Tilted Dirac cone effects and chiral symmetry breaking in a planar four-fermion model Y. M. P. Gomes1, ∗ and Rudnei O. Ramos1, y 1Departamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil We analyze the chiral symmetry breaking in a planar four-fermion model with non-null chemical potential, temperature and including the effect of the tilt of the Dirac cone. The system is modeled with a (2 + 1)-dimensional Gross-Neveu-like interaction model in the context of the generalized Weyl Hamiltonian and its phase structure is studied in the meanfield and large-N approximations. Possible applications of the results obtained, e.g., in connection to graphene, are discussed. We also discuss the effect of an external magnetic field applied to the system, which can give rise to the appearance of the anomalous Hall effect and that is expected to arise in connection with two-dimensional Weyl and Dirac semimetals. I. INTRODUCTION the properties of the first detected Weyl semimetal (for reviews, see, e.g., Refs. [29, 30]). Condensed matter systems, in particular those with We study in this work how this gapless property as- linearly dispersing fermionic excitations, have been serv- sociated with Weyl fermions would hold against possible ing as perfect platforms for testing predictions made by chiral symmetry breaking. Here, the usual Weyl Hamilto- quantum field theories. This interplay between high en- nian used in the description of Weyl semimetals [23, 24] ergy physics and condensed matter has been producing is extended by the introducing of a four-fermion inter- new insights about the quantum phenomena. This in- action. The properties of this system are then studied cludes since the parallel between the Ginzburg{Landau under the effects of both a finite chemical potential and model of superconductivity and the spontaneous symme- temperature. try breaking in the Higgs sector of the standard model Let us recall that four-fermion interacting models, par- of particle Physics to the relation between topological ticularly Gross-Neveu (GN) type of models [31] in (2+1)- insulators and axion electrodynamics, with applications, dimensions, are of particular interest. This is because e.g., in two-dimensional systems [1{7]. The physics of of their simplicity and ability to capture the relevant graphene [8], is of particular relevance in this context, physics of planar fermionic systems in general and, thus, where the electrons can be treated in a quasirelativistic these type of models have been extensively considered way through the Dirac equation in (2 + 1)-dimensions. in the literature. These models can have either a dis- Although the relativistic character of the electrons in crete chiral symmetry, ! γ5 , or a continuous one, graphene is intrinsically connected with the Lorentz sym- ! exp(iαγ5) . These type of models have been metry, this is not the case in most materials in condensed employed to study, e.g, low energy excitations of high- matter. In these material, the dispersion in the vicin- temperature superconductors [32], while analogous mod- ity of band touching points, though they can be generi- els with four-fermion interactions have also been used to cally linear and to resemble the Weyl equation, they lack study the quantum properties of graphene [33]. Taken Lorentz invariance. Despite Lorentz symmetry breaking as an effective low-energy description of the intrinsic in high energy physics may emerge in specific contexts [9{ physics of many relevant condensed matter systems of in- 15], they face strong constraints in general [16{18], De- terest, the four-fermion models of the GN type have then spite this, models exhibiting Lorentz symmetry break- served as motivation for many previous studies (see, e.g., ing have been recently applied in the condensed matter Refs. [34{39] for examples of application). In particular, area, e.g., as a path to model three-dimensional Weyl these models can make a role in understanding possible semimetals [19{22], quasi-planar organic materials and dynamical gap generation in the planar condensed mat- hetero-structures made of a combination of topological ter systems, e.g., in graphene [33, 40, 41]. and normal insulators [23{26]. The aim of this paper is then to study a Weyl-like These new systems are described by Weyl-like Hamil- Hamiltonian, augmented by a four-fermion interaction tonians and, therefore, these systems have quasiparticles of the GN type. We study the interplay between the that behave like Weyl fermions [27]. These quasiparti- arXiv:2106.09239v1 [cond-mat.mes-hall] 17 Jun 2021 characteristic anisotropy and tilting of the Dirac cone, as cles are by construction massless and are more stable typically considered in these models, with both tempera- against gap formation in comparison with Dirac ones. ture and chemical potential (i.e., the effects of doping) in After several theoretical predictions, the first experimen- the phase structure of the resulting system. This study tal measure of this kind of material [28] shed light on is performed in the context of the effective potential in the meanfield and large-N approximation. It is expected that this study will be of interest for learning some of the analogous effects that might be important when consid- ∗ [email protected] ering fermionic quasiparticles and excitonic bound states y [email protected] in planar fermionic systems of relevance. 2 The remainder of this paper is organized as follows. In Sec. II, we show the relevant properties of two- dimensional Dirac and Weyl semimetal systems. In Sec. III, we extend the model by considering a four- fermion interaction and analyze the effective potential at the meanfield and large-N level when taking into account the effects of both the anisotropy, tilting of the Dirac cone, chemical potential and temperature. In Sec. IV, we discuss some expected features for this Weyl-type model when an external magnetic field is applied to the system. Finally, in Sec. V, we give some additional discussions concerning our results and also our concluding remarks. FIG. 1. Illustration of a generic Dirac cone for an isotropic Throughout this work, we will be considering the natural Dirac semimetal (a), type I Weyl semimetal (b) and type II units where ~ = kB = c = 1. Weyl semimetal (c). II. TWO-DIMENSIONAL WEYL SEMIMETALS where λ = ±1 defines the conduction and valence bands, respectively. In the untilted case tx = ty = 0 and The generalized Hamiltonian which describes the elec- isotropic one, ξx = ξy = 1, we recover the spectrum of trons in the two-dimensional Weyl semimetal (2D WSM) isotropic graphene. The Hamiltonian Eq. (2.1) commutes is given by [23, 24], with the chirality operator given by 0 x y H (p) = v (t · p) τ + (ξ p ) τ + (ξ p ) τ ; (2.1) (ξxpx)τx + (ξypy)τy t F x x y y C = ; (2.4) p 2 2 0 (ξxpx) + (ξypy) where vF is the Fermi velocity, τ = I the 2 × 2 identity matrix, τ x and τ y are the Pauli matrices, t is the vec- and the eigenvalues of C are given by α = ±1. Due to 0 tor defining the tilt of the Dirac cone and ξ = (ξx; ξy) the two-fold degeneracy of the Dirac cones D and D , is the vector that describes the anisotropic character of with representative index ρ = ±1, the band index can be the crystalline structure. The tilt vector t is related properly identified as λ = ρα. Taking these degeneracies to the separation between the Dirac cones in the Weyl into account, we can write a four-component Weyl spinor semimetal. A consequence of the non-null tilt term in and a Dirac-like Lagrangian density can be written as 0 Eq. (2.1) is that the Dirac points, denoted by D and D , " # no longer coincide with the Brillouin corners K and K0 X L = i ¯ (@ − v t · @) γ0 − v (ξ @ )γi (see, e.g., Ref. [24] for details and a review). In particu- 0 t F F i i lar, type-I Weyl semimetals are characterized by jtj < 1, i ¯ µν while type-II ones by jtj > 1 (see Fig. 1 for an illustra- = i M γµ@ν ; (2.5) tion). Although jtj is not restricted, in order to maintain where the physical meaning of the Hamiltonian Eq. (2.1), it is 1 0 imposed that γµ = τ µ ⊗ ; (2.6) 0 −1 s 2 2 tx ty + = j~tj < 1; (2.2) τ µ = (τ ; iτ ; iτ ) and ¯ = yγ0. The γ-matrices re- ξ ξ 3 x y x y spect the algebra fγµ; γν g = 2ηµν (for details of this rep- where j~tj is called the effective tilt parameter. When the resentation for fermions in (2 + 1)-dimensions see, e.g., condition given by Eq. (2.2) is not respected, the isoen- Ref. [42]). We also have that ergetic lines are no longer ellipses, but instead they are 0 1 1 −vF tx −vF ty hyperboles [23, 24]. The value of j~tj depends on the ma- µν M = @0 −vF ξx 0 A : (2.7) terial and, e.g., in quinoid-type deformed graphene [23], 0 0 −v ξ it is of order of 0:6, with the relative strain parameter F y < 0:1 for moderate deformations. The degree of free- One should note that M µν contains the parameters that dom described by the Hamiltonian Eq. (2.1) is that of a explicitly break the Lorentz symmetry. Note also that we (massless) Weyl fermion. do not introduce the two-fold spin-degeneracy here and, in case of interest, we have to double the spinors ! s for s = ±1. It is easy to show that the Lagrangian The Hamiltonian spectrum obtained from Eq.