Renormalization of the band gap in 2D materials through the competition between electromagnetic and four-fermion interactions
Luis Fernández, Van Sérgio Alves, Leandro O. Nascimento, Francisco Peña, M. Gomes, E. C. Marino
aa r X i v : . [ h e p - t h ] M a r Renormalization of the band gap in 2D materials through the competition betweenelectromagnetic and four-fermion interactions
Luis Fern´andez, ∗ Van S´ergio Alves, † Leandro O. Nascimento, ‡ Francisco Pe˜na, § M. Gomes, ¶ and E. C. Marino ∗∗ Faculdade de F´ısica, Universidade Federal do Par´a, 66075-110, Bel´em, PA, Brazil Faculdade de Ciˆencias Naturais, Universidade Federal do Par´a, C.P. 68800-000, Breves, PA, Brazil Departamento de Ciencias F´ısicas, Facultad de Ingenier´ıa y Ciencias,Universidad de La Frontera, Avda. Francisco Salazar 01145, Casilla 54-D, Temuco, Chile Instituto de F´ısica, Universidade de S˜ao Paulo Caixa Postal 66318, 05315-970, S˜ao Paulo, SP, Brazil Instituto de F´ısica, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil (Dated: March 4, 2020)Recently the renormalization of the band gap m , in both WSe and MoS , has been experimen-tally measured as a function of the carrier concentration n . The main result establishes a decreasingof hundreds of meV, in comparison with the bare band gap, as the carrier concentration increases.These materials are known as transition metal dichalcogenides and their low-energy excitations are,approximately, described by the massive Dirac equation. Using Pseudo Quantum Electrodynamics(PQED) to describe the electromagnetic interaction between these quasiparticles and from renormal-ization group analysis, we obtain that the renormalized mass describes the band gap renormalizationwith a function given by m ( n ) /m = ( n/n ) C λ / , where m = m ( n ) and C λ is a function of thecoupling constant λ . We compare our theoretical results with the experimental findings for WSe and MoS , and we conclude that our approach is in agreement with these experimental results forreasonable values of λ . In addition we introduced a Gross-Neveu (GN) interaction which couldsimulate an disorder/impurity-like microscopic interaction. In this case, we show that there existsa critical coupling constant, namely, λ c ≈ ,
66 in which the beta function of the mass vanishes,providing a stable fixed point in the ultraviolet limit. For λ > λ c , the renormalized mass decreaseswhile for λ < λ c it increases with the carrier concentration. PACS numbers: 11.10.Hi, 11.15.-q, 11.15.Pg, 71.10.Pm
I. INTRODUCTION
The interest in two-dimensional materials has been in-creased due to several new applications, in particular,the control of charge, spin, and valley of electrons in thehoneycomb lattice. The understanding of the materialproperties and its fundamental interactions have beendiscussed in several experimental and theoretical stud-ies, aiming to applications and development of electronicdevices with these materials, in particular, for graphene[1], silicene [2], and transition metal dichalcogenides [3].Although a full description of the material propertieswould require the inclusion of several microscopic inter-actions, such as the lattice and impurities, it is possible tofocus on a low-energy description of the electrons close tothe Dirac points, also called the valleys of the honeycomblattice. In this case, a quantum-electrodynamical ap-proach is derived which is expected to describe electronicproperties at low temperatures [4–6]. Within this regime,electrons obey a Dirac-like equation with two main pa- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected] br rameters, namely, the Fermi velocity v F and the mass(band gap) m . In graphene, for instance, the energy gapvanishes and the Fermi velocity reads v F ≈ c/
300 where c is the light velocity. In other materials, like siliceneand TMDs, the breaking of sublattice symmetry yields anonzero energy gap of the order 1-2 eV. Effects of inter-actions, nevertheless, may renormalize these quantitiesyielding results that are dependent on the coupling con-stants and the electron density, as it has been shown inthe case of the renormalization of the Fermi velocity inclean graphene due to a static Coulomb potential .As it is well-known, the renormalization of v F ingraphene shows that, at very low densities, one finds anultra-relativistic regime where v F → c , recovering theso-called Lorentz symmetry [7]. Despite the experimen-tal difficulty to actually reach this regime, it is remark-able that the effect of interactions yields a possible real-ization of massless Dirac fermions in a two-dimensionalcrystal. It is worth to mention that PQED [8] appliedto graphene yields a suitable description for both caseseither for v F ≪ c or v F ≈ c [9]. Therefore, the renor-malization of v F is straightforward within this quantum-electrodynamical approach. Indeed, in similar systems,the use of quantum field theory techniques has beenshown very useful for describing electronic properties[4, 5, 10–15]. Nevertheless, the effects of GN interactionscoupled to PQED have been less discussed [16].In this work, we shall investigate the effect of elec-tromagnetic interaction on the quasiparticle mass renor-malization of two-dimensional systems via analysis of therenormalization group in the dominant order at 1/N.We shall use PQED, sometimes called reduced quantumelectrodynamics [17], to describe the interaction betweenthese quasiparticles through the Gauge field. We com-pare our theoretical results with the recent experimen-tal findings for WSe [18] and MoS [19]. Next, usingGN-type interaction to simulate some impurity/disorderpresent in the sample of a 2D-Dirac material [20], weshall investigate the influence of this interaction on thePQED renormalization group functions.This paper is organized as follow. In Sec. II, we presentthe model, Feynman’s rules and obtain photon propaga-tor and electron self-energy both in the dominant orderof 1/N. In Sec. III, we analyze the renormalization groupfunctions and we obtain the renormalized mass whichdescribes the band gap renormalization and we compareour theoretical results with the experimental findings forWSe and MoS . In Sec. IV we investigated the influenceof GN interaction in large-N expansion in the renormal-ization group functions obtained in the previous section.In Sec. V, we review the main results obtained in thispaper and in the App. A-C we show some details of thecalculations. II. ELECTROMAGNETIC INTERACTIONSFOR MASSIVE ELECTRONS INTWO-DIMENSIONS
In this section, we calculate some effects of the electro-magnetic interactions for two-dimensional materials witha band gap. This band gap may be described as a massterm at the level of Dirac equation. This is a conse-quence of the tight-binding approximation for electronsin the honeycomb lattice at low energies (See Ref. [4] fora full derivation in the supplementary material).Let us consider a Lagrangian in Euclidean space givenby L = 12 F µν F µν ( − (cid:3) ) + ˙ ı ¯ ψ a (cid:0) γ ∂ + v F γ i ∂ i − m (cid:1) ψ a − ξ A µ ∂ µ ∂ ν ( − (cid:3) ) A ν + e ¯ ψ a γ µ ψ a A µ , (1)where v F is the bare Fermi velocity, e is the electromag-netic coupling constant, and m is the bare mass of theelectron. A µ is the pseudo-electromagnetic field and F µν is its usual field-intensity tensor. ψ a is the Dirac fielddescribing the electrons of the p-orbitals in the honey-comb lattice that are relevant for describing electronicproperties. Furthermore, a = 1 , ..., N is a flavor label forthis matter field that aims for describing both valley andspins indexes (or any other internal symmetry). Here,our matter field reads ψ † a = ( ψ ∗ A ↑ , ψ ∗ A ↓ , ψ ∗ B ↑ , ψ ∗ B ↓ ) a , where( A, B ) and ( ↑ , ↓ ) are the sublattices and spin of the hon-eycomb lattice, respectively. Therefore, a = K, K ′ and N = 2 describe the valley degeneracy [9, 13]. ξ is theGauge fixing parameter. From Eq. (1), we obtain the en-ergy dispersion E ± ( p ) = ± p v F p + m , where the sign ± means either the valence band ( − ) or the conductionband (+). Note that we are using the natural system ofunits, where ~ = c = 1.We may conclude from the Dirac Lagrangian in Eq.(1) that the characteristic exponent of the space-timeanisotropy is given by z = 1. This is given by the ex-ponent of the higher-order derivative term that breaksLorentz symmetry, i.e, ( v F ∂ i ) z . Therefore, as it hasbeen shown in Ref. [21], this means a soft breaking ofthe Lorentz symmetry. Higher-order terms would implyhigher-order derivatives, which, in principle, could de-scribe the behavior of electrons far from the Dirac point.We consider the large- N expansion at one-loop approx-imation and in this case of a trilinear interaction likethat of Eq.(1) this can be done through the substitution e → e/ √ N , for fixed e . Thus, the Feynman rules, in theEuclidean space, are S F ( p, m ) = γ p + v F γ i p i + mp + v F p + m , (2)which is the Fermion propagator,∆ (0) µν ( p ) = 12 ǫ p p (cid:20) δ µν − (cid:18) − ξ (cid:19) p µ p ν p (cid:21) , (3)for the Gauge-field propagator, where ǫ is the dielectricconstant of the medium, and e √ N γ µ , and γ µ → µ = 0 , γ µ = i, v F γ i , (4)describing the electromagnetic interaction. A. The Gauge-field propagator
The full propagator of the Gauge field is calculated, inmomentum space, from∆ µν ( p ) = ∆ (0) µν ( p ) + ∆ (0) µα ( p )Π αβ ( p )∆ (0) βν ( p ) + · · · , (5)which is a geometric series. In the large- N approxima-tion, the quantum corrections may be expressed as a sumover diagrams of the same order in the parameter N , asit is shown in Fig. 1, since e → e/ √ N . = m n m n + m a b n + m a b d g n + ... FIG. 1:
The full propagator of Gauge field in the dominantorder of /N . The polarization tensor isΠ µν ( p ) = − e Tr Z d k (2 π ) γ µ S F ( k ) γ ν S F ( k + p ) . (6)Next, we assume that the interaction vertex is justgiven by γ (this means we are assuming a static regime).Using the dimensional regularization we obtain the timecomponent of the polarization tensor, which is given by(See App. A for more details)Π ( p ) = − e " p p p + v F p − p m ( p + v F p ) . (7)Thereafter, we use Eq. (7) and the free photon propaga-tor, given in Eq. (3), for calculating the full propagatorof the Gauge field. This is given by∆ ( p ) = ǫ p p + e " p p p + v F p − p m ( p + v F p ) − , (8)where the static approximation has been implemented,which consists of taking p = 0 at the free photon prop-agator. B. The fermion self-energy
The fermion propagator with the self-energy correc-tions, in the dominant order of 1/N, is shown in Fig. 2. = + (a) (b)
FIG. 2:
The full fermion propagator . ( a ) represents the freefermion propagator, ( b ) is the 1-loop correction due to the fullphoton propagator in the dominant order 1 /N . Let us first calculate the self-energy of the fermion dueto the electromagnetic interaction, represented in Fig. 2( b ). Using the static approximation, the electron self-energy readsΣ A µ ( p ) = e N Z d k (2 π ) γ S F ( p − k ) γ ∆ ( k ) . (9)Because we are interested in the small momentum behav-ior of this expression (similar to the approximation usedin Ref. [22]), we can writeΣ A µ ( p ) = Σ (0) A µ + γ p Σ (1 a ) A µ + v F γ i p i Σ (1 b ) A µ , (10) where Σ (0) A µ , Σ (1 a ) A µ , and Σ (1 b ) A µ are the lowest-order termswithΣ (0) A µ = − e N Z d k (2 π ) " mk + v F k + m ∆ ( k ) , (11)Σ (1 a ) A µ = − e N Z d k (2 π ) " v F k − k + m ( k + v F k + m ) ∆ ( k ) , (12)andΣ (1 b ) A µ = e N Z d k (2 π ) " m + k ( k + v F k + m ) ∆ ( k ) . (13)Next, we perform a variable change v F k i → k i and,using spherical coordinates, the full photon propagatorcan be written as∆ ( k ) = v F ǫ k sin θ (cid:20) e ǫv F (cid:18) − m k (cid:19) sin θ (cid:21) − . (14)Using the small-mass limit ( m ≪ k ) the term 4 m /k can be neglected in Eq. (14). Furthermore, since we arestudying the model in the static approximation, with λ = e / ǫv F <
1, hence, we haveΣ A µ ( p ) = − λπ N [ mf ( λ ) + γ p f ( λ ) − v F γ i p i f ( λ )] ln (cid:18) ΛΛ (cid:19) + FT , (15)where FT stands for finite terms, Λ and Λ are ultravio-let and infrared cutoff respectively. The functions f ( λ ), f ( λ ), and f ( λ ) are given by (See App. B for more de-tails and Ref.[22]) f ( λ ) = 2 cos − ( λ ) √ − λ , (16) f ( λ ) = − λ (cid:20) π − λ + ( λ − √ − λ cos − ( λ ) (cid:21) , (17)and f ( λ ) = 1 λ h π − λ − p − λ cos − ( λ ) i . (18) III. RENORMALIZATION GROUP
In principle the renormalization group (RG) equa-tion presents two anomalous dimensions correspondingto each field ψ and A µ . However, since the polarizationtensor for the Gauge is finite, within the dimensional reg-ularization, we may conclude that γ A µ = 0, and, there-fore, β e = 0. Hence, the RG equation reads (cid:20) Λ ∂∂ Λ + β v F ∂∂v F + β m ∂∂m − N F γ F (cid:21) Γ ( N F ,N A ) ( p i ) = 0 , (19)where Γ ( N F ,N A ) ( p i = p , ..., p N ) means the renormalizedvertex functions. N F and N A are the number of ex-ternal lines of fermion and Gauge fields, respectively. β v F = Λ ∂v F ∂ Λ and β m = Λ ∂m∂ Λ are the beta functionsof the parameters v F and m , respectively. The func-tion γ F is the anomalous dimension of the fermion, givenby γ F = Λ ∂∂ Λ (cid:0) ln p Z ψ (cid:1) , where Z ψ is the wavefunctionrenormalization. For our purpose, it is sufficient to con-sider only the vertex function for the fermion, i.e, Γ (2 , .Therefore, we can writeΓ (2 , = (cid:0) γ p + v F γ i p i − m (cid:1) + Σ A µ ( p ) . (20)Now we must replace Eq. (20) in Eq. (19). Note that,despite our notation, the parameters v F and m insideEq. (20) are the renormalized parameters in agreementwith Eq.(19). We write the betas functions as a series,such that β a = N β (0) a + N β (1) a + ... for a = v F , m , and γ F = N γ (0) F + N γ (1) F + ... , thus, we can write β v F = − π N v F (cid:20) − ( λ ) λ √ − λ (cid:21) + 2 πN v F λ , (21) β m = − π N m (cid:20) − ( λ ) λ √ − λ − πλ (cid:21) , (22)and for the anomalous dimension we have γ F = − π N (cid:20) − λ λ √ − λ cos − ( λ ) (cid:21) + 2 πN λ . (23)The coupling constant λ is defined as λ ≡ e / ǫv F = πα/
4, where α is the dimensionless fine-structure con-stant. A. Fermi velocity renormalization and anomalousdimension
Using the definition of the beta function for v F ,namely, β v F = Λ ∂v F ∂ Λ in Eq. (21), we may find therenormalized Fermi velocity depending on the energyscale Λ. We may replace the energy scale by the carrierconcentration n by performing the following transformΛ / Λ → ( n/n ) / . After doing this, it has been shownthat Eq. (21), with an effective dielectric constant, yieldsa very good agreement with the experimental data forgraphene [7, 23]. The main effect is that the value of v F increases as we decrease the value of n . This maybe improved by producing more and more clean samples.Here, we conclude that the presence of the mass does notchange this result. Therefore, a similar renormalizationof the Fermi velocity, in other two-dimensional materials,is expected to occur.Eq. (23) yields the anomalous dimension of the modelat one-loop approximation. Note that for a very large number of fermionic species, N → ∞ , the anomalousdimension vanishes. N Γ F FIG. 3:
Anomalous Dimension . We plot Eq. (23) with λ = 0 . N . In the asymptotic limit N → ∞ , γ F → B. Mass Renormalization
Using the definition of the beta function for m , namely, β m = Λ ∂m∂ Λ in Eq. (22), we may find the renormalizedmass depending on the energy scale Λ. This, neverthe-less, may be described in terms of n as discussed before.After a simple calculation, we find m ( n ) = m (cid:18) nn (cid:19) C λ / , (24)where m ≡ m ( n ) is a reference value (which must beprovided by experiments) and C λ = − π N (cid:20) − ( λ ) λ √ − λ − πλ (cid:21) (25)is a known constant fixed by the coupling constant λ and N = 2. Within the realm of two-dimensional materials,Eq. (24) shows that the band gap is tunable by changingthe carrier concentration n . The renormalization of m has been experimentally measured in Ref. [18], where theauthors have shown that, by changing the carrier concen-tration from n ≈ . × cm − to n ≈ . × cm − ,the energy gap decreases approximately 400meV of itsbare value for tungsten diselenide WSe . Beyond sev-eral kind of applications, this effect could be useful forstudying the electric-field tuning of energy bands withnontrivial topological properties. In Ref. [19], a similarresult has been found for Molybdenum disulfide MoS . n H cm - L m H e V L FIG. 4:
Renormalization of the band gap for WSe . The thickline is the plot of Eq. (24) with C λ = − .
17 ( λ ≈ .
96) and m = 2 . n = 1 . × cm − . The common lineis the plot of Eq. (24) with C λ = − .
15 ( λ ≈ .
76) and m = 1 . n = 3 . × cm − . For both curves wehave assumed N = 2 (spin degeneracy). The small dots anderror bars have been extracted from Fig. 4 in Ref. [18]. Theyare the experimental values of the renormalized band gap atdifferent values of the carrier concentration n for WSe at T =100K. The two different colors are related to the twodifferent devices with different thickness of the boron nitridesubstrate. The red point is device 1 with d BN ≈ . d BN ≈ . λ constants in our theoreticalresult in Eq. (24). In FIG. 4, we compare our analytical result with theexperimental data for WSe , using the corresponding er-ror bars for each point [18]. This experimental data hasbeen obtained by putting the monolayer of WSe abovetwo different substrates made of boron nitride. It is well-known that the substrate changes the fine-structure con-stant α , because of its effects on the dielectric constant ǫ .Within our result, in Eq. (24), this may be described byconsidering a different value of the constant λ = πα/ m = 2 . n = 1 . × cm − , which plays therole of “bare” mass for our purposes. This is a manda-tory step in the renormalization procedure in general.Thereafter, we choose a best fitting parameter C λ that,essentially, provides the value of the coupling constant λ .Using these results, we conclude that the expected val-ues for α are between α ≈ [0 . , . α ≈ .
65 (the small difference is due to the use of differentsubstrates for each device). We obtain that the decreas-ing of the energy gap, whether n → . × cm − , is ofthe order of 400meV in comparison with the bare gap, asexpected [18]. Furthermore, besides one point (red pointbelow the thick line in Fig. 4), the other experimentaldata are within our theoretical result. In particular, allof the black points are described by our result. In Fig. 5,we repeat the same procedure, but for MoS . In this case, the experimental data are within our theoretical result inEq. (24). n H cm - L m H e V L FIG. 5:
Renormalization of the band gap for MoS . The thickline is the plot of Eq. (24) with C λ = − .
13 ( λ ≈ .
63) and m = 2 . n = 5 . × cm − . We have assumed N = 2 (spin degeneracy). The small dots and error barshave been extracted from Fig. 4 in Ref. [19]. They are theexperimental values of the renormalized band gap at differentvalues of the carrier concentration n for MoS at T =295K. Although small deviations are expected to occur, be-cause the experimental data has been obtained at tem-peratures of 100K, nevertheless, this effect is small. In-deed, note that the activation temperature T ∗ from theminimum of the valence band to the minimum of the con-duction band is of the order of the bare gap, i.e, ≈ T ∗ ≈ K roughly speaking. Hence, theeffects of the thermal bath are relevant for temperaturesclose to 10 K, far beyond the room temperature. We be-lieve that either higher-order corrections or inclusion ofmore interactions at one-loop level are likely to improvethis comparison. The measurement of more experimentalpoints are also relevant for a more precise comparison. Itis worth to mention that both materials, WSe and MoS ,have an excitonic spectrum that has been accurately de-scribed by PQED in Eq. (1), in particular, with a goodagreement with the experimental findings [4]. IV. EFFECTS OF FOUR-FERMIONINTERACTIONS
To include a four-fermion interaction in our modelgiven by Eq. (1), we star, just for the fermion sector,the following Lagrangian L = ˙ ı ¯ ψ a (cid:0) γ ∂ + v F γ i ∂ i (cid:1) ψ a − g (cid:0) ¯ ψ a ψ a (cid:1) , (26)where g is the bare coupling constant of the self-interaction between fermions. Because the GN interac-tion can be related to the disorder/impurity, which mod-ifies the density of states in monolayer graphene [24, 25],a more realistic model for describing transport proper-ties in these systems should include four-fermions inter-actions. Note that although we have performed m = 0in Eq. (26), we know that the GN interaction generatesa mass for the fermion in 1/N expansion by the chiralsymmetry breaking. Also, it well-known that the GN in-teraction [26] is non-renormalizable in the coupling con-stant g . Nevertheless, it is renormalizable in the contextof the 1/N expansion [26]. In this expansion, we mustperform the following transformations g → gN , for fixed g .Next, we introduce an auxiliary field in Eq. (26) in or-der to convert the Gross-Neveu interaction into a trilinearone. This auxiliary field may represent the presence ofdisorder/impurities [20, 27] in the model. Hence, L = L + N g (cid:16) σ − gN ¯ ψ a ψ a (cid:17) = L + N g σ − σ ¯ ψ a ψ a . (27)This new field σ does not change the dynamics of system,because it represents only a constrained that is derivedfrom the Euler-Lagrange equation as σ = gN ¯ ψ a ψ a . (28)Eq. (27) shows that a mass term is generated forthe fermion field whether h σ i 6 = 0. This implies aspontaneous breaking of the discrete chiral symmetry ψ a → γ ψ a and we shall be working in this broken phase.This is the reason why we have assumed m = 0 in Eq.(26). Indeed, we may define h σ i = σ , hence, σ repre-sents the mass generated for the electrons [26]. There-after, we replace σ → σ + σ √ N for ensure that the theoryhas a ground state. Hence, the Lagrangian reads L = ¯ ψ a (cid:0) ˙ ıγ ∂ + ˙ ıv F γ i ∂ i − σ (cid:1) ψ a + N g σ + √ Ng σ σ + 12 g σ − √ N σ ¯ ψ a ψ a . (29)The free propagator of fermion field is given by the eq.(2) where m → σ and the auxiliary-field propagatorreads ∆ σ = (cid:18) g (cid:19) − , (30)and the vertex interaction is given by − √ N , describingthe GN interaction. A. The auxiliary-field Propagator
The quantum correction for the auxiliary-field propa-gator in the lowest order of 1 /N can be calculated fromthe functional integral method. We can rewrite Eq. (29)and obtain L = ¯ ψ a Kψ a + N g σ + √ Ng σ σ + 12 g σ , (31) where K = ˙ ıγ ∂ + ˙ ıv F γ i ∂ i − σ − √ N σ . Integrationover ψ in Eq. (31) yields the effective action S eff [ σ ] forthe auxiliary field, given by S eff = N Tr ln K ≈ √ N S + N S + ... , where the last equality has been obtained forlarge- N . Furthermore, S = − Tr h(cid:0) ˙ ıγ ∂ + ˙ ıv F γ i ∂ i − σ (cid:1) − σ i + Z d x g σ σ, (32)and S = Tr (cid:20)n(cid:0) ˙ ıγ ∂ + ˙ ıv F γ i ∂ i − σ (cid:1) − σ o (cid:21) + Z d x g σ . (33)Eq. (32) yields the so-called gap equation after we as-sume S = 0 in order to have a finite effective action.This gap equation reads1 g = 4 Z d p (2 π ) p + v F p + σ g = − πv F | σ | . (34)This equation shows that we can relate the generatedmass σ with the coupling constant g [28, 29].Eq.(33) may be written as S = 12 Z d xd y σ ( x )Γ σ ( x − y ) σ ( y ) , (35)where Γ σ is the inverse of the full Auxiliary-field propa-gator, hence,Γ σ ( p ) = 1 g + Tr Z d k (2 π ) [ S F ( k + p ) S F ( k )] . (36)Therefore,∆ − σ ( p ) = Γ σ ( p ) = (∆ σ ) − + Π σ ( p ) , (37)where Π σ ( p ) is the self-energy due to the Gross-Neveuinteraction. Note that Eq. (37) is the well-knownSchwinger-Dyson equation for the σ field. Using the Eq.(34) in the Eq. (36) and after some simplifications, wefind that (see App. C for more details)Π σ ( p ) = 1 πv F " | σ | + p + v F p + 4 σ p p + v F p × sin − s p + v F p p + v F p + 4 σ ! . (38)Using Eq. (30) and Eq. (38) we obtain the fullAuxiliary-field propagator. We consider that the gener-ated mass is much smaller than the external momentum,i.e, p ≫ σ . Therefore, the propagator reads∆ σ ( p ) = 4 v F p p + v F p . (39) B. The fermion self-energy
The fermion propagator with the self-energy correc-tions, in the dominant order of 1/N, is shown in Fig. 6. = + (a) (b)
FIG. 6:
The full fermion propagator . ( a ) represents the freefermion propagator, ( b ) 1-loop correction due to the Gross-Neveu interaction, within the auxiliary-field approach, wherethe double dashed line represents the full auxiliary-field prop-agator. This self-energy, due to the interactions with the aux-iliary field, is given byΣ σ ( p ) = 1 N Z d k (2 π ) S F ( p − k )∆ σ ( k ) . (40)Note that Eq. (40) has a similar structure in comparisonwith Eq. (9). Therefore, we follow the same steps asbefore, yielding the self-energy, namely,Σ σ ( p ) = Σ (0) σ + γ p Σ (1 a ) σ + v F γ i p i Σ (1 b ) σ , (41)where Σ (0) σ , Σ (1 a ) σ , and Σ (1 b ) σ are given byΣ (0) σ = 4 v F N Z d k (2 π ) σ k + v F k + σ q k + v F k , (42)Σ (1 a ) σ = 4 v F N Z d k (2 π ) v F k − k + σ ( k + v F k + σ ) q k + v F k , (43)andΣ (1 b ) σ = 4 v F N Z d k (2 π ) k + σ ( k + v F k + σ ) q k + v F k . (44)Using the same variable change that has been madefor obtain the Eq. (15), we obtainΣ σ ( p, σ ) = 23 π N (cid:0) γ p + v F γ i p i + 3 σ (cid:1) ln (cid:18) ΛΛ (cid:19) + FT . (45)Note that the divergent part of Eqs. (43) and (44)are equal because of rotational symmetry, and in view ofthis, only the wave function renormalization is sufficientto renormalize the two points Green function due to theGN interaction. As a result, GN interaction does notchange the renormalization of the Fermi velocity, as weshall see in the next section explicitly. C. Renormalization group function due GN andelectromagnetic interations
Since Π µν and Π σ are finite, within the dimensionalregularization, we may conclude that γ σ = γ A µ = 0, and,therefore, β e = 0. Hence, the RG equation reads (cid:20) Λ ∂∂ Λ + β v F ∂∂v F + β σ ∂∂σ − N F γ F (cid:21) Γ ( N F ,N A ,N σ ) ( p i ) = 0 , (46)where Γ ( N F ,N A ,N σ ) ( p i = p , ..., p N ) means the renormal-ized vertex functions. N F , N A , and N σ are the number ofexternal lines of fermion, Gauge, and sigma fields, respec-tively. β v F , β σ and γ F are defined similarly as in sectionIII. For our purpose, it is sufficient to consider only thevertex function for the fermion, i.e, Γ (2 , , . Therefore,we can writeΓ (2 , , = (cid:0) γ p + v F γ i p i − σ (cid:1) + Σ A µ ( p ) + Σ σ ( p ) . (47)Using Eq. (47) into Eq. (46) and using that β a = N β (0) a + N β (1) a + ... for a = v F , σ , and γ F = N γ (0) F + N γ (1) F + ... , we obtain, after some calculations, γ F = − π N (cid:20) − λ λ √ − λ cos − ( λ ) (cid:21) + 2 πN λ + 13 π N , (48) β v F = − π N v F (cid:20) − ( λ ) λ √ − λ (cid:21) + 2 πN v F λ , (49)and β σ = − π N σ (cid:20) − ( λ ) λ √ − λ − πλ (cid:21) + 8 σ π N . (50)We may conclude, from Eqs. (48)-(50), that the Gross-Neveu interaction modifies both the anomalous dimen-sion of the fermion as well as the beta function of themass. On the other hand, it does not modify the betafunction of the Fermi velocity , hence, the Fermi velocityrenormalization is insensitive to this interaction. The re-sults in Eq. (22) and Eq. (23) are obtained by neglectingthe last term in the rhs of Eq. (50) and Eq. (48), respec-tively. These terms are generated by the four-fermioninteraction. Other perturbative approach, consideringthis interaction, reveal similar conclusions, i.e, indeed,the Gross-Neveu interaction does not change the renor-malization of v F [20]. D. The Critical Coupling Constant λ c From Eq. (50) we may calculate the renormalized mass σ R as a function of the energy scale Λ. After a simplecalculation, and performing Λ / Λ → ( n/n ) / , we find σ R ( n ) = σ (cid:18) nn (cid:19) C GN λ / , (51)where σ ≡ σ ( n ) and C GN λ = − π N (cid:20) − ( λ ) λ √ − λ − πλ (cid:21) + 83 π N (52)is a known constant fixed by the coupling constant λ and N = 2. After solving C GN λ = 0 for λ , we find our criticalvalue λ c ≈ .
66. In this case, λ = λ c , the value of themass is the same for any energy scale. On the other hand,for λ = λ c , the asymptotic behavior of σ is dependent onthe sign of C GN λ . Indeed, for λ > λ c , hence, C GN λ < σ → λ < λ c , we have C GN λ > σ → ∞ as Λ diverges. These last two casesare a consequence of the competition between PQED andthe GN interaction, because the second term in the rhsof Eq. (52) is generated only due to the GN interaction,being the theoretical prediction with the presence of aconsiderable disorder. In Fig. 7, we summarize these dif-ferent asymptotic regimes. n (cid:144) n H cm - L Σ R (cid:144) Σ H e V L FIG. 7:
Effects of Gross-Neveu interactions in the renormal-ization of the band gap . The dashed line is the plot of Eq. (51)with λ < λ c ≈ .
66 and λ = 0 .
4. The common line is the plotof Eq. (51) with λ > λ c ≈ .
66 and λ = 0 . From Fig. 7, we conclude that the behavior of m ( n ) re-mains the same for σ R ( n ) whether λ > λ c . The behaviorof m ( n ) has been compared with experimental data ofboth WSe and MoS in Sec. II. Nevertheless, consider-ing that λ = πα/
4, and α = e / πǫv F , we also concludethat one would decrease the value of λ whether the factor ǫv F increases. This could be obtained either by choos-ing a proper substrate (with large ǫ ) or by increasing thevalue of v F which naturally occurs for clean samples [7].A fine tunning of λ close to λ c yields a case where theband gap remains the same at any energy scale. We be-lieve that these theoretical predictions could be useful forapplications in several two-dimensional materials (whomare described by the massive Dirac equations at low en-ergies), in particular, for studying electric-field tunningof band energies [18]. V. CONCLUSIONS
In this work, we investigate the renormalization of thebandgap, in both WSe and MoS . Since these materialshave sp hybridization, the electromagnetic interactionbetween the massive quasiparticles of these systems canbe described by Pseudo quantum electrodynamics. Us-ing the renormalization group approach, in the dominantorder in 1 /N , we show that our results are in excellentagreement with recent experimental measurements of thebandgap in these materials. A more realistic model fordescribing transport properties in these systems shouldinclude four-fermions interactions, once this interactioncould simulate a disorder/impurity-like microscopic in-teraction. Thus, we also investigate the influence of aGN-type interaction in the behavior of the renormaliza-tion group function of PQED, where initially masslessfermions acquire mass by chiral symmetry breaking. Weshow that the presence of a GN-type interaction does notchange the behavior of the renormalized Fermi velocity,which is well-known in the literature. On the other hand,the mass function has a richer behavior, which allows usto recognize a single fixed point at λ c ≈ .
66, represent-ing an ultraviolet fixed point. This renormalized massshows a different behaviors whether λ is above or belowthe critical point. This result could be relevant for appli-cations of 2D materials that rely on tunable band gaps.We hope that our result clarify the relevance andbeauty of applications of quantum field theory in thedescription of electrons in 2D materials. Because thefine-structure constant may be increased, it would be rel-evant to understand the nonperturbative effects on therenormalization of m as well as to calculate the higher-temperature effects. We shall investigate this elsewhere. Acknowledgments
L. F. is partially supported by Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior Brasil(CAPES), finance code 001; V. S. A. and L. O. N.are partially supported by Conselho Nacional de De-senvolvimento Cient´ıfico e Tecnol´ogico (CNPq) and byCAPES/NUFFIC, finance code 0112. F. P. acknowledgethe financial support from DIUFRO Grant DI18-0059 ofthe Direcci´on de Investigaci´on y Desarrollo, Universidadde La Frontera. F. P. acknowledge the hospitality of theFaculdade de F´ısica of the Universidade Federal do Par´a,Bel´em where part of this work was done. The work ofE.C.M. is supported in part by CNPq and FAPERJ. Theauthors would like to thank J. Gonz´alez for his commentsin the Ref. [23].
APPENDIX: SOME DETAILS OF THECALCULATIONSAppendix A: Gauge-self Energy
In this appendix we derive Eq. (6), i.e,Π µν ( p ) = − N T r Z d k (2 π ) Γ µ S F ( k )Γ ν S F ( k + p ) , (53)where the interaction vertex should be understood asΓ µ → γ e/ √ N due to the static approximation. wherethe trace operation is understood on Lorentz indices andinternal symmetry. Using matrix representation γ as4 ×
4, we have the following trace propertiesTr [ γ µ γ ν ] = − δ µν , Tr [ γ µ γ α γ ν ] = 0 , (54)Tr (cid:2) γ µ γ α γ ν γ β (cid:3) = 4 (cid:0) δ µα δ νβ − δ µν δ αβ + δ µβ δ να (cid:1) . Performing the trace operations and using the Feynmanparametrization, we haveΠ ( p ) = − e Z dx Z d k (2 π ) Num[Den] , (55)where Num= k ( k + p ) − δ ij v F k i ( k + p ) j − m and Den =( k + xp ) + x (1 − x ) p + v F ( k + x p ) + x (1 − x ) v F p + m .Solving the integrals over k and k , we obtainΠ ( p ) = − e µ ǫ πv F Z dx (cid:26)p ∆ − x (1 − x )( p − v F p ) v F √ ∆ + m v F √ ∆ (cid:27) , (56)where ∆ = v F (cid:2) x (1 − x )( p + v F p ) + m (cid:3) . Therefore,after of integration in the Feynman parameter, the time-component of the polarization tensor, in the small-masslimit p ≫ m , isΠ = − e " p p p + v F p − p m ( p + v F p ) . (57) Appendix B: Loop Integral Calculation
The integrals we solve in our model have a particu-lar feature due to the Lorentz symmetry breaking. Forclarifying this point, we show how to obtain the func-tion f ( λ ), given by Eq. (16). First, we made a variable change v F k i → ¯ k i .Thereafter, for solving the integrals,we use spherical coordinates, hence, k = k cos θ, | ~k | = k sin θ,d k = k sin θ dk dθ dφ. Therefore, we write Eq. (16) asΣ (0) A µ = − e σ π ) ǫN v F Z Λ0 dk kk + σ × Z π dθ
11 + λ (1 − σ k ) sin θ , (58)where the term σ /k ≈
0, thereby, Z π dθ
11 + λ sin θ = 2 cos − ( λ ) √ − λ . (59)Because Eq. (59) does not depend on the momentum ofthe loop, we can define it as f ( λ ) for λ <
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