Superconducting and Excitonic Quantum Phase Transitions in Doped Dirac Electronic Systems ∗ Lizardo H.C.M

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provided by Elsevier - Publisher Connector Physics Letters A 376 (2012) 779–784

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Superconducting and excitonic quantum phase transitions in doped Dirac electronic systems ∗ Lizardo H.C.M. Nunes a, , Ricardo L.S. Farias a, Eduardo C. Marino b

a Departamento de Ciências Naturais, Universidade Federal de São João del Rei, 36301-000 São João del Rei, MG, Brazil b Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brazil

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Article history: We investigate the effect of superconducting and excitonic interactions, as well as their competition, on Received 10 June 2011 Dirac electrons on a bipartite planar lattice. It is shown that, at half-ﬁlling, Cooper pairs and excitons Received in revised form 1 December 2011 coexist if the superconducting and excitonic coupling parameters are equal and above a threshold cor- Accepted 17 December 2011 responding to a quantum critical point. In the case where only the excitonic interaction is present, we Available online 21 December 2011 obtain a critical chemical potential, as a function of the interaction strength. Conversely, if only the su- Communicated by A.R. Bishop perconducting interaction is considered, we show that the superconducting gap displays a characteristic Keywords: dome as charge carriers are doped into the system. We also show that, as the chemical potential in- Dirac electrons creases, superconductivity tends to suppress the excitonic order parameter. Superconductivity © 2011 Elsevier B.V. Open access under the Elsevier OA license. Excitons Quantum criticality

1. Introduction electronic excitations appearing in the conduction band will have the dispersion relation of a relativistic massless particle and their Several condensed matter systems have been discovered in re- properties, accordingly, will be determined by the Dirac equation. cent years whose active electrons have their kinematics governed In the high-Tc cuprate superconductors, Dirac points appear in by the Dirac equation, rather than by the Schrödinger equation. the intersection of the nodes of the d-wave superconducting gap Their dispersion relation, accordingly, has the same form as that of and the 2D-Fermi surface. Again, the low-energy excitations will a relativistic particle. The reason for this unusual behavior of elec- correspond exclusively to these points [4]. trons whose speed is at least two orders of magnitude less than Quite recently, it has been experimentally found that the the speed of light can be ascribed to a particular influence of the newest high-Tc superconductors, namely the iron pnictides [5,6] lattice background on the electronic properties. also present electronic excitations whose properties are governed One early example of such Dirac electrons in condensed matter by the Dirac equation. Theoretical results also support the exis- can be found in polyacetylene [1,2].Theundopedsystemisinthe tence of Dirac electrons in the pnictides [7,8]. half-filling regime. If we expand the one-dimensional lattice tight- For many technological applications, it would be interesting if binding energy about the two Fermi points we obtain the linear graphene were a semiconductor, instead of a semimetal. Therefore, dispersion relation associated to a massless relativistic particle, the the possibility of an insulator state or the appearance of an ex- two components of the Dirac field corresponding to left and right citonic gap due to short range electronic interactions in a system moving electrons. Within a large temperature range, any electronic of massless Dirac fermions on the honeycomb lattice has been the excitations will have such a dispersion relation and consequently object of intense investigation [9–13]. Nonetheless, early propos- will obey the Dirac equation. als that electron–electron interactions could generate an electronic It is graphene [3], however the most important system pre- gap were investigated even before the discovery of graphene [14, senting Dirac electrons. In this case the tight-binding energy cor- 15], where the gap opening is an analogue of the “chiral symme- responding to an electron in the honeycomb lattice presents a try” breaking process that occurs in massless quantum electrody- band structure such that the valence and conduction bands touch namics (QED) in two dimensions [16]. This is the type of excitonic precisely in the vertices of two inequivalent Dirac cones and any instability that we consider in the present Letter. The fact that some of the above materials become superconductors upon doping provides a vast phenomenological motivation for * Corresponding author. the investigation of the superconducting phase diagram of Dirac E-mail address: [email protected] (L.H.C.M. Nunes). electrons systems.

Indeed, we consider a quasi-two-dimensional model consisting is the excitonic order parameter [14] and λexc is also regarded as of Dirac electrons both with superconducting and excitonic in- a free parameter. The phase transition corresponding to an exci- teractions. In particular, we study the competition between these tonic instability results in the opening of an insulating gap in the two interactions and the resulting different situations that arise as electronic spectrum. It is associated to a nonzero value for σ . we independently vary the coupling parameter of each interaction The excitonic interaction introduced above can be generated by term and as we add charge carriers to the system. different physical mechanisms. The Coulomb interaction, for in- Condensed matter systems with such competing interactions stance, can generate it, precisely as it occurs in the case of the display phenomena that are analogous to the onset of the chiral Hubbard model, where the on-site density-density repulsion term condensate and of color superconductivity in dense quark matter, can be derived from the static Coulomb interaction. Eq. (5) is a which is an active subject of investigation [17]. The phase transi- mean-field version of the repulsive U -term in the Hubbard model. tions obtained in such systems can be compared with the ones oc- Alternatively, the excitonic interaction term could be produced, for curring in analog condensed matter systems with Dirac fermions. instance, by an electron–phonon interaction such as the one oc- curring in the Peierls mechanism that leads to an insulating gap in 2. The model polyacetylene. The excitonic interaction would be then obtained by integration over the phonon field. A further (magnetic) mechanism Consider a system presenting a bipartite lattice formed of sub- for generating this interaction is described in [18]. lattices A and B,suchasingraphene [3] or in the pnictides [6]. The superconducting and excitonic Hamiltonians resemble the † · † † · † = ik ri = ik ri ones appearing in the so-called g-ology [19,20]. Since our system is Let ai,σ k e ak,σ and bi,σ k e bk,σ be, respectively, electron creation operators, with spin σ ,onsitei of sublattices A in 2D, however, the bosonization techniques, which are employed and B. We take the non-interacting Hamiltonian as [3] for obtaining an exact solution thereof, can no longer be used and thereforewehavetoresorttoapproximatemethods. =− † + † Combining Eq. (6) with Eq. (1) and Eq. (4), we obtain our mean- Ht μ ak,σ ak,σ bk,σ bk,σ field model Hamiltonian, which except for constant terms, reads k,σ † − † + H = Φ A Φ , (7) t skak,σ bk,σ h.c. , (1) k k k k,σ k A where the μ is the chemical potential and the last term above is where the matrix k is given by the momentum space version of the Hamiltonian describing the ⎛ ⎞ μ + σ −tsk 0 hopping of electrons between adjacent sites of different sublat- ∗ ⎜ −ts μ − σ 0 ⎟ tices A and B. A = ⎝ k ∗ ∗ ⎠ , (8) k 0 − ts ≈ σ μ k For the case of graphene, we would have t 2.8 eV and, for ∗ − − ik·a1 ik·a2 0 tsk (μ σ ) the corresponding honeycomb√ lattice, sk = 1 + e + e where ˆ ˆ ˆ a1 = aex and 2a2 = a(ex − 3e y),witha as the lattice parameter. † = † † and Φ a ↑b ↑b−k,↓a−k,↓. We assume the interaction has two terms leading, respectively, k k, k, In the case of graphene, there are six Dirac points at the cor- to superconducting and excitonic instabilities. The isotropic s-wave ners of the first Brillouin zone in the honeycomb lattice; how- pairing state is described by a BCS-type interaction as ever, only two of them are non-equivalent. We denote them by † † K =−4π/3aeˆ and K = 4π/3aeˆ and we expand Eq. (7) in their =− x x Hsc λsc a a− − a−k ,−σ ak ,σ k,σ k, σ vicinity. Collecting only the linear terms for sk, and taking the con- k,k,σ tinuum limit of our model, we obtain † † + b b− − b−k ,−σ bk ,σ . (2) 2 2 2 k,σ k, σ d k † || σ HCL = Φ (k)A Φ (k) − − , (9) 2 α α α The value of λsc is to be determined by some underlying micro- α (2π) λsc λexc scopic theory, which is not considered here. In the present Letter, the strength of the superconducting interaction is a free parameter where α = K , K denotes the Dirac points and the matrix Aα is − − − + of the Hamiltonian and can take any arbitrary positive value. given by Eq. (8),replacingtsk by vF(kx iky) or vF(kx iky) The superconducting order parameter, by definition, is given by in the√ vicinity of K or K . Moreover, the Fermi velocity is given by vF = 3ta/2, with h¯ = 1fromnowon. − = † † = † † λsc ak,↑a−k,↓ λsc bk,↑b−k,↓ . (3) k k 3. The effective potential Using a mean-field approximation, we insert the above expres- The partition function in the complex time representation can sion in Eq. (2), thereby rewriting the superconducting interaction be written as as ∗ † † † † β MF = + + 2 Hsc ak,σ a−k,−σ bk,σ b−k,−σ h.c., (4) 1 d k Z = DΠ DΦ exp dτ iΠ∂ Φ − H[Π,Φ] , k 2 τ Z0 (2π) except for a constant term. 0 In analogous way, we introduce the mean-field excitonic Hamil- (10) tonian as where Φ is an arbitrary set of fields, Π is their conjugate momen- MF = † − † tum and Z the vacuum functional. Hexc σ akσ ak,σ bk,σ bk,σ , (5) 0 k,σ Inserting Eq. (9) in the above expression, we can perform the Gaussian integral over fermionic fields to obtain an effective po- where tential. Conversely, by integrating on and σ we would reobtain † † the original fermionic quartic interaction Hamiltonian by means of σ = λexc a ak,σ − b bk,σ (6) k,σ k,σ a Hubbard–Stratonovitch transformation. σ L.H.C.M. Nunes et al. / Physics Letters A 376 (2012) 779–784 781

The effective potential per Dirac point reads 4. Zero chemical potential ∞ ||2 σ 2 d2k We analyze in this section the zero temperature phase diagram V eff = + − T λ λ (2π)2 of our model in the absence of doping or impurities, namely, at sc exc n=−∞ zero chemical potential. 4 [ − ] For μ = 0, Eq. (15) becomes j=1 iωn E j × log , (11) 4 [ − = = ] 3 ˜ 2 2 j=1 iωn E j(σ 0, 0) Λ σ˜ 2 3 ˜ 2 ˜ 2 2 V eff = + − + σ + 1 = + α λ˜ λ˜ 3 where ωn (2n 1)π T is the Matsubara frequency for fermions sc exc (with the Boltzmann constant kB = 1) and the four E j ’s are 3 ˜ 2 2 2 − + σ˜ − 1 , (17) 2 2 2 2 E j =± || + σ + (vFk) ± μ . (12) where σ˜ = σ /Λ and ˜ =||/Λ. Notice that Eq. (12) is the typical dispersion relation observed First, let us assume the following situation: λ = λ = λ .In − sc exc 2 2 2 in effective models for quantum chromodynamics, where E j cor- such case, we can define a new order parameter, ζ =|| + σ , responds to the energy required to create a particle or a hole which is inserted in Eq. (17), and we get exactly the same func- + above the Fermi surface and E j is the corresponding antiparti- tional form of the effective potential given by Eq. (16). Therefore, cle term [21]. Moreover, the effective potential in Eq. (11) is the analyzing the values of ζ that minimize the effective potential, we same obtained in [22], where the formation of chiral and diquark arrive at the following, condensates involving two quark flavors in QCD was investigated. 0,λ<λc, In their case, the values for the quark–antiquark and quark–quark = | |2 + 2 = ζ0 0 σ0 αλ 1 − 1 (18) interactions strengths were set equal by construction, via a Fierz 2 ( 2 2 ), λ λc λc λ transformation applied to the Nambu–Jona-Lasinio model, which is not the case in the present Letter, since λsc and λexc are taken where ζ0, 0 and σ0 denote the solutions for the minimum of to be free parameters. V eff. As in the case when we had just a superconducting interac- We shall restrict our analysis in this work, to the case of zero tion and μ = 0, we find a quantum phase transition at the critical = temperature. In this case, Eq. (11) becomes coupling λc α/Λ. The minima of the V eff presents a radial symmetry, in agreement to the result in Eq. (18), and it is possible to | |2 2 find simultaneously nonzero values for σ and which minimize = + σ V eff the effective potential, indicating that excitons and Cooper pairs λsc λexc ∞ coexist in the system. d2k dω γ (ω,k,,σ ) The same result was obtained in the framework of hadronic − log , (13) (2π)2 2π γ (ω,k,= 0, σ = 0) physics [22], where the formation of chiral and diquark conden- −∞ sates involving two quark flavors in QCD was investigated. In their case, the values for the quark–antiquark and quark–quark interac- where tions strengths were set equal by construction, via a Fierz transfor- 2 mation applied to the Nambu–Jona-Lasinio model. In our case, on γ (ω,k,,σ ) = ||2 + σ 2 + v2k2 + ω2 + μ4 = F the other hand, we are not restricted to the relation λsc λexc and + | |2 − 2 − 2 2 + 2 2 we shall analyze in what follows the situation for which λsc = λexc. 2 σ vFk ω μ . (14) We first assume that λsc >λexc. Then, Eq. (17) can be written Calculating the integral over ω and k in Eq. (13), we obtain the as effective potential ˜ 2 α ζ 2 3 ˜ ˜ 2 ˜ 2 2 ˜ 3 V eff ≡ V eff = + δσ − ζ + 1 − ζ − 1 , (19) ||2 σ 2 2Λ3 1 3 ˜ = + + − 3 2 − 3 Λ λsc 3 V eff El Λ El (0) λsc λexc 3α 3α l where 1 Λ2 ˜ ˜ 2 λsc − λexc + lμ ξl(x)El(x) +|| log ξl(x) + El(x) , = 2α 0 δ ˜ ˜ > 0, (20) l λscλexc

(15) with λ˜ = λ /λ and λ˜ = λ /λ . exc exc c sc sc c √ Analysis of the second derivatives of the effective potential 2 2 2 2 where α = 2π v , El(x) = || + ξ (x), and ξl(x) = x + σ + lμ, ˜ F l yields the minimum of V eff at the point with l =+1or−1. Notice that we have introduced the momentum ˜ 2 cutoff Λ/v (Λ = 2πhv¯ /a), which is naturally provided by the λ − 1 F F ( , σ ) = sc , 0 , (21) lattice. 0 0 ˜ 2λsc In particular, for μ = σ = 0, Eq. (15) is reduced to when λsc > 1 and δ>0 for positive values of ζ0 and σ0.Hence, 2 3 || 2 ζ0 = 0 and we have only superconductivity in the system pro- V = − ||2 + Λ2 2 −||3 − Λ3 , (16) eff vided λ >λ . There is a single minimum for positive values of λsc 3α sc exc and σ exactly at the point given by Eq. (21). which is exactly the same result obtained by some of us for a the- Accordingly, when λsc <λexc, the minimum becomes ory describing a single layer of Dirac electrons interacting via a BCS-type superconducting interaction [23]. Analyzing the minima λ˜ 2 − 1 = 0 exc (22) of the above effective potential, we found a quantum phase tran- (0, σ0) , ˜ , 2λexc sition leading to the superconducting state at the critical coupling ˜ λc = α/Λ [23]. when λexc > 1 and we only have excitons in the system. 782 L.H.C.M. Nunes et al. / Physics Letters A 376 (2012) 779–784

5. Finite chemical potential and Eq. (24) yields

˜ 2 Consider now the effective potential in Eq. (15), in the case of (2) σ 2 V˜ = − μ˜ + . (29) a nonzero chemical potential. Making the change of variable x = eff ˜ λexc 3 (v k)2,weget F √ • σ˜ < μ˜ < 1 + σ˜ 2: ||2 σ 2 2Λ3 V eff = + + λsc λexc 3α At this intermediary condition, Eq. (24) yields Λ2 1 − | |2 + + 2 + 2 1 dx x σ lμ . (23) ˜ 2 2α (3) σ 2 1 V˜ = + − dy y + ˜ 2 + ˜ l 0 eff ˜ σ μ λexc 3 2 0 We notice that V eff(μ) = V eff(−μ). Therefore, given λsc and 2 2 λexc, the minima for μ > 0 are exactly the same for μ < 0 and μ−σ we constrain our analysis to positive values of the chemical poten- 1 − dy y + σ˜ 2 − μ˜ tial in this section. 2 In the following, we consider superconductivity and excitonic 0 fluctuations separately, and then we combine our results to study 1 the superconducting and excitonic interactions together as the + + ˜ 2 − ˜ chemical potential increases. dy y σ μ , (30) μ2−σ 2 5.1. λexc = 0, λsc = 0 and this equation becomes In the absence of the superconducting interaction, the super- ˜ 2 3 3 conducting order parameter is zero and Eq. (23) becomes (3) σ 2 μ V˜ = − ˜ 2 + 1 2 − 1 − + ˜ 2. (31) eff ˜ σ μσ α λexc 3 3 V˜ ≡ V (λ = 0) eff 3 eff sc Λ (1) In the first case, the function V˜ is minimized by 1 eff 2 σ˜ 1 2 = − dy y + σ˜ 2 + lμ˜ + , (24) ˜ λ˜ 2 3 0, λexc < 1, exc l σ˜ = (32) 0 0 1 ˜ − ˜ −1 ˜ 2 (λexc λexc), λexc 1. ˜ where λexc = λexc/λc and the quantities within the modulus are divided by Λ and are denoted by σ˜ and μ˜ , since we have made Notice that this result does not depend of a given chemical po- the replacement y = x/Λ2 in the above integral, as in the previous tential, since the effective potential is μ-independent. In the sec- section. ˜ (2) ˜ = ond case, V eff is minimized by σ 0. In the third case, we do | + ˜ 2 + ˜ |= + ˜ 2 + ˜ (3) Since μ > 0, we have that y σ μ y σ μ,for not find a nonzero minimum for V˜ as well, but only a max- every positive value of σ˜ , μ˜ , y ∈[0, 1] and l in the above equation. eff imum at the physical value of (λ−1 + μ)2 − 1. Therefore, the For l =−1, on the other hand, according to the modulus definition, √ ˜ + ˜ 2 we have system does√ not present excitonic fluctuations when μ > 1 σ ˜ ˜ + ˜ 2 or σ < μ < 1 σ . y + σ˜ 2 − μ˜ , μ˜ y + σ˜ 2, Combining all the above results, we can obtain the phase dia- + ˜ 2 − ˜ = y σ μ (25) gram of the system for the excitonic fluctuations. −( y + σ˜ 2 − μ˜ ), μ˜ > y + σ˜ 2. ˜ Let us start by the case λexc < 1 and μ˜ = 0, in this case, ˜ = Hence, we have three different results for the effective potential: σ0 0, according to Eq. (32). Therefore, as the chemical potential increases, we can only get the second or the third situations for • ˜ ˜ ˜ (2) ˜ (3) μ < σ : the effective potential, V eff or V eff respectively. Hence, we never reach a nonzero minimum for the excitonic order parameter when- ˜ In this case, the modulus becomes ever λexc < 1 for all values of μ˜ . ˜ When λexc > 1, the analysis is subtler, but also straightforward. y + σ˜ 2 − μ˜ = y + σ˜ 2 − μ˜ , ∀y ∈[0, 1], (26) We start by the assumption that μ˜ = 0.Inthiscase,wedohavea nonzero minimum σ˜0 given by Eq. (32). Now, if we take a small and Eq. (24) yields value for μ˜ ,sayμ˜ = μ¯ ,asσ˜ increases, the effective potential (2) (1) passes from the second situation, V˜ , to the first situation, V˜ , ˜ 2 3 eff eff (1) σ 2 ˜ = ¯ V˜ = − ˜ 2 + 1 2 − ˜ 3 − 1 , (27) at the point σ μ and the effective potential can only reach a eff ˜ σ σ λexc 3 minimum if σ˜0 > μ¯ as σ˜ continues to increase. However, even if σ˜0 > μ¯ , the effective potential may or may not possess a mini- which is exactly the same expression given by Eq. (16) if one re- (1) ˜ ˜ mum, because the minimum for V˜ has to be smaller than any places σ˜ and λexc for ζ and λsc respectively. eff ˜ (2) ˜ ¯ ˜ ˜ (2) √ value of V , when σ < μ.Sinceλexc > 1, we have that V is a eff ˜ ¯ eff • μ˜ > 1 + σ˜ 2: monotonically increasing function for all σ0 < μ. Therefore, we get the following condition for the appearance of a minimum for the ˜ (1) ˜ = ˜ ˜ (2) ˜ = The modulus becomes effective potential: V eff (σ σ0) 1 and μ˜ > 0. L.H.C.M. Nunes et al. / Physics Letters A 376 (2012) 779–784 783

Fig. 1. Phase diagram of the system taking into account only the excitonic interac- ˜ Fig. 2. Plots of the superconducting gap 0 as a function of the chemical potential tion. The area below the curve indicates the presence of a nonzero excitonic order ˜ μ˜ for several values of λsc < 1. parameter in the system. The solid curve in the plot is given by the critical chemical potential in Eq. (33) ˜ optimum value of the chemical potential and 0 decreases as μ˜ ˜ (1) increases even further. Moreover, notice that the particular case when V (σ˜ = σ˜0)< ˜ eff For the case λsc < 1, starting at μ˜ = 0,thesystemisinthe ˜ (2) ˜ = ˜ V eff (σ 0) provides a critical value for the chemical potential, normal state. However, as μ increases, the system asymptotically becomes superconducting and the order parameter also increases 3 1 ˜ 1 3 2 1 2 3 up to a maximum value at an optimal chemical potential. As μ μ = −8 + − + 2 + + λ˜ 2 − λ˜ 3 . ˜ c 2 ˜ ˜ 3 ˜ 2 exc increases even further, 0 decreases and the energy gap displays 2 3 λexc λexc λexc a dome-shaped plot. Our results are shown in Fig. 2, which are (33) consistent with [25], where chiral and diquark condensates are cal- culated for two-color and two-flavor QCD in three dimensions. For As ˜ increases above the threshold ˜ , we can no longer find μ μ > μc a certain choice of parameters, the authors obtain the same dis- a minimum for the effective potential. Hence, provided ˜ 1, λexc > persion relation given by Eq. (12) and their numerical results for ˜ = is also a quantum critical point. μ μc also display a dome-shaped plot, as can be seen in Fig. 1 of [25], Fig. 1 shows the phase diagram as a function of ˜ and ˜ .The λexc μ for the choice of parameters I, referred as the weak-coupling case. area below the curve for ˜ indicates the region where the μc(λexc) A dome-like structure of the superconducting phase for strongly system presents excitonic fluctuations. To our best knowledge, this interacting two-dimensional Dirac fermions has been previously is the first time that this phase diagram was obtained. obtained in [26], where the authors observed a dome-shaped plot of the superconducting phase at intermediate filling fractions, sur- = = 5.2. λsc 0, λexc 0 rounded by the normal phase for fillings close to unity or zero, which is consistent to our results. In this section we analyze the phase diagram of the system tak- Since the energy gap and the superconducting critical temper- ing into account only the presence of superconducting interactions. ature Tc are proportional, our results qualitatively reproduce the In this case, Eq. (23) becomes superconducting phase diagram of several compounds, like 122 α pnictides and cuprate superconductors, where the critical temper- V˜ ≡ V (λ = 0) ature displays a characteristic dome as charge carriers are doped eff 3 eff exc Λ into the system. 1 We conclude this section with a final remark: contrary to what |˜ |2 2 1 √ = + − dy |˜ |2 + ( y + lμ˜ )2, (34) happens at μ = 0, where superconductivity appears in the system ˜ 3 2 λsc α only when λsc >λc , at finite chemical potential, we could always l 0 find a finite μ˜ that provided nonzero superconducting gaps, even where the relevant quantities are divided by Λ,asintheprevious at small values of λsc, as can be seen in Fig. 2, where plots of ˜ ˜ sections. This problem has been previously studied by some of us 0 as a function of μ˜ for small values of λsc are shown. This re- for μ = 0 [23], and some results for μ = 0 have been previously sult suggests that even for very small superconducting interaction reported considering the s-wave and an exotic p-wave pairing [24]. strengths, one can always find superconductivity as charge carriers For μ = 0, the system undergoes a quantum phase transition at are added to the system. the critical coupling λsc = α/Λ ≡ λc and the system becomes superconducting whenever λsc >λc . 5.3. λsc = 0, λsc = 0 ˜ Given values for μ˜ and λsc, we can look for positive numerical values of ˜ that minimize the effective potential, given by In the previous sections we have analyzed the conditions for Eq. (34). Once again, our analysis is constrained to positive values the appearance of the excitonic and superconducting order param- ˜ ˜ ˜ ˜ ˜ of μ,sinceV eff(μ) = V eff(−μ). eters considering each interaction separately. In this section, we ˜ We start our discussion considering the case λsc > 1. For μ˜ = 0, study the minima of the effective potential taking into account ˜ the minimum of the effective potential 0 is given by Eq. (18) both the interactions simultaneously and also the variation of the ˜ ˜ ˜ implying ζ0 → 0.Asμ increases, 0 reaches a maximum at an chemical potential. From Eq. (23), V eff can be written as 784 L.H.C.M. Nunes et al. / Physics Letters A 376 (2012) 779–784 α V˜ ≡ V (λ = 0,λ = 0) chemical potential, indicating the appearance of superconductivity eff 3 eff sc sc Λ as charge carriers are added to the system. We emphasize that this | ˜ |2 ˜ 2 → = + σ + 2 curve actually vanishes exponentially as μ 0, hence supercon- ˜ ˜ ductivity persists down to μ = 0, below some temperature. This is λsc λexc 3 in agreement with the fact that Cooper’s theorem must be valid Λ2 for μ = 0, when a Fermi surface builds up. 1 ˜ 2 − dy ||2 + y + σ˜ 2 + lμ˜ (35) Finally, we have analyzed the possibility of coexistence between 2 = l 0 Cooper pairs and excitons for μ 0 and we show that, even if the excitonic interaction strength is greater than the superconducting and, once again, we constrain ourselves to positive values of the interaction, as the chemical potential increases, superconductivity chemical potential, since V eff is even with respect to μ. tends to suppress the excitonic order parameter, even if λ˜ > λ˜ . ˜ = ˜ ˜ exc sc We start our analysis with μ 0, assuming that λexc > λsc > 1. We have considered here exclusively the zero temperature situ- As we have seen in Section 4, since the interaction strengths are ation. We are presently investigating finite temperature effects on different we cannot have coexistence of superconductivity and ex- ˜ ˜ this system and will report the results elsewhere. citonic fluctuations. Actually, since λexc > λsc,wehavezerosu- ˜ = ˜ 2 − ˜ perconducting order parameter and σ0 (λexc 1)/2λexc.When Acknowledgements μ˜ increases, we regain the radial symmetry of the minima for the effective potential; in other words, we recover the coexis- This work has been supported in part by CNPq, FAPEMIG and tence of superconductivity and excitonic order fluctuations even if ˜ ˜ FAPERJ. We would like to thank H.C.G. Caldas and A.L. Mota for λexc > λsc. As the chemical potential increases even more, an inter- discussions on related matters. ˜ ˜ esting result shows up: the minimum of V eff becomes σ0 = 0and ˜ afinite0, which means that the excitonic fluctuations were sup- References pressed and only superconductivity remains. 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