-|{\bf\tilde{t}}|^{2})\Delta_{0}^{2}}\neq
0$, one finds that the contribution for the charge density from the function
$I$ is non-null only for $\langle\Delta_{0}\rangle<\Delta_{\bf t}$. Thus,
based on Fig. 2(a), this contribution is non-null only for $\mu<\mu^{*}_{<}$.
In the case where $g_{2}>0$, on the other hand, from Fig. 2(b), the density
will receive extra contributions only for $\mu>\mu^{*}_{>}$.

##  V Phase structure for $\mu\neq 0$

Previous works Klimenko:2012tk ; Ebert:2016ygm ; Zhukovsky:2017hzo ;
Klimenko:2012tk  have shown that it is sufficient to analyze the chiral-
superconducting phase structure by comparing the vacuum properties in the
$\sigma_{0}=0$ and $\Delta_{0}=0$ axes. Here we follow the same strategy.
Through this analysis of the local minimum in each axis, we can compare them
and find the global minimum which defines the real phase of the system. For
instance, as shown in the previous section, for fixed $g_{1}<0$, there is a
chemical potential for coexistence, $\mu_{c}(g_{2})$. The value of
$\mu_{c}(g_{2})$ defines the lower bound for the chemical potential such that
for $\mu>\mu_{c}(g_{2})$ the system is in the superconducting phase (phase
III), for $\mu<\mu_{c}(g_{2})$ the system is in the chiral symmetry-breaking
phase (phase II), and for $\mu=\mu_{c}(g_{2})$ both phases II and III
coexists. This coexistence point $\mu=\mu_{c}(g_{2})$ defines a first-order
transition between phases II and III. In the case of $g_{2}<0$, there is
another particular value for the chemical potential, $\mu_{<}^{*}$, as
discussed in the previous section, such that for $\mu>\mu_{<}^{*}$ the
superconducting phase stops to drive the system in favor of the chiral phase.
The opposite happens when $g_{2}>0$, in which case there is now a value for
the chemical potential, $\mu=\mu_{>}^{*}$ that becomes an upper bound and, for
$\mu<\mu_{>}^{*}$, it is when the superconducting phase stops to drive the
system in favor of the chiral phase. Finally, the chiral symmetry will be
restored for $\mu>\sqrt{1-|\bf\tilde{t}|^{2}}\bar{\sigma}_{c}$. Let us now
show the different phase portraits that will display the above structure
relating the chemical potential with the superconducting coupling constant
$g_{2}$ of the system when assuming $g_{1}<0$, which is the relevant situation
for nontrivial chiral and superconducting gaps.

Figure 5: Phase portrait for the normalized chemical potential ($|g_{1}|\mu$)
vs $g_{2}/|g_{1}|$ for $g_{1}<0$ and when (a)
$|{\bf\tilde{t}}|=0.3<\tilde{t}^{*}$ and (b)
$|{\bf\tilde{t}}|=0.9>\tilde{t}^{*}$. The thin dashed lines represents the
value for the chemical potential of coexistence in the nontilted case. Phases
I, II and III represent the metallic phase (with $\sigma_{0}=\Delta_{0}=0$),
the insulating phase (with $\sigma_{0}\neq 0$ and $\Delta_{0}=0$) and the
superconducting phase (with $\sigma_{0}=0$ and $\Delta_{0}\neq 0$),
respectively.

For illustration, in Fig. 5(a) we show the phase portrait when
$|{\bf\tilde{t}}|=0.3<\tilde{t}^{*}$ and in the region ranging from negative
to positive values for $g_{2}$, while in Fig. 5(b) the phase portrait is shown
for the case $|{\bf\tilde{t}}|=0.9>\tilde{t}^{*}$. For reference, in both
Figs. 5(a) and 5(b), the phase portrait in the nontilted case,
$|{\bf\tilde{t}}|=0$, is shown by the light gray dashed line, which matches
the result previously obtained in Ref. Klimenko:2012tk . Note that in the non-
tilted case, $|{\bf\tilde{t}}|=0$, the lines of coexistence separates the
phase II, which lies below the dashed line, from the phase III, which lies
above it. There is no phase I (where the chiral and superconducting phases are
absent) in this case. Looking at the region where $g_{2}<0$ in the case
$|{\bf\tilde{t}}|<\tilde{t}^{*}$ shown in Fig. 5(a), it is apparent that the
presence of the effective tilt parameter $|{\bf\tilde{t}}|$ does not
qualitatively change the phase portrait with respect to that of the non-tilted
case. The structure of the phase transition can be summarized as a first-order
phase transition between the insulating phase and the superconducting phase
for a given $\mu_{c}(g_{2})$ represented by the black line. In this case, the
superconducting phase is present for $\mu>\mu_{c}(g_{2})$ in the same manner
as in the non-tilted situation. Looking now at the region where $g_{2}>0$ for
the case $|{\bf\tilde{t}}|<\tilde{t}^{*}$ also shown in Fig. 5(a), one can
notice that the superconductivity induced by the chemical potential still
exists, but in a smaller area when compared to the non-tilted case (dashed
line). We recall that from the results shown in the previous section, for
$\mu<\mu^{*}_{>}$ and when
$\mu>\sqrt{1-|{\bf\tilde{t}}|^{2}}\bar{\sigma}_{c}$, phase I takes place.
Thus, in this case, one finds a point of coexistence, $(\mu^{t},g_{2}^{t})$,
which separates phases I-III, which is given by

$(\mu^{t},g_{2}^{t})|_{g_{2}>0,\,|{\bf\tilde{t}}|<\tilde{t}^{*}}=\left(\frac{\sqrt{1-|{\bf\tilde{t}}|^{2}}}{|g_{1}|},\frac{\sqrt{1-|{\bf\tilde{t}}|^{2}}}{\mu_{>}^{*}}\right)~{}.$
(39)

The presence of the coexistence point as a consequence of the tilt of the
Dirac cone is one of our main results, showing a quite different behavior when
compared to the results in the nontilted case Klimenko:2012tk .

Going further, looking at the case for $g_{2}<0$ and
$|{\bf\tilde{t}}|>\tilde{t}^{*}$, which is shown in Fig. 5(b), one notices a
much stronger change in the phase portrait as compared to the region with
$g_{2}<0$ shown in Fig. 5(a). The presence of the tilt effectively causes the
superconducting gap to stop to drive the system for $\mu>\mu^{*}_{<}$ and
phase I now takes place for
$\mu>\sqrt{1-|{\bf\tilde{t}}|^{2}}\bar{\sigma}_{c}$. The phase portrait in
this case displays a much restricted area for the superconducting phase. The
superconducting phase occurs only for values of $|g_{2}|/|g_{1}|\lesssim 1$.
In this case, one coexistence point also appears and it is found to be given
by

$(\mu^{t},g_{2}^{t})|_{g_{2}<0,\,|{\bf\tilde{t}}|>\tilde{t}^{*}}=\left(\frac{\sqrt{1-|{\bf\tilde{t}}|^{2}}}{|g_{1}|},-\frac{\sqrt{1-|{\bf\tilde{t}}|^{2}}}{\mu_{<}^{*}}\right).$
(40)

Finally, looking at the region where $g_{2}>0$ shown in Fig. 5(b), the
induction of a superconducting phase due to the chemical potential is ruled
out for any value of the chemical potential and the phase transition occurs
between phases I and II. Through the increase of the chemical potential and in
the presence of a tilt satisfying $|{\bf\tilde{t}}|>\tilde{t}^{*}$, both
effects work in favor of the chiral symmetric phase. This can be seen by the
enlarged region for phase I shown in Fig. 5(b) when compared to the nontilted
case. This is our other main result that is extracted from the phase portrait.
It shows once more the effect of the tilt on hindering the formation of gaps
in the system and, in this case, the formation of an induced gap due to the
presence of the chemical potential. The role of the threshold value for the
effective tilt parameter $\tilde{t}^{*}$ becomes quite evident when
contrasting the two panels in Fig. 5.

##  VI Concluding remarks

In this paper, we have investigated the phase diagram of the Weyl fermion
system with four-fermion interactions that introduce the effects of both
chiral and superconducting gaps. Furthermore, we have focused on the effect of
the tilt factor of the Dirac cone. As one of our main results, it is the
demonstration, both analytically and numerically, of the presence of a
threshold value for the effective tilt parameter $\tilde{t}^{*}$ beyond which
the value of the tilting of the Dirac cone strongly affects the
superconducting gap. More specifically, one explicitly finds that
$\tilde{t}^{*}\simeq 0.55$. The stability of the superconducting phase is also
found to be much different, whether the tilting factor is lower or higher than
$\tilde{t}^{*}$. At this value for the effective tilt parameter the system
behaves completely differently under the formation of the chiral and
superconducting gaps when compared to the nontilted case. In the case where
$|{\bf\tilde{t}}|<\tilde{t}^{*}$, the superconducting phase persists for a
negative superconducting coupling constant, which is responsible for the
attractive interaction in the Cooper channel. A first-order phase transition
occurs for a chemical potential for coexistence, as seen by the black curve in
Fig. 5(a). This feature is similar to the results for graphene and other two-
dimensional materials Klimenko:2012tk . One also sees that for $g_{2}>0$ the
induction of a superconducting gap due to the presence of a chemical potential
exists. This induction, however, only happens for stronger values of the
coupling constant $g_{2}$, since the metallic phase appears for small values
of the superconducting coupling constant. Due to the presence of a metallic
phase, we were able to find the expression for the point of coexistence, which
is given by Eq. (39).

While for values of $|{\bf\tilde{t}}|<\tilde{t}^{*}$ the changes to the phase
portrait seen in Fig. 5(a) are of a qualitative nature, when the effective
tilt exceeds the threshold value, the changes now become quantitative. When
$|{\bf\tilde{t}}|$ exceeds the value $\tilde{t}^{*}$, the superconducting
phase now becomes restricted to a smaller area in the phase portrait. Indeed,
in this case, the superconducting phase occurs only for regions with small and
negative superconducting coupling constant. Through a first-order phase
transition, the metallic phase takes place for a sufficient large
superconducting coupling constant and chemical potential. One also finds the
analytic expression for the point of coexistence in this case, which is given
by Eq. (40). Finally, one shows that our qualitative analysis point to the
fact that for $g_{2}>0$, the superconducting gap induced by the chemical
potential is ruled out and a first-order phase transition occurs between
phases I and II at the chemical potential for coexistence, which is
represented by the black curve in Fig. 5(b). The presence and role of the
threshold value for the effective tilt parameter represent one of the main
important results shown in this paper.

We can try to explore the consequences of the results we have obtained for
some known planar systems and which have been currently studied in laboratory
experiments. For example, using the experimental data obtained from the two-
dimensional (2D) organic conductor $\alpha-\text{(BEDT-TTF)}_{2}I_{3}$ ex1 ;
ex2 , the estimated effective tilt parameter is found to be
$|{\bf\tilde{t}}|\simeq 0.76$ (see, e.g., Ref. Gomes:2021nem ). This case
occurs in the situation where $|{\bf\tilde{t}}|>\tilde{t}^{*}\simeq 0.55$ and
which we have discovered in this paper. From our results, this implies that
the inducing of a superconducting gap should be absent in this material. It
would be interesting to probe this prediction using this type of material in
the laboratory. By also accounting for the results obtained from the analysis
of Ref. Gomes:2021nem , we can also conclude that this same system should
exhibit a metallic phase, which would become very strong under doping. On the
other hand, we can also compare with the predictions that our results would
imply for the case of quinoid-type graphene under uniaxial strain goerbig1 .
In this case, the estimated values for the effective tilt parameter are such
that $|{\bf\tilde{t}}|\lesssim 0.06$ for moderate deformations. From our
results, we can conclude that for this material the properties of the
superconducting gap should be similar to the graphene case, which includes the
induction of a superconducting gap by the chemical potential. To the authors’
best knowledge, we are not aware of other materials where the value of the
tilt parameter has been provided, at least as far as two-dimensional materials
are concerned. We are hopeful that as new two-dimensional materials are
experimentally probed and fabricated, new data from those experiments will
help to shed light on the results we have presented here.

The study of possible two-dimensional fermionic systems where our results can
be of interest can be exploited in several directions. First, since the
evaluation of $\tilde{t}^{*}$ is based on the large-$N$ limit of the effective
thermodynamical potential, it is possible that this result receives quantum
corrections beyond the large-$N$ approximation. This can be an interesting
extension of the present work. Going further, the presence of an anomalous
Hall effect Gomes:2021nem ; AHE ; AHE2  in the 2D Weyl semimetal indicates the
possibility that the tilt of the Dirac cones could modify the superconducting
gap under the presence of an external magnetic field. Moreover, since the tilt
of the Dirac cone introduces a special direction in the system, the analysis
of the $p$-wave superconducting gap properties in this context could bring new
features. This can be another problem of interest that can be a target of
further investigation. These problems are possible lines of study that our
results motivate and we hope to address them in the future.

###### Acknowledgements.

Y.M.P.G. is supported by a postdoctoral grant from Fundação Carlos Chagas
Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ). R.O.R.
acknowledges financial support of the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES) - Finance Code 001 and by research grants
from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq),
Grant No. 307286/2021-5, and from Fundação Carlos Chagas Filho de Amparo à
Pesquisa do Estado do Rio de Janeiro (FAPERJ), Grant No. E-26/201.150/2021.

##  Appendix A  Performing the path integral over the fermion in Eq. (10)

Here we show some of the details of the path integral over the fermions in Eq.
(10), which leads to the effective thermodynamic potential. Adopting the
procedure described in Ref. Klimenko:2012tk , we assume two anti-commuting
four-component Dirac spinor fields $q(x)$ and $\bar{q}(x)$. Then, Eq. (10) can
be rewritten as

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int DqD\bar{q}~{}e^{i\int
d^{3}x\left[\bar{q}\mathcal{O}q-\frac{\Delta}{2}q^{T}Cq-\frac{\Delta^{*}}{2}\bar{q}C\bar{q}^{T}\right]},$
(41)

where $\mathcal{O}=iM^{\mu\nu}\gamma_{\mu}\partial_{\nu}+\mu\gamma^{0}-\sigma$
and $C=i\gamma^{2}$ is the charge conjugation matrix. Using the Gaussian path
integral identities

$\displaystyle\int Dpe^{i\int
d^{3}x\Big{[}-\frac{1}{2}p^{T}Ap+\eta^{T}p\Big{]}}$ (42) $\displaystyle=$
$\displaystyle(\det A)^{\frac{1}{2}}e^{-\frac{i}{2}\int
d^{3}x\eta^{T}A^{-1}\eta}~{},$

and

$\displaystyle\int D\bar{p}e^{i\int
d^{3}x\Big{[}-\frac{1}{2}\bar{p}A\bar{p}^{T}+\eta\bar{p}^{T}\Big{]}}$ (43)
$\displaystyle=$ $\displaystyle(\det A)^{\frac{1}{2}}e^{-\frac{i}{2}\int
d^{3}x\bar{\eta}A^{-1}\bar{\eta}^{T}},$

and by also considering $A=\Delta C$, $\bar{q}\mathcal{O}=\eta^{T}$,
$\mathcal{O}^{T}\bar{q}^{T}=\eta$, one finds, after integrating over $q$ and
$\bar{q}$, the result

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int DqD\bar{q}~{}e^{i\int
d^{3}x\Bigg{[}\bar{q}\mathcal{O}q-\frac{\Delta}{2}q^{T}Cq-\frac{\Delta^{*}}{2}\bar{q}C\bar{q}^{T}\Bigg{]}}$
$\displaystyle=$ $\displaystyle(\det\Delta C)^{\frac{1}{2}}\int
D\bar{q}e^{\frac{i}{2}\int
d^{3}x\Bigg{\\{}\bar{q}\left[\Delta^{*}C+\mathcal{O}(\Delta
C)^{-1}\mathcal{O}^{T}\right]\bar{q}^{T}\Bigg{\\}}}$ $\displaystyle=$
$\displaystyle(\det\Delta
C)^{\frac{1}{2}}\left[\det(\Delta^{*}C+\mathcal{O}(\Delta
C)^{-1}\mathcal{O}^{T})\right]^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\left[\det\left(\Delta^{2}+\mathcal{O}C^{-1}\mathcal{O}^{T}C\right)\right]^{\frac{1}{2}},$

where we have assumed $\Delta=\Delta^{*}$ in the last step (we are not
interested in the phase of the superconducting order parameter, but solely on
its absolute (modulus) value). Using the relations
$C^{-1}\gamma_{\mu}^{T}C=-\gamma_{\mu}$ and
$\partial_{\mu}^{T}=-\partial_{\mu}$ one finds that

$I=[\det(-\Delta^{2}+\mathcal{O}_{+}\mathcal{O}_{-})]^{1/2}=(\det
B)^{\frac{1}{2}},$ (45)

with
$\mathcal{O}_{\pm}=iM^{\mu\nu}\gamma_{\mu}\partial_{\nu}\pm\mu\gamma^{0}-\sigma$.
Finally, using the identity $\det B=\exp({\rm Tr}\ln B)$ one finds

$\ln I=\frac{1}{2}{\rm tr}(\ln B)=\int
d^{3}x\sum_{i=1}^{2}\int\frac{d^{3}p}{(2\pi)^{3}}\ln\lambda_{i}(p),$ (46)

where

$\displaystyle\lambda_{1,2}$ $\displaystyle=$
$\displaystyle\sigma^{2}+\left[p_{0}-v_{F}({\bf t}\cdot{\bf
p})\right]^{2}-v_{F}^{2}{\bf\tilde{p}}^{2}-\mu^{2}-|\Delta|^{2}$
$\displaystyle\pm$ $\displaystyle
2\sqrt{\sigma^{2}\left\\{\left[p_{0}-v_{F}({\bf t}\cdot{\bf
p})\right]^{2}-v_{F}^{2}{\bf\tilde{p}}^{2}\right\\}+v_{F}^{2}\mu^{2}{\bf\tilde{p}}^{2}},$

are the eigenvalues of $B$.

##  Appendix B  The effective thermodynamic potential

In this section one shows some of the details for the derivation of the
effective thermodynamic potential. From Eq. (LABEL:veffsup), we obtain

$\displaystyle\Omega^{ren}(0,\Delta_{0},\mu)=\frac{\Delta_{0}^{2}}{2g_{2}v_{F}}+\frac{\Delta_{0}^{3}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}$ (48)
$\displaystyle-\int\frac{d^{2}p}{(2\pi)^{2}}\left\\{\frac{}{}|\mathcal{E}^{+}_{\Delta}|+|\mathcal{E}^{-}_{\Delta}|\right.$
$\displaystyle\left.-2\left[v_{F}({\bf t}\cdot{\bf
p})+\sqrt{v_{F}^{2}|{\bf\tilde{p}}|^{2}+\Delta_{0}^{2}}\right]\right\\},$

where $\mathcal{E}^{\pm}_{\Delta}=v_{F}({\bf t}\cdot{\bf
p})+\sqrt{(v_{F}|{\bf\tilde{p}}|\pm\mu)^{2}+\Delta_{0}^{2}}$. The effective
thermodynamic potential depends on momentum integrals of the form

$\displaystyle i_{\pm}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}p}{(2\pi)^{2}}\left\\{|\mathcal{E}^{\pm}_{\Delta}|-\left[v_{F}({\bf
t}\cdot{\bf
p})+\sqrt{v_{F}^{2}|{\bf\tilde{p}}|^{2}+\Delta_{0}^{2}}\right]\right\\}.$

Then, $\Omega^{ren}(0,\Delta_{0},\mu)$ can be written as

$\Omega^{ren}(0,\Delta_{0},\mu)=\frac{\Delta_{0}^{2}}{2g_{2}v_{F}}+\frac{\Delta_{0}^{3}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}-i_{+}-i_{-}.$ (50)

It can now be shown that for $\mu>0$, $\mathcal{E}^{+}_{\Delta}>0$ for all
$p>0$. Hence, after some algebraic steps, one finds

$\displaystyle i_{+}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\int_{0}^{\infty}dpp\left[\sqrt{(p+\mu)^{2}+\Delta_{0}^{2}}-\sqrt{p^{2}+\Delta_{0}^{2}}\right].$

For $\mathcal{E}^{-}_{\Delta}$, one has that $\mathcal{E}^{-}_{\Delta}>0$ only
for $p<p_{-}$ and for $p>p_{+}$, where

$p_{\pm}=\frac{1}{(1-|{\bf\tilde{t}}|^{2})}\Big{[}\mu\pm{\rm
Re}\sqrt{|{\bf\tilde{t}}|^{2}\mu^{2}-(1-|{\bf\tilde{t}}|^{2})\Delta_{0}^{2}}\Big{]},$
(52)

where ${\rm Re}$ means the real part. From the above expressions, then, it
follows that

$\displaystyle\int\frac{d^{2}p}{(2\pi)^{2}}|\mathcal{E}^{-}_{\Delta}|=\frac{1}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}$
$\displaystyle\times\int_{0}^{\infty}dpp\int_{0}^{2\pi}\frac{d\theta}{(2\pi)}\left[\left||{\bf\tilde{t}}|p\cos\theta+\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}\right|\right]$
$\displaystyle=\frac{1}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\int_{0}^{\infty}dpp\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}$
$\displaystyle\times\Theta\left[\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}-|{\bf\tilde{t}}|p\right],$
(53)

where we have used the identity
$\int_{0}^{2\pi}\frac{d\theta}{2\pi}|a\cos\theta+b|=b\Theta(b-a)$, for $a>0$
and $b>0$ in the last step and $\Theta(x)$ is the Heaviside function. Finally,
the inequality $\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}-|{\bf\tilde{t}}|p>0$ is
respected when $p<p_{-}$ and $p>p_{+}$. In particular, one notes that in the
limit $|{\bf\tilde{t}}|\rightarrow 0$, one finds $p_{+}=p_{-}=\mu$. Therefore,
using the fact that
$\int_{0}^{p_{-}}+\int_{p^{+}}^{\infty}=\int_{0}^{\infty}-\int_{p_{-}}^{p_{+}}$
it follows that

$\displaystyle i_{-}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\int_{0}^{\infty}dpp\left[\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}-\sqrt{p^{2}+\Delta_{0}^{2}}\right]$
(54) $\displaystyle-$ $\displaystyle\frac{1}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\int_{p_{-}}^{p_{+}}dpp\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\Bigg{(}\int_{0}^{\infty}dpp\left[\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}-\sqrt{p^{2}+\Delta_{0}^{2}}\right]$
$\displaystyle-$ $\displaystyle I(\Delta_{0},\mu)\Bigg{)},$

where

$\displaystyle I(\Delta_{0},\mu)$ $\displaystyle=$
$\displaystyle\int_{p_{-}}^{p_{+}}dpp\sqrt{(p-\mu)^{2}+\Delta_{0}^{2}}$ (55)
$\displaystyle=$ $\displaystyle\left(2\Delta_{0}^{2}-\mu^{2}+2p_{+}^{2}-\mu
p_{+}\right)\sqrt{\Delta_{0}^{2}+(p_{+}-\mu)^{2}}$ $\displaystyle+$
$\displaystyle
3\Delta_{0}^{2}\mu\tanh^{-1}\left[\frac{p_{+}-\mu}{\sqrt{\Delta_{0}^{2}+(p_{+}-\mu)^{2}}}\right]$
$\displaystyle-$ $\displaystyle\left(2\Delta_{0}^{2}-\mu^{2}+2p_{-}^{2}-\mu
p_{-}\right)\sqrt{\Delta_{0}^{2}+(p_{-}-\mu)^{2}}$ $\displaystyle-$
$\displaystyle
3\Delta_{0}^{2}\mu\tanh^{-1}\left[\frac{p_{-}-\mu}{\sqrt{\Delta_{0}^{2}+(p_{-}-\mu)^{2}}}\right].$

In normalized units $x=\Delta_{0}/\bar{\Delta}_{c}$, $y=\mu/\bar{\Delta}_{c}$,
$I(x,y)$ is given by

$\displaystyle I(x,y)$ $\displaystyle=$
$\displaystyle\left(2x^{2}-y^{2}+2z_{+}^{2}-yz_{+}\right)\sqrt{x^{2}+(z_{+}-y)^{2}}$
(56) $\displaystyle+$ $\displaystyle
3x^{2}y\tanh^{-1}\left[\frac{z_{+}-y}{\sqrt{x^{2}+(z_{+}-y)^{2}}}\right]$
$\displaystyle-$
$\displaystyle\left(2x^{2}-y^{2}+2z_{-}^{2}-yz_{-}\right)\sqrt{x^{2}+(z_{-}-y)^{2}}$
$\displaystyle-$ $\displaystyle
3x^{2}y\tanh^{-1}\left[\frac{z_{-}-y}{\sqrt{x^{2}+(z_{-}-y)^{2}}}\right],$

with $z_{\pm}$ defined as

$z_{\pm}=\frac{1}{(1-|{\bf\tilde{t}}|^{2})}\Big{[}y\pm
Re\sqrt{|{\bf\tilde{t}}|^{2}y^{2}-(1-|{\bf\tilde{t}}|^{2})x^{2}}\Big{]}.$ (57)

Finally, after integration over the momentum $p$, one can write the
renormalized effective thermodynamic potential for the superconducting phase
(when $\sigma_{0}=0$) as

$\displaystyle\Omega^{ren}(0,\Delta_{0},\mu)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{0}^{2}}{2g_{2}v_{F}}+\frac{(\mu^{2}+\Delta_{0}^{2})^{3/2}}{3\pi
v_{F}^{2}\xi_{x}\xi_{y}}$ $\displaystyle-$
$\displaystyle\frac{\mu^{2}\sqrt{\mu^{2}+\Delta_{0}^{2}}}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}$ $\displaystyle-$
$\displaystyle\frac{\mu\Delta_{0}^{2}}{2\pi
v_{F}^{2}\xi_{x}\xi_{y}}\ln\left(\frac{\mu+\sqrt{\mu^{2}+\Delta_{0}^{2}}}{\Delta_{0}}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2\pi\xi_{x}\xi_{y}v_{F}^{2}}I(\Delta_{0},\mu).$

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