Tilted Dirac superconductor at quantum criticality: Restoration of Lorentz symmetry

: Lorentz symmetry appears as a quite robust feature of the strongly interacting Dirac materials even though the lattice interactions break such a symmetry. We here demonstrate that the Lorentz symmetry is restored at the quantum-critical point (QCP) separating the tilted Dirac semimetal, breaking this symmetry already at the noninteracting level, from a gapped s − wave superconducting instability. To this end, we employ a one-loop ϵ = (3 − D ) − expansion close to the D = 3 upper critical dimension of the corresponding Gross-Neveu-Yukawa field theory. In particular, we show that the tilt parameter is marginally irrelevant and ultimately vanishes at the QCP separating the two phases. In fact, as we argue here, such a Lorentz symmetry restoration may be generic for the strongly interacting tilted Dirac semimetals, irrespective of whether they feature mirror-symmetric or mirror-asymmetric tilting, and is also insensitive to whether the instability represents an insulator or a gapped superconductor. The proposed scenario can be tested in the quantum Monte Carlo simulations of the interacting tilted Dirac fermion lattice models.


Introduction
The Dirac crystals featuring Dirac quasiparticles as their low-energy excitations exhibiting a linear dispersion opened up a new paradigm in the modern condensed matter physics [1][2][3].They also bridge the gap between the seemingly unrelated phenomena in the realms of high-energy and condensed-matter physics through the emergence of the Dirac and Weyl quasiparticles, being the prime examples in this respect.Such Dirac materials feature an emergent relativistic-like Lorentz symmetry because of the Dirac quasiparticles' linear energy-momentum dispersion, E(k) = v F k, with the Fermi velocity, v F , typically a few hundred times smaller than the speed of light, playing the role of an effective velocity of light in their crystalline universe.Such a relativistic-like Lorentz symmetry turns out to be quite a robust feature of the Dirac materials even when the electron-electron interactions that explicitly break it are taken into account.For instance, the Lorentz symmetry emerges in the deep infrared regime for the two-dimensional (2D) Dirac fermions coupled via the long-range Coulomb interaction, where the speed of light takes over as the common velocity for both the Dirac fermions and the photons [4].Furthermore, for Yukawa (short-range) interacting Dirac fermions, the quantum-critical point (QCP) separating a semimetallic and a strongly-coupled gapped (insulating or superconducting) phase is Lorentz symmetric [5][6][7][8][9].This issue has been also recently addressed beyond the realm of the usual Hermitian Dirac materials, by taking into account the effects of the non-Hermiticity, that may arise, e.g. from the coupling with an environment, resulting in the Lorentz symmetry emerging at low energies [10,11].These results point towards a ubiquitous emergence of the Lorentz symmetry at low energy in the Dirac materials, even though such a symmetry is absent at the lattice (UV) energy scale.
To obtain further insights into the ubiquity of such an emergent Lorentz symmetry, we here consider tilted Dirac materials, featuring low-energy quasiparticles with tilted Dirac dispersion, explicitly breaking this symmetry at the lattice scale already at the noninteracting level.Such an explicit symmetry breaking is realized by the direction of the tilt term in the dispersion of the tilted Dirac fermions (TDFs), which is dictated by the underlying crystalline lattice, and is represented by a single term ∼ α, with α as the tilt parameter, in a Dirac Hamiltonian, as shown in Eqs.(2.3) and (2.4).Introduction of such a tilt term may be considered as possibly the simplest Lorentz-symmetry-breaking deformation of an otherwise Lorentz-symmetric noninteracting Dirac Hamiltonian.Its effects have been studied in various contexts, for instance, it was argued that TDFs may represent a platform for the simulation of the curved space-time in condensed-matter systems [12][13][14][15][16][17][18][19][20][21], while the quantum transport [22][23][24][25][26][27][28], the effects of the long-and short-range components of the Coulomb interaction [29][30][31][32], charged impurities [33][34][35][36], and the magnetic field [37][38][39] have also been addressed.We emphasize that the tilted Dirac dispersion has been theoretically investigated in the materials using first-principle techniques [40][41][42].In fact, the TDFs have been realized in a plethora of quantum materials [43][44][45][46][47][48][49][50], both with the subcritical tilt (type-1), featuring the point-like Fermi surfaces, and in overtilted materials (type-2), with the electron-and hole-like Fermi surfaces, putting the tilted Dirac crystals within the landscape of the quantum materials.
In this article, we explore the effects of the s−wave superconducting instability in the type-1 tilted Dirac materials.Our main result is that at the strong-coupling QCP separating the tilted Dirac semimetal and the s−wave SC phase, the tilt operator becomes irrelevant and therefore the Lorentz symmetry gets restored.To address this problem, we employ the Gross-Neveu-Yukawa (GNY) theory [51][52][53][54][55][56][57][58], since this instability necessarily occurs at strong coupling, as can be seen already at the mean-field level [Eq.(3.5)], due to a vanishing density of states (DOS) at zero energy [Eq.(3.6)].To account for the effects of both fermionic and order-parameter (OP) fluctuations in the vicinity of such a QCP, since both the Yukawa and Φ 4 −couplings [Eq.(5.1)] are marginal in three spatial dimensions (D = 3), we use the renormalization group (RG) analysis within the ϵ = (3−D)-expansion.We demonstrate that the tilt parameter in the mirror-symmetric [Eq.(2.3)] and mirrorsymmetry-breaking [Eq.(2.4)] tilted Dirac semimetals are irrelevant at the superconducting QCP, with the corresponding RG β−function given by Eq. (5.9) [see also Fig. 1], and the flow shown in Fig. 2, therefore implying the restoration of the Lorentz symmetry at such a QCP.To further corroborate the ubiquity of this scenario, we show that the transition into a CDW phase also features the emergent Lorentz symmetry at low energy, albeit with a slower rate of the restoration, as shown in Eq. (5.18), see also Fig. 3 and Fig. 1.We argue that such an emergence of the Lorentz symmetry is generic for the TDFs interacting via a short-range Coulomb interaction, with the rate that depends on whether the OP commutes or anticommutes with the tilt term.Finally, the anticommutation makes the rate of the restoration faster, as can be seen explicitly in the case of the SC and CDW instabilities.
The paper is organized as follows.In Sec. 2, we introduce the low-energy theory for the 2D TDFs, while in Sec. 3, we analyze the short-range interactions at the mean-field level.We then proceed with the setup of the quantum-critical GNY field theory in Sec. 4, and in Sec. 5 we perform its RG analysis.Our conclusions are presented in Sec.6, and additional technical details are relegated to the Appendix A.
2 Tilted Dirac superconductor: The low-energy theory We consider a low-energy effective theory for the Dirac fermion quasiparticles around the two inequivalent valleys, as for instance, two Dirac points at the two inequivalent corners of the Brillouin zone in graphene.To construct the Nambu-doubled basis for the Dirac superconductor, we first write an eight-component Dirac spinor given by [59] Ψ ω,k = c where a v s and b v s are the fermionic fields corresponding to two different (A and B) sublattices with valley index v = (±), and spin projection s =↑, ↓.Then, we introduce a particle-hole degree of freedom and write a doubled, sixteen-component Dirac-Nambu spinor as We conveniently write matrices acting on the sixteen-dimensional Dirac-Nambu spinors as Here, {η}, {σ}, {τ }, and {α} are the sets of Pauli matrices operating on particle-hole, spin, valley and sublattice degrees of freedom, respectively, with µ, ν, λ, ρ = 0, 1, 2, 3, and the index 0 corresponding to the 2 × 2 unity matrix.Notice that the lower component of the Dirac-Nambu spinor in Eq. (2.2) is a time-reversal partner of the upper one, since the matrix Γ 210 = σ 2 ⊗ τ 1 ⊗ α 0 is the unitary part of the time reversal operator in the undoubled basis.Furthermore, the form of the Γ−matrices dictates that any two of them either commute or anticommute.To simplify notation, we suppress subscript "Nam" hereafter, Ψ ≡ Ψ Nam and Ψ † ≡ Ψ † Nam .To make a more direct connection to the relativistic Dirac field theory, we notice that in this way defined 16(= 2 × 8)-component Euclidean Dirac fermions in one imaginary-time and two spatial dimensions correspond to 8 copies of two-component spin-1/2 relativistic Dirac fermions living in the (2 + 1)−dimensional Minkowski space, with the sublattice degrees of freedom directly translating into the (real) spin degrees of freedom of the relativistic fermions [? ], therefore justifying the label "pseudo-spin" for the sublattice index.
Given the Dirac-Nambu representation in Eq. ( 2.2), we can write the noninteracting low energy effective Hamiltonian for the tilted Dirac fermions in the form Ĥ = k Ψ † ( Ĥ(k) − µ tot Γ 3000 )Ψ, with µ tot as the chemical potential, and where v is the Fermi velocity and α represents the (dimensionless) tilt parameter.We here consider only case with |α| < 1 (type-1) featuring the point-like Fermi surfaces with the TDF quasiparticle excitations, while the overtilted (type-2) system with |α| > 1 hosts a particle-and a hole-like Fermi surfaces.Notice that the tilt term can be thought of as a momentum-dependent chiral chemical potential, and it commutes with the (total) chemical potential represented by the matrix Γ 3000 .We hereafter set v = 1 and consider the system at half filling, µ tot = 0, unless otherwise stated.The form of the single-particle Hamiltonian implies its invariance under time reversal, generated by T = Γ 0210 K, with K as the complex conjugation, SU(2) spin rotations generated by Γ 0s00 , with s = {1, 2, 3}, as well as the valley (sublattice) exchange, represented by [60].Finally, the unitary particle-hole symmetry is represented by C = Γ 1000 , so that {C, Ĥ(k)} = 0, implying that its anti-unitary counterpart is represented by the operator CT .Notice that C = Γ 2000 can also be taken as a unitary particle-hole operator, which is a consequence of the U(1) symmetry of the noninteracting Hamiltonian generated by the matrix Γ 3000 representing the electric charge operator in the Dirac-Nambu representation employed here.Finally, we emphasize that the choice of the tilt term (∼ α) in Eq. (2.3) ensures its invariance under all the mentioned symmetries.
We point out that the mirror-symmetry breaking tilted Dirac Hamiltonian, with the sign of the tilt equal for all the valleys (or other fermion flavors) is obtained by breaking the valley exchange or mirror symmetry discussed above, and we refer to it as asymmetric tilt.The corresponding Hamiltonian then takes the form with the tilt term proportional to the matrix representing the chemical potential.Therefore, the asymmetric tilt term effectively represents a momentum-dependent chemical potential.As we show in the following, our conclusions regarding the restoration of the Lorentz symmetry in the case of the interacting fully symmetric tilted Dirac Hamiltonian [Eq.(2. 3)] equally apply to the asymmetrically tilted Hamiltonian in Eq. (2.4), as follows from the form of the corresponding propagator, Eq. (4.8).

Mean-field analysis
To set the stage for the analysis of the strong-coupling superconducting instability of the tilted Dirac semimetal, we first discuss mean-field (MF) theory for a transition from the semimetallic to a fully gapped ordered state, which is independent of the nature of the state, i.e. whether the state is an insulator or a superconductor.For concreteness, we here subscribe to the s−wave superconducting order parameter, which reads where φ is the U(1) phase of the superconducting OP.This OP anticommutes with the particle number operator N = Γ 3000 , and commutes with all the three generators of spin rotations, Γ 0s00 , as dictated by its s-wave nature.Notice that both matrices in the swave OP anticommute with the tilted Dirac Hamiltonian in Eq. ( 2.3) which ensures that the underlying ordered state is fully gapped.In the MF analysis, we set the OP to be a constant, ϕ(r, τ ) ≡ m, and fix the SC phase φ = 0. We assume that such a condensation is propelled by an effective local density-density interaction of the form Then, we formulate the MF theory for the transition into such a state by taking the Hamiltonian with the first term representing the free tilted Dirac Hamiltonian in Eq. (2.3), together with the short-range interaction part, given by Eq. (3.2).We then perform a Hubbard-Stratonovich transformation in the s−wave channel to obtain the corresponding energy of the ground state at half filling (µ = 0) in the form, Therefore, the MF ground state energy is tilt-independent, yielding the same form of the gap equation as in the untilted case, after minimization with respect to m, and implies that m ∼ (g − gc ), (3.5) with the tilt-indenpendent critical coupling gc ∼ Λ scaling with the ultraviolet cutoff, defining the energy scale up to which the Dirac dispersion holds.Therefore, the transition into the s−wave (and, more generally, any gapped) phase takes place at a strong coupling, which is a consequence of the vanishing density of states (DOS) for the tilted Dirac fermions for a single 16-component Dirac flavor.It is therefore expected that at any finite-tilt fixed point of the RG, the critical value of the interaction decreases with respect to the untilted (α = 0) value.Indeed this is the case for unstable fixed points at finite tilt corresponding to the s−wave and the CDW OPs, which can be obtained from Eq. (5.10) and (5.19), respectively, which are, however, rendered unstable, as the tilt is irrelevant for both the s−wave and the CDW OPs, see Eqs. (5.9) and (5.18).
4 Gross-Neveu-Yukawa quantum-critical field theory We now address the behavior of the (2 + 1)−dimensional tilted s−wave Dirac superconductor at the quantum-critical point within the framework of the Euclidean (imaginarytime) Gross-Neveu-Yukawa (GNY) field theory.

S−wave superconducting instability
First, since both noninteracting Hamiltonian and the OP are spin singlets, we discard the spin subspace hereafter.We therefore consider eight-component massless Dirac fermions endowed with N F copies, which interact with a complex bosonic order parameter field via Yukawa interaction (g).The s−wave superconducting OP is in the form given by Eq. (3.5), which after discarding the spin subspace, reads ϕ(r, τ ) = Ψ † (r, τ )(Γ 100 cos φ + iΓ 200 sin φ)Ψ(r, τ ) .(4.1) Hereafter, Ψ ≡ Ψ(r, τ ), Ψ † ≡ Ψ † (r, τ ) and ϕ ≡ ϕ(r, τ ).The dynamics of the system is then described by an imaginary time (Euclidean) action with the noninteracting fermionic part of the form where τ is the imaginary time and Ĥ(k) represents the Hamiltonian for noninteracting massless TDFs in Eq. (2.3).The coupling of the massless Dirac fermions to the OP fluctuations takes the characteristic Yukawa form, Lastly, the dynamics of the OP is described by the standard Ginzburg-Landau action where m 2 B is the tuning parameter for the continuous phase transition with m 2 B > 0 (m 2 B < 0) in the U (1) symmetric (symmetry broken) phase.We furthermore set the bosonic and fermionic velocities to be equal to unity in the critical region [7], see also Sec.A.1.3.
The fermionic and the bosonic propagators read, respectively, and where x .We point out that the form of the propagator corresponding to the asymmetrically tilted system, described by the Hamiltonian in Eq (2.4), preserves the form in Eq. (4.6), since the asymmetric tilt commutes with the untilted piece of the Dirac Hamiltonian.Explicitly, the propagator of the asymmetric TDFs reads and such a structure of the propagator, as we show in the following, implies that all the conclusions derived for the symmetric tilt equally apply to the asymmetric case.We therefore consider only the fully symmetric TDFs.

Renormalization group analysis
Next, we proceed with the analysis of the U(1) GNY field theory.The engineering (bare) dimensions of the couplings are dim[g 2 ] = dim[λ] = 3 − D and for the tilt parameter dim[α] = 0, implying that D = 3 is the upper critical spatial dimension of the theory.We therefore employ an ϵ-expansion about the upper critical dimension, ϵ = 3 − D, to access the quantum-critical behavior in D = 2.To this end, we define the Euclidean action S R = dτ drL R , where the renormalized Lagrangian is where Z j 's are renormalization constants, and we restore the Fermi velocity for completeness.
To make a connection with relativistic Dirac field theories more explicit, we here rewrite the above Lagrangian in the Minkowski spacetime by performing a Wick rotation from the imaginary to real time (τ → −it), set Z v = 1, anticipating this result (see App. A.1.3),and fix both fermionic and bosonic velocities to unity, which then takes a form Here, we defined Ψ = Ψ † Γ 0 , with Γ 0 = Γ 200 , while Γ 1 = −iΓ 131 , Γ 2 = −iΓ 102 , Γ 5 = Γ 300 , and the matrix Γ 3 = −iΓ 133 can be used to extend the Dirac kinetic term to the (3 + 1)−dimensional Minkowski space.This is so because these five Γ matrices, acting on the (complex) Dirac fermions, form a Clifford algebra of mutually anticommuting matrices.Furthermore, the matrix Γ 5 can be taken as the generator of chiral symmetry, which is inherited from the particle-hole symmetry in the Euclidean theory, while the valley degrees of freedom become trivial in this representation (all the five Γ−matrices are diagonal in the valley index).Finally, the Lagrangian of the tilt term, Ltilt , which can be considered as a minimal Lorentz-symmetry breaking deformation of the above fully relativistic Dirac field theory, reads Ltilt = iZ α α ΨΓ tilt ∂ x Ψ, ( with the matrix Γ tilt = −iΓ 130 .

β−functions for a tilted Dirac s−wave superconductor
To obtain RG flow of the couplings, we integrate out the fermionic and bosonic modes with frequency −∞ < ω < ∞ and use dimensional regularization in D = 3−ϵ spatial dimensions within the minimal subtraction scheme to handle the ensuing divergences.The one-loop computation of the self-energy diagrams for the fermions (App.A.1) and for the bosonic order parameter (App.A.2) then yields the fermion field renormalization (App.A.1.1), the boson field renormalization (A.2), and the renormalization constant for the tilt parameter (App.A.1.2),which, respectively, read while the renormalization constant for the velocity Z v = 1, as shown in App.A.1.3.Here, the functions are defined as The computation of the vertex diagrams to the same order gives the renormalization conditions for the Yukawa coupling constant (App.A.3) and for for the Φ 4 coupling (App.A.4) Here, the couplings with (without) the subscript are the bare (renormalized) ones.Furthermore, µ is the RG (momentum) scale, and we consider N B complex components for the bosonic OP.
Using the above renormalization conditions, we obtain the RG β functions to the leading order in the ϵ-expansion, in the critical hyperplane m 2 B = 0, see Eq. (5.1).We define the β function for a coupling constant X as β X ≡ −dX/d ln (µ)| {X 0 } , with all the bare couplings ({X 0 }) fixed.Since the renormalization constant for the velocity Z v = 1, the renormalization condition Z v v = v 0 implies that the Fermi velocity is marginal, β v = 0. Furthermore, the renormalization condition for the tilt term, Z α α = α 0 implies that with the function K(α) as strictly positive for |α| < 1, as shown in Fig. 1.Therefore, the tilt parameter is irrelevant, and flows to zero as the system approaches the critical point governing the transition into the s-wave superconducting state.Ultimately, the Lorentz symmetry is restored at the QCP and the quantum-criticality is controlled by the GNY QCP for α = 0. To confirm this scenario, using the renormalization condition in Eq. (5.7), we furthermore find the β−function for the Yukawa coupling where we rescaled g 2 /8π 2 → g 2 .The RG flow in the (g 2 , α) plane is shown in Fig. 2, which indeed corroborates the irrelevance of the tilt parameter, and the ensuing restoration of the Lorentz symmetry at the nontrivial QCP with the value given by Eq. (5.10), For completeness, we also show the RG flow equation for the Φ 4 −coupling for the tilted Dirac superconductor, which is obtained from the renormalization condition in Eq. (5.8), after rescaling X/8π 2 → X, with X = {g 2 , λ}.Notice that the β functions in Eqs.(5.10) and (5.9) for the untilted Dirac fermion (α = 0) coincide with the ones obtained in Ref. [54].The corresponding fixed points of the RG flow are obtained from the β−functions for the Yukawa and the λ couplings after taking α = 0, as implied by the irrelevance of the tilt term [Eq.(5.9)].Only one of them is stable in the critical plane (m 2 B = 0), and therefore represents the critical point, which is located at [54] where X = 1−2N F .Notice that λ ⋆ is positive, and therefore corresponds to a second-order phase transition, only for N F > 1/2, or equivalently, in the Dirac system featuring at least one eight-component Dirac fermion flavor.
We emphasize here that identical renormalization factors and vertex corrections are obtained for the asymmetric TDFs, since the tilt matrix in this case anticommutes with the SC OP and commutes with the untilted part of the Dirac Hamiltonian, which is the same as for the symmetric TDFs, while all the form factors in the propagators for the asymmetric [Eq.(4.8)] and symmetric [Eq.(4.6)] TDFs are identical.

Comparison with charge-density-wave instability
We here use the opportunity to revise the problem of a charge-density wave (CDW) instability breaking the sublattice symmetry of a tilted Dirac semimetal [30], in light of the restoration of the Lorentz symmetry at the superconduting QCP.The CDW OP takes the form, implying that the Yukawa action reads with the OP matrix anticommuting with the untilted part of the Dirac Hamiltonian in Eq. (2.3), and therefore gaps it out.We are interested here only in the RG flow of the tilt parameter and the Yukawa coupling, and therefore consider the fermionic self-energy, which reads which is obtained directly from the self-energy for the SC OP, Eq. (A.1), taking into account that the CDW OP commutes with the tilt matrix, Γ 330 .The functions A ± and B are defined in Eq. (2.3).This form of the fermionic self-energy, in turn, yields the renormalization constant for the field (Z Ψ ) in the form given by Eq. (5.4), while the factor Z α reads with the function G(α) [I(α)] defined in Eq. (5.6) [Eq.(A.14)].The corresponding RG flow equation for the tilt is then straightforwardly obtained from the renormalization condition Z α α = α 0 , yielding with the strictly positive function K(α) shown in Fig. 1.Therefore, the Lorentz symmetry gets restored for the CDW instability, but much slower than in the SC case, since K(α) < K(α), for |α| < 1, as can be seen in Fig. 1.Furthermore, the flow of the Yukawa coupling in this case is given by with the stable QCP realized at The corresponding RG flow in the (g 2 , α) plane is displayed in Fig. 3, which corroborates the restoration of the Lorentz symmetry at the QCP governing the transition into the CDW phase, and making the line of critical points found in Ref. [30] eventually unstable.
We point out that the sign in front of the strictly negative function I(α) (for |α| < 1) in Eq. (5.18), is a direct consequence of the commutation of the tilt matrix (Γ 330 ) and the CDW OP Γ 303 , which is diagonal in the Nambu space.This overall positive contribution is responsible for slowing down the RG flow of the tilt with respect to the SC case, implying that the restoration of the Lorentz symmetry takes longer RG time.
In general, we can observe that the rate of the RG flow for the tilt parameter, and therefore the rate of the Lorentz symmetry restoration is directly related to the (anti)commutation relation of the OP with the tilt matrix.Namely, since all the Γ−matrices square to unity and either commute or anticommute with each other, an OP that commutes with the tilt matrix is expected to make the tilt parameter less irrelevant than an anticommuting OP.In particular, an insulating (particle-hole) instability diagonal in other flavor subspaces (valley, sublattice, etc.) should render the tilt less irrelevant than a superconducting (particleparticle) instability, as the examples of the CDW and the s−wave instabilities explicitly demonstrate.Finally, the form of the propagator of the asymmetric TDFs implies that the β−functions at CDW QCP for the asymmetric TDFs are of the same form as for the symmetric TDFs.

Conclusions and outlook
In this paper, we have shown that the Lorentz symmetry is restored at the strongly-coupled QCP separating the tilt Dirac semimetal from a gapped s−wave SC instability through the irrelevance of the tilt operator and its ultimate vanishing at the QCP.As we further argued such a symmetry restoration is a quite generic feature for the strongly interacting TDFs, irrespective of whether they break a discrete mirror lattice symmetry at the noninteracting level, i.e. whether they feature symmetric [Eq.(2.3)] or asymmetric [Eq.(2.4)] tilt.
Our results can be routinely tested in the lattice quantum Monte Carlo simulations [61] of the Hubbard interacting TDFs.In light of this, we emphasize that even though the tilt parameter is irrelevant, and ultimately vanishes at the QCP, it may take long RG time to reach this value.Therefore in any finite-size system the effects of a finite tilt may still be appreciable, as given by the solution of the flow equations (5.9), (5.10) and (5.12).See also Fig. 2.
Directly pertinent to the SC instability of the TDFs is whether the coupling to a dynamical U (1) gauge field can lead to the breaking of the Lorentz symmetry, which is an issue that we plan to investigate in future.In particular, it would be interesting to see whether such a coupling could help stabilization of a charged QCP at a finite coupling.
Our work should motivate further studies of the interaction effects in the non-Hermitian tilted Dirac materials with the coupling to the environment explicitly taken into account through non-Hermitian terms in the Hamiltonian-like operator.In particular, it will be important to address whether the restoration of the Lorentz symmetry is also featured in the non-Hermitian tilted Dirac semimetals, similarly to the case of untilted non-Hermitian Dirac fermions [10].
Finally, we would like to point out that these results may be relevant for future studies of the strong coupling instabilities of type-2 Dirac semimetals with electron-and hole Fermi surfaces, and should spur further research efforts in this direction.It is convenient to rewrite the coupling of the bosonic order parameter to the fermionic field explicitly in terms of the real and imaginary parts of the OP, where the matrices P ± are defined as and we employ eight-dimensional representation of the Γ−matrices.
To address the quantum-critical behavior in D = 2, we employ ϵ = 3−D, the deviation from the upper critical D = 3 spatial dimensions, as an expansion parameter.

Figure 1 :
Figure 1:The form of the functions K(α) and K(α) governing the renormalizationgroup flow of the tilt parameter at the transition from the tilted Dirac semimetal to the s−wave superconducting (SC) and the charge-density wave (CDW) phases, respectively.The corresponding β function is given by Eq. (5.9) [Eq.(5.18)] for the s−wave SC [CDW] instability.

Figure 2 :
Figure 2: One-loop RG flow of the tilt parameter α and the Yukawa coupling g 2 at fixed ϵ = 1 in the case of the s−wave superconducting instability.The red dot corresponds to the Gross-Neveu-Yukawa superconducting quantum-critical point (QCP), for different number of eight-component Dirac flavors (N F ). RG flow is shown for (a) N F = 1; (b) N F = 2; (c) N F = 3.The analytical form of the Lorentz-symmetric QCP (α = 0) is given by Eq. (5.13).

Figure 3 :
Figure 3: One-loop RG flow of the tilt parameter α and the Yukawa coupling g 2 at fixed ϵ = 1 in the case of the charge-density-wave (CDW) instability.The red dot corresponds to the Gross-Neveu-Yukawa CDW quantum-critical point (QCP), for different number of eight-component Dirac flavors (N F ). RG flow is shown for (a) N F = 1; (b) N F = 2; (c) N F = 3.The analytical form of the Lorentz-symmetric QCP (α = 0) is given by Eq. (5.20).

Figure 4 :
Figure 4: Feynman diagrams to the leading order in the ϵ expansion: (a) fermionic selfenergy, (b) bosonic self-energy, (c) correction to the Yukawa vertex arising from fermion excitations, (d) correction to the λ vertex from the boson self-interaction, and (e) correction to the λ vertex from the Yukawa interaction.The solid line represents the fermionic propagator, while the dashed line stands for the bosonic propagator.