Excitonic gap generation in thin-film topological insulators
EExcitonic gap generation in thin-film topological insulators
Nat´alia Menezes , C. Morais Smith and Giandomenico Palumbo Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena,Utrecht University, Princetonplein 5, 3584CC Utrecht, the Netherlands (Dated: July 22, 2018)In this work, we analyze the excitonic gap generation in the strong-coupling regime of thin filmsof three-dimensional time-reversal-invariant topological insulators. We start by writing down theeffective gauge theory in 2+1-dimensions from the projection of the 3+1-dimensional quantumelectrodynamics. Within this method, we obtain a short-range interaction, which has the form ofa Thirring-like term, and a long-range one. The interaction between the two surface states of thematerial induces an excitonic gap. By using the large- N approximation in the strong-coupling limit,we find that there is a dynamical mass generation for the excitonic states that preserves time-reversalsymmetry and is related to the dynamical chiral-symmetry breaking of our model. This symmetrybreaking occurs only for values of the fermion-flavor number smaller than N c ≈ .
8. Our resultsshow that the inclusion of the full dynamical interaction strongly modifies the critical number offlavors for the occurrence of exciton condensation, and therefore, cannot be neglected.
I. INTRODUCTION
Topological materials are, nowadays, a rich and welldeveloped research field in condensed-matter physics.The study of two-dimensional (2D) topological systemsstarted in the early 80’s, with the experimental discoveryof the integer quantum Hall effect in GaAs [1]. There-after, the deep relation between this novel phase andthe topological invariant induced by a non-trivial Berryphase was theoretically unveiled [2]. An essential featureof these quantum states is that time-reversal symmetryis broken in the bulk. However, the recent discovery of2D two-dimensional topological insulators (TIs) [3–6] hasopened the way to the exploration and classification ofa vast number of novel materials, also in higher dimen-sions. In 3D, similar versions of 2D TIs have been firstlytheoretically formulated [7] and then experimentally dis-covered [8, 9]. These systems support surface gaplessmodes, topologically protected by the non-trivial topo-logical number in the gapped bulk.Although the free-fermion topological phases havebeen completely classified for all dimensions in terms oftheir symmetries [10, 11], much less is known about thecomplete classification and characterization of interact-ing systems, where a variety of quantum phenomena andquasi-particles emerge in the low-energy regime. Thisis the case of anyons in fractional quantum Hall states[12–14] and fractional topological insulators [15, 16],which carry fractional electric charge and spin, Cooperpairs (bound states of spin-up and spin-down electrons)in topological superconductors [17], and excitons, i.e.particle-hole bound states in bilayer systems [18–22]. Atthe microscopic level, Hubbard-like Hamitonians havebeen employed in the study of exciton condensation inmonolayer [23] and bilayer graphene [24], bilayer quan-tum Hall systems [18, 25, 26] and in 3D thin-film TIs inthe class AII [27–29]. In the latter case, the electron-holepairs residing on the surface states can condense to forma topological exciton condensate. This kind of condensa-tion can be seen as an electronic superfluid with dissipa- tionless electronic transport and could enable ultra-low-power and energy-efficient devices, as already proposedin Ref. [30]. At a theoretical level, mean-field theorystudies show the presence of an excitonic gap induced bythe short-range part of the Coulomb interaction betweenthe surface states [27].
FIG. 1. The surfaces of a 3D TI separated by a distance d . In this paper, we propose a precise and self-consistentderivation of the gauge theory describing the short-rangeinteraction in thin films of TIs. In these materials,the free-surface states are defined in terms of masslessDirac fermions and the corresponding interactions areencoded in quantum electrodynamics (QED). Our theo-retical model is based on the fact that the massless Diracfermions are confined on the 2D surfaces, while the vir-tual photons that mediate their quantum electromagneticinteractions are free to propagate in the 3D surroundingspace. This approach has been already successfully em-ployed in the study of several quantum systems, such asgraphene [31, 32], transition-metal dichalcogenides [33],and the edge modes of 2D TIs [34]. The local part ofour effective-field theory is given by a generalized 2+1-D Thirring model, which has important applications inboth condensed-matter and particle physics [35–38], andrepresents one the main results of this paper. Impor-tantly, our approach fixes uniquely the value of its cou-pling constant, which turns out to be proportional to theelectric charge and the width of our thin-film TI. a r X i v : . [ c ond - m a t . s t r- e l ] J un Moreover, if on one hand our work reproduces the ef-fective local Hubbard-like model proposed in Ref. [27], onthe other hand it does not require any mean-field theoryapproximation for the identification of the exciton massgap. By solving the Schwinger-Dyson equation [39] forthe 2+1-D effective field theory in the strong-couplingregime, we show that the mass generation in the exci-ton condensation is induced dynamically. The dynam-ical mass generation is due to the breaking of the chi-ral symmetry [40–43], and represents a non-perturbativephenomenon, beyond the standard mean-field theory.
II. THE MODEL
We start our analysis with the description of two gap-less surface states in 3D thin-film TIs in class AII. Theysupport an odd number of topologically protected helicalmassless Dirac fermions, which are described by a 2+1-DDirac theory. We then consider the interactions in andbetween the two surfaces by including a quantum dy-namical U(1) gauge field coupled to the Dirac fermions.This is encoded in the standard QED by introducing aminimal coupling between the gauge potential A µ andthe fermionic current J µ . Importantly, while the maslessfermions are confined on the surfaces of the material, thevirtual photons that carry the electromagnetic interac-tion are free to propagate in the 3D space. This is thecrucial assumption that will allow us to derive an effec-tive 2+1-D projected theory. Thus, for simplicity, weconsider a single Dirac fermion per surface, such thatour system is described by the following QED-like action S = i (cid:126) (cid:90) d r (cid:0) ¯ ψ t ¯ σ µ ∂ µ ψ t + ¯ ψ b σ µ ∂ µ ψ b (cid:1) − (cid:90) d r (cid:16) ε c F αβ F αβ + eJ α A α (cid:17) , (1)where ψ b and ψ t denote fermionic fields with ¯ ψ i = ψ † i σ ,which are constraint to propagate on the top (t) and bot-tom (b) surfaces of the TI, respectively. Here, σ µ are 2 × µ = 0 , ,
2, and we adopt ¯ σ µ = − σ µ ,meaning that the two fermions have opposite helicity.The differential elements are given by d r = v dx dy dt and d r = c dx dy dz dt , with v and c the Fermi velocityand the speed of light, respectively. The coupling con-stant between the matter current and the gauge field e is the electric charge carried by each fermion. ε is thevacuum dielectric constant, F αβ = ∂ α A β − ∂ β A α is thefield-strength tensor, J α = j α t + j α b = ¯ ψ t σ α ψ t + ¯ ψ b σ α ψ b ,and α, β = 0 , , , ψ t , b , which in our context representquasi-particles and quasi-holes confined on two differentsurfaces. As illustrated in Fig. 1, the surfaces of the 3DTI are separated by a distance d , which is the width of thethin-film, and we describe the surface Dirac fermions by imposing the following constraints on the matter current j α t , b ( t, x, y, z ) = (cid:40) j µ t ( t, x, y ) δ ( z − d/ ,j µ b ( t, x, y ) δ ( z + d/ . (2)Because the fermions interact with a dynamical quantumelectromagnetic field, we can integrate out the gauge fieldto obtain the effective non-local interaction term S effint = − e (cid:90) d rd r (cid:48) J α ( r ) 1( − (cid:3) ) J α ( r (cid:48) ) . (3)By imposing the constraints given in Eq. (2) we areeffectively describing the system as a single surface livingin the middle of the thin-film. Hence, Eq. (3) becomes S effint = − e (cid:90) d rd r (cid:48) j µκ ( r ) V κρ ( r − r (cid:48) ) j ρµ ( r (cid:48) ) , (4)where V κρ ( r − r (cid:48) ) = [1 / ( − (cid:3) )] ξ κρ , κ, ρ = t , b and ξ κρ rep-resents the different values at which the Green’s functionhas to be evaluated.Although the system from now on may be treated asan effectively two-dimensional surface, the informationabout the thin-film width d is carried within the projec-tion. As known in the literature [28, 29, 44], the excitoncondensation in thin-films may only occur when the inter-surface distance d is smaller than an in-plane distance a ,i.e. d/a <
1. We introduce this minimal in-plane dis-tance a in our model by shifting the coordinates of thequasiparticles as follows: r → r − a/ r (cid:48) → r (cid:48) + a/ S effint = − e (cid:90) d rd r (cid:48) j µκ ( r − a/ V κρ ( r − r (cid:48) − a ) j ρµ ( r (cid:48) + a/ , (5)and now the effective interaction carries the informationabout the length a .The explicit values of ξ κρ are ξ tt : z = z (cid:48) = d/ , ξ tb : z = d/ z (cid:48) = − d/ ,ξ bt : z (cid:48) = d/ z = − d/ , ξ bb : z = z (cid:48) = − d/ , where, after the projection, the top and bottom compo-nents represent two different flavors in the effective mid-dle plane. For both ξ tt and ξ bb , we obtain similar resultsas found in Ref. [45], namely (cid:20) − (cid:3) ) (cid:21) ξ ii = 12 (cid:90) d k (2 π ) e ik · ( r − r (cid:48) − a ) √ k = 14 π ( | r − r (cid:48) − a | + a ) , (6)where a settles a minimum distance between the quasi-particles, implying a cutoff on the momenta k max = 1 /a .The terms ξ tb and ξ bt yield (cid:20) − (cid:3) ) (cid:21) ξ ij = 12 (cid:90) d k (2 π ) e − d √ k e ik · ( r − r (cid:48) − a ) √ k . (7)Now, by considering that d | k | < − d | k | ) ≈ − d | k | and perform theintegration over k to find (cid:20) − (cid:3) ) (cid:21) ξ ij ≈ π ( | r − r (cid:48) − a | + a ) − d δ ( r − r (cid:48) − a ) . (8)Here, we used the approximation (cid:90) d k (2 π ) e ik · ( r − r (cid:48) − a ) ≈ δ ( r − r (cid:48) − a ) . (9)We can finally summarize the results for the effectiveinteraction V κρ after the projection, V tt = V bb = 14 π | r − r (cid:48) − a | ,V tb = V bt ≈ π | r − r (cid:48) − a | − d δ ( r − r (cid:48) − a ) . where we neglected terms proportional to a ≈
0. Byplugging back the interactions above into Eq. (5), we maywrite down S effint as a long and a short-range contribution(see Appendix A for details). III. SINGLE-SURFACE DESCRIPTION
The aim of this section is to describe a two-surface sys-tem in terms of a single effective surface with two speciesof fermions. Our 2+1-D effective action after the projec-tion is given by S eff = i (cid:126) (cid:90) d r (cid:0) ¯ ψ t σ µ ∂ µ ψ t − ¯ ψ b σ µ ∂ µ ψ b (cid:1) − e ε c (cid:90) d r (cid:48) (cid:90) d r j µκ V κρ j ρµ . (10)where κ, ρ = t , b represent the different surfaces. Now,we can rewrite the action (10) in terms of a single spinorΨ = ( ψ t , ψ b ) (cid:62) . For the kinetic part, we obtain¯ ψ t σ µ ∂ µ ψ t − ¯ ψ b σ µ ∂ µ ψ b = ¯Ψ γ µ ∂ µ Ψ , (11)where the 4 × γ -matrices are defined as [38] γ µ = (cid:18) σ µ − σ µ (cid:19) , with γ = (cid:18) σ − σ (cid:19) , γ τ = i (cid:18) σ τ − σ τ (cid:19) . Here, τ = 1 , γ µ ≡ σ ⊗ σ µ , and ⊗ represents the tensorproduct. The fermionic currents can be written in termsof the new spinors j µ t = 12 ¯Ψ( + σ ) ⊗ σ µ Ψ , (12) j µ b = 12 ¯Ψ( − σ ) ⊗ ¯ σ µ Ψ , (13) where ⊗ σ µ = − iγ µ γ γ , with γ = i (cid:18) − (cid:19) , γ = iγ γ γ γ = (cid:18) (cid:19) . Once we have expressed all contributions to the effectiveaction (10) in terms of four-component spinors ¯Ψ and Ψ,we can write down the following single-surface action S eff [ ¯Ψ , Ψ] = e ε c (cid:90) d r (cid:48) (cid:90) d r J µ π | r − r (cid:48) | J µ + (cid:126) (cid:90) d r (cid:20) i ¯Ψ γ µ ∂ µ Ψ + e d (cid:126) ε c (cid:0) J µ J µ + J µ J µ (cid:1)(cid:21) , (14)where J µ ≡ ¯Ψ γ µ Ψ and J µ ≡ ¯Ψ γ µ γ γ Ψ. IV. DYNAMICAL GAP GENERATION
In the previous section, we derived an effective single-surface interacting model (see Eq. (14)), which involvesboth a short- and a long-range interaction. The formercorresponds to a generalized Thirring model [37, 42],while the latter is similar to the non-local field theorystudied in Refs. [31, 33, 43]. These kind of interactionshave been already studied separately in the context ofdynamical mass generation in Refs. [40–43]. This mech-anism is relevant in interacting quantum-field theoriesand is related to the dynamical breaking of a classicalsymmetry due to quantum effects. In fact, all the threeinteraction terms in our effective action (14) are invari-ant under chiral symmetry, which is dynamically brokenat the quantum level. In the first part of this section, wewill focus on the short-range interactions J µ J µ + J µ J µ .By following the approach developed in Ref. [40], we willshow that in the strong-coupling regime both Thirring-like terms yield the same mass generation, and their com-bined action leads to a larger critical number of fermionflavors N c , as compared to a single Thirring term. Atlast, we will add the long-range interaction and showthat the excitonic gap is then enhanced, in agreementwith the results found in Refs. [23, 47] for the case ofGross-Neveu theory. A. Short-range interactions
Firstly, let us focus on the dynamical mass generateddue to the Thirring-like interactions of Eq. (14). In thelarge- N approximation, we can write down the effectiveLagrangian as L eff [ ¯Ψ , Ψ] = i (cid:126) ¯Ψ a γ µ ∂ µ Ψ a + g N (cid:0) ¯Ψ a γ µ γ γ Ψ a ¯Ψ ¯ a γ µ γ γ Ψ ¯ a + ¯Ψ a γ µ Ψ a ¯Ψ ¯ a γ µ Ψ ¯ a (cid:1) , where g = e dN/ ε c . Here the indexes a, ¯ a denote asum over N fermion flavors.Through a Hubbard-Stratonovich transformation, weintroduce two auxiliary vector fields W µn ( n = 1 ,
2) andtwo scalar fields φ n in a way to preserve gauge symmetry.Thus, we obtain L eff [ ¯Ψ , Ψ , W , W , φ , φ ] = i (cid:126) ¯Ψ a γ µ D µ Ψ a − (cid:88) n =1 , g (cid:16) W µn − √ N ∂ µ φ n (cid:17) , (15)where D µ = ∂ µ − ( i/ √ N ) γ γ W µ − ( i/ √ N ) W µ . By fol-lowing a similar procedure as adopted in Ref. [40], weintroduce a non-local gauge-fixing term of the form − (cid:20) ∂ µ W µ + √ N ζ ( ∂ ) g φ (cid:21) ζ ( ∂ ) (cid:20) ∂ ν W ν + √ N ζ ( ∂ ) g φ (cid:21) for each gauge field W µn in the Lagrangian (15). As aresult, we obtain L eff [ ψ, ¯ ψ, W , W ] + L eff [ φ , φ ] = i (cid:126) ¯Ψ a γ µ D µ Ψ a − g W nµ W µn − ∂ µ W µn ζ ( ∂ ) ∂ ν W νn − g (cid:2) ζ ( ∂ ) φ n (cid:3) φ n − ∂ µ φ n ∂ µ φ n , (16)where the gauge-fixing term decoupled the φ -boson fields,which have also been rescaled as (cid:112) N/gφ n → φ n . Thedouble index n indicates a summation over the fields.Notice in Eq. (16) that only the strong-coupling regime g → ∞ preserves gauge symmetry, leading to a masslessgauge boson. We shall return to this point later in theSchwinger-Dyson analysis.Once we have obtained the gauge theory in Eq. (16), weproceed by defining the Feynman rules needed for calcu-lating the mass generation. The full fermion propagatorreads S ( p ) = iA ( − p ) γ µ p µ − B ( − p ) , (17)where A represents a correction to the fermion-field wavefunction, and B is the order parameter of the chiralsymmetry, which preserves parity in 2+1 dimensions.The Schwinger-Dyson equation for the fermion two-pointfunction is given by S − ( p ) = S − ( p ) − i Σ( p ) , (18)where S = i/γ µ p µ is the free-fermion propagator. Theself-energy Σ contains the contribution from both typesof local interaction, and it is determined by − i Σ = − N (cid:90) d k (2 π ) γ µ γ γ S ( k )Γ ν γ γ G µν ( p − k ) − N (cid:90) d k (2 π ) γ µ S ( k )Γ ν G µν ( p − k ) . (19)Γ ν and G nµν are the full-vertex function and the fullgauge-boson propagators, respectively. Here, we willadopt the bare-vertex approximation, i.e. Γ ν = γ ν . Theexplicit expression for the full gauge-boson propagatorreads G nµν ( k ) = iG n ( − k ) (cid:18) g µν − η ( − k ) k µ k ν k (cid:19) , (20) where G = 1 / ( g − − Π), G = 1 / ( g − + Π), and η is anon-trivial function of the momentum related to the non-local gauge approximation [40]. The function Π( − k )emerges from the one-loop polarization tensor, inducingdynamics to the gauge fields W nµ through interaction ef-fects.In the strong-coupling regime ( g → ∞ ), both contri-butions in Eq. (19) reduce to a single term. By replacingthe respective Γ ν and G nµν functions into Eq. (19) andusing that [ γ µ , γ γ ] = 0, we obtain[ A ( p ) − γ µ p µ − B ( p ) =2 N (cid:90) d k (2 π ) γ µ ( Aγ α k α + B ) γ ν ( A k + B )Π( q ) (cid:18) g µν − η q µ q ν q (cid:19) , (21)where q = p − k . We also performed a transformation tothe Euclidean space ( k → ik E ).By taking the trace over γ -matrices in Eq. (21), weobtain two coupled equations: one related to the renor-malization of the fermion wavefunction and another re-lated to the generation of the fermionic mass. Within thenon-local gauge-fixing picture, the fermion wavefunctionis not renormalized. This means that A ( p ) = 1, and itleads to both0 = 2 N p (cid:90) d k (2 π ) k + B )Π [( η − p · k − η ( k · q )( p · q ) q (cid:21) , (22)and B = 2 N (cid:90) d k (2 π ) B (3 − η )( k + B )Π , (23)where Eq. (22) is used to determine η ( q ). After somecalculations, one finds that in the massless gauge bo-son limit g → ∞ , η = 1 / q ) (cid:54) = 0, as seen in Eq. (23). Hence,the quenched approximation Π( q ) = 0 sometimes usedin the literature [43] to simplify calculations can only beused here in the case of a massive gauge boson.We proceed with the computation by considering themassless gauge boson limit with η = 1 /
3, which yields B = 1283 N (cid:90) d k (2 π ) B ( k + B ) (cid:112) ( p − k ) , (24)where we used Π( q ) = (cid:112) q /
8. The integrals over k inEq. (24) are performed in spherical coordinates. We firstintegrate over the solid angle, and then split the remain-ing integral over positive values of k into two regions, B = 643 π N (cid:26)(cid:90) p dk k B ( k ) k + B ( k ) 1 | p | + (cid:90) Λ p dk k B ( k ) k + B ( k ) 1 | k | (cid:41) , (25)where the virtual-momentum k is, respectively, less orgreater than the external momentum p . Here, Λ is acutoff and p = | p | . Now, we transform the integralEq. (25) into a differential equation, and by considering p + B ( p ) ≈ p , we obtain p d Bdp + 2 p dBdp + 643 π N B = 0 . (26)The solution of Eq. (26) reads B ( p ) = (cid:114) mp (cid:104) C cos (cid:16) λ ln pm (cid:17) + iC sin (cid:16) λ ln pm (cid:17)(cid:105) , (27)where we have introduced the infrared parameter m suchthat the ratio p/m is dimensionless and the solutionobeys the normalization condition B ( m ) = m . C and C are coefficients to be determined according to the ul-traviolet (UV) and infrared (IR) boundary conditions.The parameter λ indicates the behavior of the solutionsof Eq. (26), and it is given by λ = 12 (cid:114) π N − . (28)We see in Eq. (28) that there is a critical value N c =256 / π ≈ . with a non-localgauge fixing. For values of N > / π , the solutions inEq. (27) are real exponentials, with a contribution thatincreases in the UV limit. Hence, the only possible solu-tion in this regime is B ( p ) = 0 (trivial solution; no massgeneration) [48]. For N < / π , we obtain the oscil-latory solutions (27). This implies that B ( p ) (cid:54) = 0, andconsequently, the chiral symmetry has been broken bythe dynamical generation of a fermion mass.The IR and UV boundary conditions are, respectively, (cid:20) dB ( p ) dp (cid:21) p = m = 0 , and (cid:20) p dB ( p ) dp + B ( p ) (cid:21) p =Λ = 0 . (29)The IR condition yields a relation between the coeffi-cients C and C , C = 2 iλC . By using this result inthe UV condition, we obtain an expression for mm = Λ exp (cid:20) − λ arctan (cid:18) λ λ − (cid:19)(cid:21) . (30)The solution (27) can be rewritten as B ( p ) = m F (cid:16) pm , λ (cid:17) , (31)with F (cid:16) pm , λ (cid:17) = (cid:114) mp (cid:20) cos (cid:16) λ ln pm (cid:17) + 12 λ sin (cid:16) λ ln pm (cid:17)(cid:21) . So far, we have shown that the Thirring-like interac-tions derived within the dimensional-reduction methodbreak the chiral symmetry and generate a mass in thefermionic sector with a critical number N c that is twice the value of the standard Thirring model derived inRef. [40]. This makes sense in the strong-coupling regimebecause the contributions of both Thirring-like inter-actions sum up, yielding the multiplicative factor 2 inEq. (21). B. Long-range interaction
At last, we investigate the effect of the long-range in-teraction in the strong-coupling regime. First, we rewritethe long-range interaction of Eq. (14) in terms of a gaugetheory, e.g. H µν √ (cid:3) H µν + ¯ gh µ J µ , (32)where H µν = ∂ µ h ν − ∂ ν h µ and ¯ g is the coupling constant.This non-local gauge theory is similar to the one studiedin Ref. [43], where the authors also showed the breakingof chiral symmetry.By adding the contribution of the long-range interac-tion to Σ( p ) and following a standard procedure, we ob-tain a differential equation similar to Eq. (26), but witha different coefficient multiplying the fuction B ( p ). Inother words, we obtain a different parameter λ , namely λ (cid:48) = 12 (cid:115) N (cid:18) π + 8 π (cid:19) − , (33)where 32 /N π is the long-range contribution. The newparameter λ (cid:48) leads to a critical number N c = 352 / π ≈ .
8. Thus, the difference between the effect caused bythe short- and the long-range interaction is mainly as-sociated to the critical number of fermions (or criticalcoupling) below which the symmetry is dynamically bro-ken.Our results show that the short-range interactionyields the major contribution to the dynamical mass gen-eration when compared to the long-range one. However,both interaction effects add up in a way to increase thevalue of the critical fermion flavor N c for the occurrenceof exciton condensation. This dynamical mechanism isdriven mainly by the presence of electronic interactions between the surfaces of 3D TI thin-films, and is robustonly when the surfaces are strongly interacting. The re-sulting gap is time-reversal invariant and represents asignature of excitonic bound states. C. Application: Bi Se thin-film Here, we apply our theoretical results about the dy-namical gap generation to Bi Se thin films. This ma-terial is one of the most investigated three-dimensionaltopological insulators [9, 49], together with Bi Te [50].Experimentally, the size of the gap depends on the ma-terial, on the thickness of the film, and on the substratewhere the material is grown. In particular, the widthof the sample drives the transition from a trivial insula-tor to a quantum spin Hall insulator, up to the limit inwhich the material presents the characteristics of a truethree-dimensional topological insulator. This transitionhas been theoretically and experimentally investigated inRef. [9].In our manuscript, to describe these thin films, weadopted the regime where the distance between the sur-faces d – the width of the 3D TI – is smaller than the in-plane average separation a between electrons and holes.In general, one would not expect interactions betweenthe surfaces of a 3D TI because of the high values ofthe bulk dielectric constant. However, the bulk dielectricconstant depends on the thickness of the material anddecreases for thinner samples [51, 52]. In this limit, theeffect of electronic interactions becomes relevant. As wehave shown, in the strong coupling regime there is a gapgeneration in each of the surfaces.Within these assumptions, by using Eq. (30) we areable to estimate the excitonic gap generated at zero tem-perature. This estimative depends on the material anddielectric constant of the substrate via the cutoff Λ, whichin the case of Bi Se , for a single Dirac mode ( N = 1),is 0 . λ (cid:39) .
65 and determine the maximumvalue for the gap, m ≈ .
07 eV, arising from the elec-tronic interactions. Interestingly, this value agrees withthe gap measured through ARPES for a thin-film thick-ness of 4 nm in Bi Se [9]. V. CONCLUSIONS
It was theoretically proposed that the excitonic boundstates at zero magnetic field may have important techno-logical applications such as for dispersionless switchingdevices [53], or in the design of topologically protectedqubits [54], or in heat exchangers [30]. It is also wellknown that TI-based electronic devices are attractive asplatforms for spintronic applications. In this work, weprovide further theoretical support for exciton condensa-tion in thin-film 3D TIs by investigating the influence ofelectromagnetic interactions in these systems.We started by considering that the photons propa-gate through the 3D surrounding space where the ma-terial is immersed, while the mobile electrons propagateon the two 2D surfaces of the 3D TI. Upon project- ing the photon dynamics to these two 2D surfaces, wefound the effective intra- and inter-surfaces interactionin the system. The problem was then mapped into asingle surface one, in which the top and bottom layersappear as flavors of a single fermionic spinor. Within asingle-surface picture, we showed that the fermions in-teract via two effective short-range and one long-rangeinteraction terms. By using a Hubbard-Stratonovichtransformation, we introduced the corresponding effec-tive gauge theory and analyzed the dynamical gap gener-ation through the Schwinger-Dyson equation. This gapterm is time-reversal invariant and is associated to thechiral symmetry breaking.Our results indicate that the combined effect of short-and long-range interactions that emerge from project-ing QED enhance the value of the critical fermion fla-vor number N c in comparison to models that only in-clude short- or long-range interaction. They also confirmthe existence and robustness of excitonic bound states inthin-film TIs in the non-perturbative regime. Notice thatthese results are achieved in the strongly-coupling regime,which is usually difficult to access with analytic tech-niques due to the failure of the standard perturbation-theory approach.The method used here can be extended to multi-layer systems, which involve a larger number of fermionspecies. This will allow one to analyze the chiral-symmetry breaking and dynamical mass generation in ex-perimentally available samples of multi-layered Dirac ma-terials. At present, the multi-layer samples are of higherquality than the corresponding single-layer ones, and itis therefore essential that theoretical investigation tacklethose more complex, multi-flavor systems. Furthermore,the same method can be used to study lower-dimensionalexcitonic bound states, which have been recently pro-posed in two parallel nanowires [55]. This problem willbe analyzed in future work. ACKNOWLEDGMENTS
This work was supported by CNPq (Brazil) throughthe Brazilian government project Science WithoutBorders. We are grateful to S. Kooi, S. Vandoren, E. C.Marino for fruitful discussions.
Appendix A: Effective interactions after projection
After the projection, we obtain the following interaction terms V tt = V bb = 14 π | r − r (cid:48) − a | ,V tb = V bt ≈ π | r − r (cid:48) − a | − d δ ( r − r (cid:48) − a ) . where a ≈
0. By plugging back these results into Eq. (5), we find S effint = − e (cid:90) d rd r (cid:48) j µt,b ( r − a/
2) 14 π | r − r (cid:48) − a | j t,bµ ( r (cid:48) + a/ − e (cid:90) d rd r (cid:48) j µt,b ( r − a/ (cid:20) π | r − r (cid:48) − a | − d δ ( r − r (cid:48) − a ) (cid:21) j b,tµ ( r (cid:48) + a/ − e (cid:90) d rd r (cid:48) j µt,b ( r − a/
2) 14 π | r − r (cid:48) − a | j t,bµ ( r (cid:48) + a/ (cid:124) (cid:123)(cid:122) (cid:125) r → r + a/ r (cid:48) → r (cid:48) − a/ − e (cid:90) d rd r (cid:48) j µt,b ( r − a/
2) 14 π | r − r (cid:48) − a | j b,tµ ( r (cid:48) + a/ (cid:124) (cid:123)(cid:122) (cid:125) r → r + a/ r (cid:48) → r (cid:48) − a/ + e d (cid:90) d rj µt,b ( r + a/ j b,tµ ( r + a/ (cid:124) (cid:123)(cid:122) (cid:125) r → r − a/ = − e (cid:90) d rd r (cid:48) j µt,b ( r ) 14 π | r − r (cid:48) | j t,bµ ( r (cid:48) ) − e (cid:90) d rd r (cid:48) j µt,b ( r ) 14 π | r − r (cid:48) | j b,tµ ( r (cid:48) ) + e d (cid:90) d rj µt,b ( r ) j b,tµ ( r ) . (A1) Appendix B: η -function in the strong coupling regime By rewriting Eq. (22) of the main text in spherical coordinates, we obtain0 = 1
N p (cid:90) ∞ k dk (2 π ) k + B (cid:90) π dθ sin θ × (cid:2) f ( q , k , p ) cos θ − f ( q , k , p ) sin θ (cid:3) , (B1)where f ( q , k , p ) ≡ ˜ G ( q )( η + 1) (cid:112) k p , and f ( q , k , p ) ≡ ˜ G ( q )2 ηk p q . Here, we denote ˜ G = lim g →∞ G , in the massless gauge boson limit. Now, we integrate by parts the first integralover θ in Eq. (B1), which yields (cid:90) π dθ sin θ cos θf = − (cid:90) π dθ sin θ d ˜ f dq , (B2)where we used that q = p + k − (cid:112) k p cos θ and ˜ f = (cid:112) k p f . Replacing the result (B2) into Eq. (B1), we find0 = 1 N (cid:90) ∞ dk (2 π ) k k + B × (cid:90) π dθ sin θ (cid:40) d [( η + 1) ˜ G ] dq + 2 η ˜ G q (cid:41) , (B3)with d [( η + 1) ˜ G ] dq + 2 η ˜ G q = 1 q (cid:34) d ( η ˜ G q ) dq + q d ˜ G dq (cid:35) . Thus, η satisfies the following differential equation d ( η ˜ G q ) = − q d ˜ G dq dq , and η ( q ) = 2˜ G ( q ) q (cid:90) q ˜ G ( ζ ) ζ dζ −
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