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https://doi.org/10.1038/s41467-018-08203-9 OPEN Prediction of a topological p + ip excitonic with parity anomaly

Rui Wang1,2, Onur Erten3, Baigeng Wang1 & D.Y. Xing1

Excitonic insulators are insulating states formed by the coherent condensation of and hole pairs into BCS-like states. Isotropic spatial wave functions are commonly considered for excitonic condensates since the attractive interaction among the and the holes s 1234567890():,; in usually leads to -wave . Here, we propose a new type of excitonic insulator that exhibits order parameter with p + ip symmetry and is characterized by a chiral

Chern number Cc = 1/2. This state displays the parity anomaly, which results in two novel topological properties: fractionalized excitations with e/2 charge at defects and a sponta- neous in-plane magnetization. The surface state is a promising platform to realize the topological excitonic insulator. With the -momentum locking, the interband optical pumping can renormalize the surface electrons and drive the system towards the proposed p + ip instability.

1 National Laboratory of State Microstructures, Collaborative Innovation Center of Advanced Microstructures, and Department of , Nanjing University, Nanjing 210093, China. 2 Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China. 3 Department of Physics, Arizona State University, Tempe, AZ 85257, USA. Correspondence and requests for materials should be addressed to O.E. (email: [email protected]) or to B.W. (email: [email protected])

NATURE COMMUNICATIONS | (2019) 10:210 | https://doi.org/10.1038/s41467-018-08203-9 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08203-9

assless Dirac particles preserve both parity and time- where we show that a p-wave-type interaction component emerges Mreversal symmetry (TRS). However, an infinitesimal intrinsically, which stabilizes the p + ip excitonic in a sig- mass will inevitability break these symmetries nificant parameter region of the phase diagram. Moreover, we also through the radiative correction1–3, resulting in an Abelian or consider the effects of interband optical processes and we nonAbelian Chern–Simons theory of the external gauge field. demonstrate that the electron– coupling can effectively This effect is called parity anomaly. with linear drive the system into the proposed p + ip excitonic insulator as a dispersion in have recently drawn result of the spin-flip mechanism demonstrated in Fig. 1b. broad attention4–13, and they provide a quintessential platform to test these effects. Indeed, the gapless Dirac of topological Results insulator (TI) surface states have been predicted to exhibit the Model. We start from a spontaneous symmetry breaking of 2D parity anomaly and a number of exotic phenomena when dif- Dirac fermions with spin-momentum locking and a momentum- ferent types of mass terms are considered, including the p + ip dependent mass term, H ¼ H þ M^ ðkÞ, where XDirac type topological superconductor induced by the superconductor y 14,15 ¼ Φ σ Á Φ : (SC) proximity effect , and the quantum anomalous Hall HDirac kvF k k ð Þ – 1 effect in the TI surface with a Zeeman field16 19. k In this work, we expand the realm of two-dimensional (2D) Φ = ψ ψ T topological states by devising a study on the particle–-hole k is the 2D wave vector, k [ k,↑, k,↓] is the fermionic spinor ψy ψ instability of Dirac fermions. We show that the formation and and k;σ ( k,σ) denotes the creation (annihilation) operator of condensation of excitons can lead to a novel topological excitonic electronsP with spin σ.Wefirstly focus on a concrete mass term ^ ð Þ = Φy ðÞΔ θσz þ Δ 2θσx À Δ θ θσy Φ insulator that exhibits the parity anomaly. Furthermore, in sharp M k k k c cos c sin csin cos k, contrast to conventional semiconductors, the 2D helical Dirac while under which conditions M^ ðkÞ can be spontaneously gen- θ Δ with spin-momentum locking (e.g., the TI surface), has an erated will be investigated below. is the polar angle of k, and c inherent tendency towards the formation of spin-triplet excitons. is the overall gap scale. It is straightforward to verify that M^ ðkÞ This remarkable property manifests itself most clearly by the breaks TRS, T^M^ ðkÞT^À1≠M^ ðÀkÞ, where T^ ¼ iσyK being the interband optical absorption processes, as shown by Fig. 1. time-reversal operator, with K the complex conjugation operator. For conventional semiconductors with absence of the In contrast with the TI surface with a Zeeman field, where the spin–orbit coupling, the direct interband excitation generates a z 23 diagonal term mσ breaks TRS , here the diagonal term Δc cos conduction electron and a valence hole with the same momentum θσz in M^ ðkÞ respects the TRS. The TRS is however violated by the k and the opposite spins (Fig. 1a), resulting in a spin-singlet two off-diagonal terms in M^ ðkÞ. after they form into a pair. The situation is, however, One can unveil the physical meaning of M^ ðkÞ by making a different in TI surfaces, as shown by Fig. 1b. Due to the spin- unitary transformation onto the chiral basis with H being momentum locking of TI surface, the direct interband excitation is Dirac diagonalized: C ¼ R^ðkÞΦ , where R^ðkÞ is a 2 by 2 rotation accompanied by the spin-flip of the electron; the resultant exciton k k pffiffiffi pffiffiffi ^ ¼À^ ¼ iθ= ^ ¼ ^ ¼ = = matrix with R11 R21 e 2, R12 R22 1 2, and Ck consists of an electron and a hole with the same spin polarization, T and therefore a spin-triplet state. This unique optical property of [ck,+, ck,−] is the spinor composed of the conduction and valence TI surface, as will be shown below, can be used for generation and band fermions. In the chiral basis, H becomes + !y ! ! stabilization of a unique p ip excitonic insulator. The natural X ÀΔ Àiθ + ck;þ vFk ce ck;þ topological p-wave excitonic insulator has p ip symmetry H ¼ : ð2Þ fi ÀΔ iθ À inherent to their order parameters. We rstly explore the salient k ck;À ce vFk ck;À feature of the p + ip excitonic state by studying a minimal theo- Δ −iθ retical model. In contrary to the conventional excitonic insula- The off-diagonal ce describes the process where an electron tors20 described by a s-wave Bardeen–Cooper–Schrieffe (BCS)-like in the valence band is scattered up to the conduction band, theory21,thep + ip state mathematically resembles the spinless leaving a particle–hole excitation on top of the many-body + 22 + Δ ≠ – p ip SC in the chiral basis. However, unlike the p ip SC, it ground state. For c 0, the particle hole pairs condense and gap does not break the global U(1) gauge symmetry but only violates out the Dirac cone, generating an excitonic insulator. Another ±iθ = the TRS. To characterize such exotic states, we propose a chiral interesting feature of the off-diagonal term is the factor e (kx 24 + 25 Chern number Cc and unveil the underlying parity anomaly by ±iky)/k , similar to the spinless p ip-wave SC , but we note deriving an effective Chern–Simons gauge theory associated with that the order parameter resides in the particle–hole rather than topological defects. This is further responsible for (i) fractionalized the particle–particle basis. The model Hamiltonian can be excitations with charge e/2 at vortices and (ii) a spontaneous in- analyzed similarly as the BCS superconductor theory. The ground plane magnetization of ground state. Then, we investigate the state wave function |Ψ〉 is obtained after a Bogoliubov excitonic instability of a strongly correlated TI surface model transformation as, P Y y gk ck;þck;À ji¼Ψ k ji; ð Þ uke 0 3 ab k

with g = v /u , u = Δ (k − ik )/kξ , v = v k/ξ , where ξ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik k k k c x y k k F k k spin-singlet, s-wave spin-triplet, p-wave Δ2 þ 2 2 Ψ〉 c vFk is the energy spectrum of quasiparticles. | clearly describes a coherent state of electron–hole pairs with the Semiconductors TI surface distribution factor gk of the p + ip symmetry. Fig. 1 Schematic plot of the direct interband optical absorption process. a The s-wave excitons are formed in the conventional semiconductors. b The Topological classification and chiral Chern number.To helical spin texture of topological insulator (TI) surface state favors p-wave examine the topological classification of the p + ip excitonic rather than s-wave excitons insulator, we write Eq. (2) into the compact form as,

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a 1.0 b 1.0 0.5 0.5 dy 0.0 dy 0.0 –0.5 –0.5 1.0 –1.0 1.0 –1.0 0.5 0.9

d dz z 0.0 0.8 –0.5 –1.0 0.7 0.0 0.5 1.0 –1.0 –0.5 –1.0 –0.5 0.0 0.5 1.0 dx dx ^ Fig. 2 The mapping of dðkÞ. a The mapping in the original spin basis. b The mapping in the chiral basis. The difference between the two situations shows that the two topological invariants are needed to completely characterize the p + ip excitonic insulator, i.e., (C, Cc) = (0, 1/2) P H¼ 3 ð Þτi τi fi + i¼1 d k , with being the Pauli matrices de ned in the Recalling the p ip SCs where Majorana fermions are encoded in chiral basis. For the continuum model, where k → ∞ is envisioned vortices22, we shift our attention to topological defects in the p + = −Δ θ −Δ θ as a single point, the three-vector d(k) ( c cos , c sin , ip excitonic state. To this end, we can solve the Bogoliubov de fi 2 vFk)denes a mapping from the k-space (S ) to the d-vector Gennes (BdG)-type equations looking for zero-energy solutions. space. Since the z-component v k ≥ 0, the normalized d-vector In real space, the equations can be derived as (see Methods for ^ F ^ dðkÞ¼dðkÞ=jjdðkÞ resides in a hemisphere surface. For dðkÞ details): covering the hemisphere n times, the winding number becomes ÀÁ iα À ∂ þ À1∂ ð ÞÀ 1 Δ ð Þ ð Þ¼ ; ð Þ n/2. This winding number is a topological invariant defined in the e i r r α u r c r v r 0 4 = T vF chiral basis Ck [ck,+, ck,−] , which we term as the chiral Chern number Cc. fi ÀÁ Recalling that for the TI surface with a Zeeman eld, a Àiα À ∂ À À1∂ ð Þþ 1 ΔÃð Þ ð Þ¼ ; ð Þ fi 16,23 e i r r α v r c r u r 0 5 conventional Chern number C is de ned in the same way but vF Φ = ψ ψ T in the spin basis, k [ k,↑, k,↓] . The difference between C and Cc becomes clear from the perspective of Berry phase as where α is the polar angle in real space and u, v are the two following. In the original spin basis, following a closed loop in components of the spinor wave function. Similar equations have momentumH space, the accumulation of Berry phase is been studied in refs. 22,29,30. However, note that the physics in the ϕ ¼À Á hj∇ ji ji i dk uk k uk , where uk is the eigenstate of the present model is different from previous works. Read and valence band in the spin basis. For a Bloch state defined in the Green22 derive the equations for SCs, where charge conservation chiral basis,H a similar geometric phase can be defined as is broken and Majorana fermions are the emergent excitations. ϕc ¼À Á c ∇ c c i dk uk k uk , where uk is the eigenstate of the For our system, the U(1) gauge symmetry is preserved, therefore valence band in the chiral basis. The Bloch states in the above two no Majorana modes are allowed. Hou et al.29 investigate graphene formalisms are related to each other by the unitary transforma- with a mixing texture which induces the mass term, and c ¼ ^ð Þji 30 tion uk R k uk . It thenH becomes obvious that the two Berry Seradjeh et al. discuss the exciton condensation between two ϕc − ϕ = À Á hj^yð Þ∇ ^ð Þji phases satisfy H i dk uk R k kR k uk , leading to opposite TI surfaces. In contrary, here we focus on the excitonic − = Àð = πÞ Á hj^yð Þ∇ ^ð Þji= Cc C i 2 dk uk R k kR k uk 1/2, after insert- order in a Dirac liquid state and the mass term is generated by the ^ð Þ ing the R k matrix we used to diagonalize HDirac. condensation of p + ip excitons. To solve the equations in the To explicitly demonstrate the topological invariants, we plot 31 Δ = Δ iα ^ presence of a , we assume c(r) c(r)e .A the normalized vector dðkÞ with traversing the whole k-space. 32,33 ^ unique normalized zero-energy bound state can be obtained Figure 2a, b shows the mapping of the dðkÞ vector in the spin and = i(α+π/2)/2 = * ^ as, u Af(r)e and v u , where A is the normalization chiral basis, respectively. It is found that dðkÞ in latter basis = − Δ ^ constant and the function f(r) exp[ | c|r/|vF|] decays completely covers the hemisphere while dðkÞ in the former does exponentially. = = not, leading to Cc 1/2 and C 0. Therefore, two topological We note that the single particle Hamiltonian h in Eq. (2) invariants are needed to completely characterize the p + ip satisfies (iτ2)h(iτ2) = h. This allows one to define a time-reversal- = excitonic state, namely, (C, Cc) (0, 1/2). The zero Chern type operator Θ = iτ2K, with which it is clear that for any = + ψ Θψ number C 0 means that the p ip excitonic insulator is eigenstate E with energy E, E is also a eigenstate with energy topologically distinct from the TI surface with a Zeeman field that −E. Besides,R the total number of states are conserved, = 16 1 has C 1/2 . This is clear from the Landau quantization of the namely, À1 dEδρðEÞ¼0, where δρ(E) is the change of density two states. For the TI surface withp a Zeemanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifield, the Landau of states. Applying this formula to the case where a topological ≥ ¼ 2 jjþ 2 level (LL) for |n| 1 reads as En ± 2evFBn m , where m is defect is in presence, and then taking into account the fact that ρ the Zeeman gap. In addition, there is an unconventional zeroth (E) is a symmetric function with respect to E = 0, one comes to = 26–28 LL E0 sgn(m)m which is located exactly at the gap edge . the conclusion that both the conduction and valence band lose This zeroth LL is a manifestation of the nontrivial Zeeman mass half a state. This state is compensated by the zero mode that is as a result of C = 1/227,28. For the p + ip excitonic insulator bound to the topological defect. Consequently, at half filling, the phase, however, one cannot find the unconventional zeroth LL electron charge associated to the defect is either −e/2 or e/2, located at the gap edge, consistent with C = 0. depending on whether the zero mode is occupied or empty. We have shown that the p + ip excitonic insulator is = characterized by Cc 1/2 and displays fractional charge e/2 at Chern–Simons theory and . The nonzero chiral vortices. However, it is still unclear how the two observations are = + Chern number Cc 1/2 indicates that the p ip excitonic insu- related to each other. This can be answered by the parity lator must be topologically distinct from normal insulators. anomaly34 of the p + ip excitonic insulator and the effective gauge

NATURE COMMUNICATIONS | (2019) 10:210 | https://doi.org/10.1038/s41467-018-08203-9 | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08203-9 theory associated with the topological defects. First, we couple an As has been discussed before, 〈σz〉 does not break TRS. The external U(1) gauge field Aμ (with μ = 0, 1, 2) to the p + ip integral of 〈σz〉 over the whole k-space vanishes. The in-plane excitonic state35, and we introduce the Dirac matrices defined as spin texture (〈σx〉, 〈σy〉) is however nonzero, and we plot it in γ0 = τ3, γ1,2 = iτ1,2. In order to extract the effective theory for the Fig. 3 for different values of β, i.e., β = 0, β = π/2, β = π, β = 3π/2. vortices, we include the vortex degrees of freedom χ in the order Interestingly, it is found that the in-plane spin configuration is parameter. Second, we perform a singular gauge transformation, tilted from the standard helical spin texture of the original TI ±iχ/2 36 c± → e c± , which results in the following Lagrangian (see surface, and the spins at two TRS-related momentum points, k Methods for details), and −k, are no longer always opposite to each other, violating the  TRS. Though being tilted, the chiral nature of electron’s spin is L¼C ~γμ∂ þ ~γμ þ C ; ð Þ x y k i μ aμ m k 6 preserved as we can see that the direction of the vector (〈σ 〉, 〈σ 〉) still undergoes a change of 2π when traversing a closed path ÀÁ = C C fi ~γμ fi ~γ0 ¼ γ0 around k 0 point. Furthermore, we calculate the total in-plane where k k is the Grassmann eld. are de ned as , 〈σx〉 〈σy〉 ~γ1 ¼ γ2, ~γ2 ¼Àγ1. In Eq. (6), the vortex degrees of freedom have magnetization M by integrating ( , ) over the momentum fi space. It is found that M always points towards the opposite been absorbed into an effective gauge eld aμ after the singular β ∂ χ = ¼ þ 1 ∂ χ direction of the global phase , as shown by the thick red arrows transformation. For a static defect with 0 0, aμ Aμ 2 μ . Finally, after one integrates out the Grassmann field, a in Fig. 3. We also note that the emergent magnetization of ground – fi state can lead to anisotropic in-plane transport of excitons which Chern Simons term with respect to the effective gauge eld aμ + can be obtained in the second order expansion: serves as an evident experimental signature of the p ip excitonic insulator. Furthermore, since β determines the direction of I μνρ magnetization of ground state, one expects to control β by L ¼ ϵ aμ∂νaρ; ð7Þ CS 2π applying a weak in-plane magnetic field during the process of TRS symmetry breaking. where the coefficient I is found to be 1/2 for the proposed p + ip excitonic insulator. The excitonic phases on interacting TI surfaces. After careful We note that for the TI surface with a Zeeman field23, a similar characterization of the topological properties of the p + ip exci- Chern–Simons Lagrangian arises as a result of the parity tonic insulator, we now study its possible realization on inter- anomaly, where the coefficient in front of the Chern–Simons acting TI surface states. Although various instabilities of term is exactly the Chern number C. Here, the coefficient I = 1/2 interacting topological insulators and topological super- fi fi 37–48 is identi ed to be the chiral Chern number Cc de ned above. The conductors have been extensively studied , we focus on the similarity between the two gauge field theories is due to the excitonic instability of the surface states and propose to realize the underlying parity anomaly; however, for the p + ip excitonic minimal model of the topological excitonic insulators via spon- insulator, it is the effective gauge field aμ rather than the external taneous symmetry breaking that enjoys a fractional chiral Chern fi – = eld Aμ that obeys the Chern Simons form. Further inserting number Cc 1/2, which is different from earlier proposals based ¼ þ 1 ∂ χ L ¼ I ϵμνρ ∂ À aμ Aμ 2 μ , the full Lagrangian CS 2π Aμ νAρ on heterostructure with an integer μ þL L 49 Aμj v consists of a Lagrangian describing the vortex v Thouless–Kohmoto–Nightingale–Nijs (TKNN) number . and a current operator associated with the vortex, The effective model Eq. (1) describing the surface state is usually modified by a higher order wrapping effect, reducing the μ ¼ I ϵμνρ∂ ∂ χ: ð Þ j π ν ρ 8 rotational symmetry of the Dirac cone down to a discrete C3 2 subgroup. Our further studies on the wrapping effect shows that the exciton gap remains very robust and therefore one can neglect For a static vortex (with vortex number n = 1) that centered at r , 0 the wrapping effect for the following analysis. We firstly consider we have ϵij∂ ∂ χ ¼ 2πδðÞr À r , inserting which into Eq. (8), the i j 0 the particle–hole symmetric case, while the generalization to finite electron charge of the defect is then obtained as R chemical potential μ is very straightforward and will be discussed Q ¼ d2xj0 ¼ I  C ¼ 1=2. Therefore, we have proved that c below. the chiral Chern number C equals to the fractional charge (in c First we start with demonstrating that a p-wave-type interac- unit of e) of a topological defect, justifying C to be the essential c tion that spontaneously generates the p + ip-wave exciton topological invariant characterizing the p + ip excitonic insulator. insulator naturally arises from the Hubbard model, which reads in k-space as: Spontaneous in-plane magnetization. The order parameter − θ X X −Δ i V y y ce in Eq. (2) breaks TRS. This suggests that the ground state ¼ ψ ψ ψ ψ : HI kþq;σ k′Àq;σ k′;σ k;σ ð9Þ carries a magnetization associated with it. Before proceeding, we 2 ; ′; σ −Δ −iθ k k q note that the order parameter ce still enjoys a redundant global phase. In the following, we take this into account and write −Δ −iθ′ θ′ = θ − β To facilitate the study of excitonic instability, we transform HI the off-diagonal term in Eq. (2)as ce , where with β being a constant angle independent of k. The global phase onto the chiral basis. Then, considering only interband forward β is analogous to U(1) phase of a superconductor. scattering channel, the low-energy effective vectorial interaction To clearly show the feature of magnetization, we calculate the component can be derived as (see Methods for details), 〈σx〉 〈σy〉 〈σz〉 P spin expectation values , , from the single particle ¼¼ y + H nvFkck;nck;n eigenstates and study the spin texture of the p ip excitonic k;n ground state |Ψ〉 (Eq. (3)). We obtain P hihið10Þ V ^ y ^ y À kc c ;À Á k′c c ′; ; ðÞΔ θ′þξ ðÞθ′þΔ 2θ′ 4 k;n k n k′;Àn k n hiσx ¼À c cos k vFk cos c sin ; k;k′;n 2 2þΔ2þΔ θ′ξ vFk c c cos k ðÞΔ θ′þξ ðÞθ′ÀΔ θ′ θ′ hiσy ¼À c cos k vFk sin c sin cos ; 2 2þΔ2þΔ θ′ξ where we assume a momentum cutoff Λ implicit in the sum of k. vFk c c cos k Δ2 2θ′ÀΔ2 2θ′ÀΔ2À Δ θ′ξ n is the band index.hi The interactionhi term in the second line, hiσz ¼À c sin c cos c 2 ccos k : P 2 2þΔ2þΔ θ′ξ ð1Þ ^ y ^ y v k c cos k ¼ÀV Á ′ F c Heff 4 k;k′;n kck;nck;Àn k ck′;Ànck′;n ,isofp-wave type,

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abky ky 0.3 0.3

kx kx –0.30.3 –0.3 0.3

M –0.3 –0.3 M  = 0  = /2

cdky ky 0.3 0.3

kx kx –0.30.3 –0.3 0.3

–0.3 M –0.3 M  =  = 3/2

Fig. 3 The in-plane spin texture of the p + ip excitonic insulator. a–d The results for different global phase of the exciton order parameter, β = 0, β = π/2, β = π, β = 3π/2, respectively. The total net magnetization M is shown by the arrow for each case

ð Þ and it arises from the spin-momentum locking of the surface result of the p-wave interaction H 1 . Moreover, we note that for fi Δ eff state. After introducing a complex bosonic vector eld and particle–hole asymmetric case with a finite chemical potential μ, performing the Hubbard–Stratonovich transformation to decou- μ ð Þ the critical Vc increases with increasing . Although the exciton ple H 1 , we arrive at a mean-field theory with the order eff P DE insulator is always stable for strong enough interaction V > Vc,it 1 ^ y is more experimentally favorable to tune μ close to the Dirac parameter given by Δ ¼ V kc ;Àc ;þ . When Δ ≠ 0, the 2 k k k point50,51. condensation of excitons gaps out the Dirac point, resulting in the In the above calculations, we temporarily focused on the p- 3 fl 25 ð1Þ excitonic insulator phase. Similar to He super uid theory , wave interaction H without considering the s-wave compo- fi eff symmetry analysis implies two possible con gurations of the nent. Nevertheless, the results clearly show that the TI surface order parameter that are the extrema of the free energy: (1) stateindeedhasastrongtendencytowardsthep + ip excitonic (chiral) p + ip excitonic insulator with Δ ¼ p1ffiffi Δ be À ibe and instability via spontaneous TRS breaking. We now take into 2 c x y (2) the polar excitonic insulator with Δ ¼ Δ be, where be is a unit account the interaction in the s-wave channel, which can also be p derived from the localP Hubbard interaction as (see Methods for vector in the xy plane. The polar excitonic phase is topologically ð2Þ V y y details), H ¼À c c ;À c c ′; .Inordertobe trivial with zero chiral Chern number. eff 4 k;k′;n k;n k n k′;Àn k n fi more general, we assume in the following two independent By self-consistently solving the mean- eld gap equation, we couplings V and V for the p-wave and s-wave interaction obtain the condensation energy and the order parameter (Δ , Δ ), p s c p respectively, and treat both termsP DE on equal footing. An s-wave which are plot in Fig. 4a, b, respectively. Figure 4b shows the fi Δ ¼ y bosonic scalarð Þ eld s Vs k ckÀckþ is introduced to of a threshold Vc, beyond which the system develops 2 + Δ ≠ decouple Heff in addition to the p ip excitonic order the excitonic instabilities with | c/p| 0. Figure 4a indicates that Δ + parameter c. The coupled self-consistent equations with even though both the p ip excitonic insulator and polar respect to Δ and Δ are obtained in the standard approach excitonic insulator are stable compared to the gapless Dirac cone s c + which leads to the calculated phase diagram shown by Fig. 4c. It for V > Vc, the p ip excitonic insulator is more energetically is found that both the s-wave and p + ip-wave excitonic favored due to the lower condensation energy. Moreover, we also fi + insulator can be spontaneously generated from the Dirac liquid performed addition mean- eld calculations by treating the p ip phase with strong enough interactions. The former and latter are and polar excitonic instability on equal footing. It is found that + stabilized for large Vs and Vp respectively. We note that the p ip channel always suppresses the polar order down to although the p + ip excitonic phase, which is absent in zero. Therefore, even though the ground state energies of the two conventional semiconductors, becomes stable in a large para- states are close, the mass term of the polar exciton insulator is meterregionofthephasediagram,itdoesnotoccurasthe negligible in the mean-field theory. Hence, the resultant mean- fi ground state for the discussed local Hubbard model, which lies eld Hamiltonian exactly matches with Eq. (2), and the proposed in the parameter line V = V that only enters into the s-wave ^ ð Þ s p mass term M k is generated by spontaneous TRS breaking as a excitonic condensation.

NATURE COMMUNICATIONS | (2019) 10:210 | https://doi.org/10.1038/s41467-018-08203-9 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08203-9

a 0.00 b 0.6 p+ip excitonic insulator –0.01 0.5 Polar excitonic insulator F F v v –0.02 0.4 Λ Λ / /

c/p 0.3 Δ cond –0.03 E 0.2 –0.04 p+ip excitonic insulator Polar excitonic insulator 0.1 Vc –0.05 0.0 0.6 0.7 0.8 0.9 1.0 0.6 0.7 0.8 0.9 1.0 Λ Λ 2 vF/V 2 vF/V

c 1.5

Vs = Vp s-wave excitonic 1.0 insulator F v Λ /2 s V 0.5 p+ip-wave excitonic Dirac liquid insulator 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Λ Vp/2 vF

Fig. 4 The results from mean-field calculations. a Condensation energy versus interaction V for p + ip excitonic insulator and polar excitonic insulator. b 2 fi p + ip Δ Δ V Δ =v2Λ2  Calculated mean- eld excitonic order parameter for excitonic insulator ( c) and polar excitonic insulator ( p) versus interaction . c=p F 1 is assumed. c The phase diagram based on unbiased mean-field with considering both the s- and p + ip-wave interaction channel. The dash-dotted red line denotes the Hubbard interaction case with Vs = Vp

The stabilization of the p + ip excitonic insulator. Now we the spin-momentum locking of TI surface states. For linearly + ¼ b ¼ b study the stabilization of the p ip excitonic insulator on TI polarized perpendicular incident , e1 ex and e2 ey. fl σx σy surfaces by going beyond the natural Dirac and taking Thus, the spin- ip matrices and take place in Hel-ph. This into account external perturbations. Recalling the interband verifies the spin-flip mechanism illustrated by Fig. 1bina  optical absorption process shown by Fig. 1b, one expects that microscopic perspective. Moreover, since c vF, the conserva-  additional photons should prefer the p-wave pairing due to the tion of energy and momentum requires jjk jjq in Hel-ph, and spin-flip mechanism that gives rise to spin-triplet excitons. Since therefore ensures the direct interband excitation with k − q ≃ k. excitons are usually generated by absorption of photons in a fi Furthermore,P the photonelds are Gaussian with the Hamilto- material, we consider a linearly polarized perpendicular incident y nian H ¼ ω a a ;λ þ 1=2 , one can therefore exactly laser with frequency ω and light intensity I. The corresponding ph qλ q q;λ q microscopic theory can be constructed by quantization of the integrate them out. This procedure leads to the effective action vector potential A(r), i.e., Sint describing the renormalization from photons: sffiffiffiffiffiffi R P P  X  2πe2v2 π y ¼À F σ Á σ Á 2 ÀiqÁr Sint dt ω αβ eλ ργ eλ AðrÞ¼ eλ a ;λ þ a e ; ð Þ q ω q Àq;λ 11 kk′q αβργλ ð Þ qλ q 13 ´ 1 ψy ψ ψy ψ :  i∂ Àω k;α k;β k′;ρ k′;γ y t q where a λ a is the annihilation (creation) operator of pho- q, q;λ P  λ = ω = x x y y tons, eλ ( 1, 2) are the polarization vectors, q c|q| is the Where λ σαβ Á eλ σργ Á eλ ¼ σαβσργ þ σαβσργ, inserting light frequency, with c being the light velocity and q the wave which it is found that S consists of an equal-time renormaliza- vector of photons. int tion term that produces the effective interaction between The coupling of the external field to the surface electrons is electrons, described within the formalism of the minimal coupling, which X – y y leads to the electron photon interaction, H ¼ÀV ψ ψ ψ ψ ; P qffiffiffiffi  int eff k;σ k;σ k′;σ k′;σ ð14Þ ¼À 2πðÞσ Á ψy ψ kk′σ HelÀph evF ω σσ′ eλ k;σ aq;λ kÀq;σ′ kqλσσ′ q ð Þ  12 where V ¼ 4πe2v2n =ω2 and n is the number density of þψy y ψ ; eff F ph ph k;σ aÀq;λ kÀq;σ′ photons that interact with the electrons, which is proportional to the light intensity I. We have utilized the fact that all the wave vectors q σ ¼ b ¼ ωb = ω fi where the term ( σσ′ · eλ) arises due to the helical spin texture and satisfy q qez ez c for photons with the frequency .The rst

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ψy ψ provided a concrete example and discussed its possible realization two operators in Hint, k;σ k;σ , are generated by the optical interband absorption process with spin-flip depicted by Fig. 1b. in strongly correlated TI surface. Interestingly enough, similar to ψy ψ the TIs due to band inversion, the predicted topological beha- Similarly, the last two operators, k′;σ k′;σ , are originated from the viors, i.e., the charge fractionalization and the emergent magne- fl interband relaxation process with spin- ip. After a unitary tization of ground state, are robust and immune to continuous transformation to the chiral basis, Eq. (14)isexactlycastinto: deformation of the Hamiltonian, as long as the exciton gap X hihi remains unclosed. This suggests that realizations of topological p- Veff ^ y ^ y H ¼À kc c ;À Á k′c c ′; : ð Þ wave excitonic insulators are likely in many other physical sys- int 2 k;n k n k′;Àn k n 15 k;k′;n tems. For example, the spectrum under study can be continuously deformed from the Dirac cone to a quadratic band crossing point ^ ^ (QBCP)63. This indicates that the instability is also possible and Interestingly, H enjoys the p-wave factor k Á k′, exactly the same int may be more feasible in interacting 2D electron with spin- as the p-wave interaction component reduced from the Hubbard ð1Þ momentum locking and QBCP. In this sense, the present results model, Heff . The interband optical processes automatically select are general and may provide new physical settings for the reali- the p-wave channel due to the spin-flip scatterings (Fig. 1b). More zation of quantum anomalies in condensed matter system, and fi importantly, Hint has a strength Veff tunable by external eld may stimulate different directions to search for more exotic parameters, ω and I. According to the mean-field study above and topological state of matters. We also note that a laser-induced the calculated phase diagram in Fig. 4, it is known that the non-equilibrium version of our proposal has been experimentally fi 55 tunable external eld can renormalize the Dirac cone state and observed recently in Bi2Se3 , where a millimeter-long transport drive it into the proposed p + ip-wave excitonic insulator. The distance of excitons up to 40 K is reported, supporting an exci- resultant state displays the predicted fractional charges associated tonic condensation state formed on the TI surface. Our study on with vortex excitations, the spontaneous in-plane magnetization the p + ip excitonic insulator stabilized by the optical pumping and thus the anisotropic transport of excitons on the TI surface. further predicts an anisotropic transport of excitons, which we believe would be of great interest for future experimental Discussion investigations. Excitonic insulators require strong interactions since the topo- logical surface states become unstable only when the interaction is Methods as large as the ultraviolet cutoff (band gap) as shown in Fig. 4b. The zero-energy BdG-type equations. From Eq. (2) in the main text, the zero- Therefore, we argue that topological Kondo insulators (TKIs) are energy eigen-equations can be written down as: À Δð Þ Àiθ ¼ ; good candidates for excitonic instability on TI surface. For vFkuk r e vk 0 À iθΔÃð Þ À ¼ : instance, in the case of SmB6, which is the archetypal TKI, the e r uk vFkvk 0 renormalized Coulomb interaction (U ~ 1 eV) is much larger than Λ Δ the direct band gap ( vF ~ 20 meV). For these parameters, our Here, we assume a real-space dependence of the order parameter (r). With a Δ = Δ iα α mean-field theory predicts an excitonic gap of order Δ ≃ 2–10 vortex,we have (r) c(r)e , where is the polar angle of r in real space. To s/c simplify the eigen-equations, one can multiply a factor eiθ from the right in the first meV. Indeed, magnetism and hysteresis in magnetotransport equation, and a factor e−iθ from the left in the second equation. Then, a Fourier 52 have been observed in SmB6 . Whether these experiments transformation yields, indeed observe an excitonic insulator require future work. We  À ∂ þ ∂ À 1 Δð Þ ¼ ; i x y ur r vr 0 would like to note that our theory is not related with the recent vF 53 1 ΔÃð Þ þÀ∂ À ∂ ¼ : works on excitonic contributions to quantum oscillations or the r ur i x y vr 0 54 vF intervalley excitonic instabilities in SmB6. In addition to the short-range interaction, we also considered After transformation to the polar coordinate, we arrive at the zero-energy BdG-like a long-range Coulomb interaction on the TI surface and make equations in real space. fi 55 iα À1 1 generalization to nite temperature . It is found that the cri- e ðÞÀi∂ þ r ∂α uðrÞÀ Δ ðrÞvðrÞ¼0; r vF c tical interaction Vc for BCS exciton condensation is sig- ÀiαðÞÀ ∂ À À1∂ ð Þþ 1 ΔÃð Þ ð Þ¼ : e i r r α v r c r u r 0 nificantly reduced for long-range interaction. At finite vF temperature, taking into account the thermal fluctuation, the strict long-range order is absent, the Kosterliz–Thouless tran- Effective gauge field theory. In the second quantized form, the effective sition temperature TKT can be calculated by evaluating the Hamiltonian of the p + ip excitonic insulator reads as phase stiffness of the exciton order based on the mean-field X Âà ¼ y τz À Δ Àiθτþ À Δà iθτÀ : calculations56. Experimentally, the driven TI surface offers a H Ck vFk ce c e Ck k rare platform for realization of new optical devices and facil- pffiffiffi !C ¼ = itates the study of possible collective instabilities of Dirac Formally, one can make transformation Ck k Ck k, while keeping the electrons. Experiments measuring bulk metallic TI materials total Hamiltonian unchanged. The action in the functional representation is then cast into: have obtained the lifetime of the excited states to be around a Z – Âà 57 59 fi À à ′à þ few picoseconds ,whichissignicantly larger than than S ¼ dτdkC i∂ þ τzv k2 þ Δ k′γ þ Δ k τ C ; that in graphene. Moreover, a remarkably long lifetime k 0 F c c k μ exceeding 4 s has been observed by experiment for the surface fi ′ = − γ0 60 where we have de ned k kx iky and introduced the Dirac matrices in 2D, states in bulk insulating Bi2Te2Se . These indicate promising = τ3, γ1 = iτ1, γ2 = iτ2, and γ± = (γ2 ±iγ1)/2. The above action formally describes 2 observation of a transient or quasi-equilibrium excitonic a 2D Dirac fermion with a mass vFk . Since the sign of mass determines the 59,61 2 ≥ insulator . Besides, the surface has a reduced dielectric of 2D Dirac systems, and vFk 0 for all k, one can continuously deform constant which can be tuned even smaller by doping or gat- the mass into a constant m while keeping the topological property unchanged. 62 With a U(1) gauge transformation of the Grassmannfield, the gauge invariance ing , and this possibly makes the long-range interaction more ~′ ¼ À À À requires the minimal coupling k kx Ax i ky Ay . With further including effective. the vortex degrees of freedom in the order parameter, the Lagrangian can be To summarize, we proposed a new type of topological p + ip written as,  excitonic insulator that achieves the parity anomaly and is L¼C γ0 f∂ þ γþ iχ~′ þ γÀ~′ Àiχ þ C ; characterized by the nontrivial chiral Chern number. We k i 0 e k k e m k

NATURE COMMUNICATIONS | (2019) 10:210 | https://doi.org/10.1038/s41467-018-08203-9 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08203-9 where χ is the vortex phase which goes from 0 to 2π when one completes a closed 16. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Δ = loop in real space. Without losing generality, we set | c| 1. The behavior of Mod. Phys. 83, 1057 (2011). vortex degrees of freedom can be extracted from the above Lagrangian after 17. Chen, Y. L. et al. Massive Dirac fermion on the surface of a magnetically integrating out the Grassmann fields. To the second order expansion, the doped topological insulator. Science 329, 659–662 (2010). Chern–Simons action in Eq. (7) is obtained. 18. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013). Mean-field analysis of the Hubbard interaction. Let us denote the four k vectors 19. Yu, R. et al. Quantized anomalous Hall effect in magnetic topological = = ′ = ′ − = + insulators. Science 329,61–64 (2010). in the Hubbard interaction Eq. (9)ask1 k, k2 k , k3 k q and k4 k q, respectively. For zero temperature, only those degrees of freedom near μ become 20. Bucher, B., Steiner, P. & Wachter, P. Excitonic insulator phase in dominant. Due to conservation of momentum in 2D, only the forward scattering is TmSe0.45Te0.55. Phys. Rev. Lett. 67, 2717 (1991). least irrelevant for repulsive interactions in the low-energy window in the renor- 21. Jérome, D., Rice, T. M. & Kohn, W. Excitonic insulator. Phys. Rev. 158, 462 = = malization group sense, which consists of two scattering channels: k1 k3 (k2 k4) (1967). = = and k1 k4 (k2 k3). Then we make a unitary transformation into the chiral basis. 22. Read, N. & Green, D. Paired states of fermions in two dimensions with = = For the scattering channel k1 k3 (k2 k4), after expansion, there are 16 different breaking of parity and time-reversal symmetries and the fractional quantum combinations in total. The requirement of conservation of energy greatly simplifies Hall effect. Phys. Rev. B 61, 10267 (2000). ð^′ Á ^Þ y y fi the expression, leaving us six terms that read as: k k ck′;þck;þck′;þck;þ, 23. Qi, X.-L., Hughes, T. & Zhang, S.-C. Topological eld theory of time-reversal Àð^′ Á ^Þ y y ð^′ Á ^Þ y y ð^′ Á ^Þ y y k k ck′;þck;Àck′;þck;À, k k ck′;þck;Àck′;Àck;þ, k k ck′;Àck;þck′;þck;À, invariant insulators. Phys. Rev. B 78, 195424 (2008). Àð^′ Á ^Þ y y ð^′ Á ^Þ y y k k ck′;Àck;þck′;Àck;þ, k k ck′;Àck;Àck′;Àck;À. Only those scattering terms that 24. Wei, H., Chao, S.-P. & Aji, V. Excitonic phases from Weyl semimetals. Phys. involve two bands can open a gap in the Dirac state, while the scatterings within Rev. Lett. 109, 196403 (2012). one band only renormalize the Fermi velocity but do not contribute to any 25. Volovik, G. E. Universe in a Helium Droplet (Clarendon, Oxford, 2003). instabilities. Moreover, the translation invariant mean-field treatment to the second 26. Jiang, Y. et al. Mass acquisition of Dirac fermions in magnetically doped fi and fth term contributes to a self-energy that only renormalizes the chemical topological insulator Sb Te films. Phys. Rev. B 92, 195418 (2015). = = 2 3 potential. With these consideration, the forward scattering channel k1 k3 (k2 27. Yoshimi, R. et al. Quantum Hall states stabilized in semi-magnetic bilayers of k4) produces the following reduced interaction, topological insulators. Nat. Commun. 6, 8530 (2015). P  ð1Þ ¼ÀV ^ Á ^′ y y : 28. Zhang, S. et al. Anomalous quantization trajectory and parity anomaly Heff 4 k k ck;nck;Ànck′;Ànck′;n ð16Þ k;k′;n in Co cluster decorated BiSbTeSe2 nanodevices. Nat. Commun. 8, 977 (2017). = = 29. Hou, C. Y., Chamon, C. & Mudry, C. Electron fractionalization in two- For the scattering channel k1 k4 (k2 k3), following the same approach above, one obtains the following reduced interaction, dimensional graphenelike structures. Phys. Rev. Lett. 98, 186809 (2007). P 30. Seradjeh, B., Moore, J. E. & Franz, M. Exciton condensation and charge ð2Þ ¼ÀV y y : fi Heff 4 ck;nck;Ànck′;Ànck′;n ð17Þ fractionalization in a topological insulator lm. Phys. Rev. Lett. 103, 066402 k;k′;n (2009). ð1Þ ð2Þ 31. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. Das Non- Both Heff and Heff , p-wave and s-wave type respectively, are reduced from the Abelian and topological quantum computation. Rev. Mod. Phys. 80, original Hubbard interaction, which are the least irrelevant channels that can 1083 (2008). possibly gap out the Dirac cone. 32. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979). Data availability 33. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Soliton excitations in polyacetylene. The data that support the findings of this study are available from the corre- Phys. Rev. B 22, 2099 (1980). sponding author upon reasonable request. 34. Chamon, C. et al. Electron fractionalization for two-dimensional Dirac fermions. Phys. Rev. B 77, 235431 (2008). 35. Kurkov, M. & Vassilevich, D. Parity anomaly in four dimensions. Phys. Rev. D Received: 2 July 2018 Accepted: 21 December 2018 96, 025011 (2017). 36. Franz, M. & Tešanović, Z. Quasiparticles in the vortex lattice of unconventional superconductors: Bloch waves or Landau levels. Phys. Rev. Lett. 84, 554 (2000). 37. Wang, C., Potter, A. C. & Senthil, T. Classification of interacting electronic References topological insulators in three dimensions. Science 343, 629–631 (2014). 1. Redlich, A. N. Gauge noninvariance and parity nonconservation of three- 38. Wang, J. et al. Evidence for electron-electron interaction in topological dimensional fermions. Phys. Rev. Lett. 52, 18 (1984). insulator thin films. Phys. Rev. B 83, 245438 (2011). 2. Redlich, A. N. Parity violation and gauge noninvariance of the effective gauge 39. Wray, L. A. et al. A topological insulator surface under strong Coulomb, fi eld action in three dimensions. Phys. Rev. D 29, 2366 (1984). magnetic and disorder perturbations. Nat. Phys. 7,32–37 (2011). 3. Semenoff, G. W. Condensed-matter simulation of a three-dimensional 40. Raghu, S., Qi, X.-L., Honerkamp, C. & Zhang, S.-C. Topological Mott anomaly. Phys. Rev. Lett. 53, 2449 (1984). Insulators. Phys. Rev. Lett. 100, 156401 (2008). 4. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. 41. Rachel, S. & Hur, K. L. Topological insulators and Mott physics from the The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009). Hubbard interaction. Phys. Rev. B 82, 075106 (2010). 5. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in 42. Wang, Z., Qi, X.-L. & Zhang, S.-C. Topological order parameters for – graphene. Nature 438, 197 200 (2005). interacting topological insulators. Phys. Rev. Lett. 105, 256803 (2010). 6. Liu, Z. K. et al. Discovery of a three-dimensional topological Dirac , 43. Slagle, K., You, Y.-Z. & Xu, C. Exotic quantum phase transitions of strongly – Na3Bi. Science 343, 864 867 (2014). interacting topological insulators. Phys. Rev. B 91, 115121 (2015). 7. Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A 44. Gurarie, V. Single-particle Green’s functions and interacting topological = Na, K, Rb). Phys. Rev. B 85, 195320 (2012). insulators. Phys. Rev. B 83, 085426 (2011). 8. Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator 45. Stoudenmire, E. M., Alicea, J., Starykh, O. A. & Fisher, M. P. A. Interaction multilayer. Phys. Rev. Lett. 107, 127205 (2011). effects in topological superconducting wires supporting Majorana fermions. 9. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi Phys. Rev. B 84, 014503 (2011). – arcs. Science 349, 613 617 (2015). 46. Wang, C. & Senthil, T. Interacting fermionic topological insulators/ 10. Lv, B. Q. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X superconductors in three dimensions. Phys. Rev. B 89, 195124 (2015). 5, 031013 (2015). 47. Yao, H. & Ryu, S. Interaction effect on topological classification of 11. Zhang, H. et al. Topological insulators in Bi2Se3,Bi2Te3 and Sb2Te3 with a superconductors in two dimensions. Phys. Rev. B 88, 064507 (2013). – single Dirac cone on the surface. Nat. Phys. 5, 438 442 (2009). 48. Baum, Y. & Stern, A. Magnetic instability on the surface of topological 12. Chen, Y. L. et al. Experimental realization of a three-dimensional topological insulators. Phys. Rev. B 85, 121105(R) (2012). – insulator, Bi2Te3. Science 325, 178 181 (2009). 49. Hao, N. et al. Chiral topological excitonic insulator in semiconductor 13. Xia, Y. et al. Observation of a large-gap topological-insulator class with a quantum wells. Phys. Rev. B 82, 195324 (2010). – single Dirac cone on the surface. Nat. Phys. 5, 398 402 (2009). 50. Chen, J. et al. Gate-voltage control of chemical potential and weak 14. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions antilocalization in Bi2Se3. Phys. Rev. Lett. 105, 176602 (2010). at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008). 51. Checkelsky, J. G., Hor, Y. S., Cava, R. J. & Ong, N. P. Bulk band gap and 15. Sun, H.-H. et al. Majorana zero mode detected with spin selective Andreev surface state conduction observed in voltage-tuned of the topological reflection in the vortex of a topological superconductor. Phys. Rev. Lett. 116, insulator Bi2Se3. Phys. Rev. Lett. 106, 196801 (2011). 257003 (2016).

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52. Nakajima, Y., Syers, P., Wang, X., Wang, R. & Paglione, J. One-dimensional Author contributions edge state transport in a topological Kondo insulator. Nat. Phys. 12, 213–217 R.W., O.E., B.W. and D.Y.X. performed the calculations, discussed the results and pre- (2016). pared the manuscript. 53. Knolle, J. & Cooper, N. Excitons in topological Kondo insulators: theory of thermodynamic and transport anomalies in SmB6. Phys. Rev. Lett. 118, Additional information 096604 (2017). 54. Roy, B., Hofmann, J., Stanev, V., Sau, J. & Galitski, V. Excitonic and nematic Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- instabilities on the surface of topological Kondo insulators. Phys. Rev. B. 92, 018-08203-9. 245431 (2015). 55. Hou, Y. et al. Emergence of excitonic superfluid at topological-insulator Competing interests: The authors declare no competing interests. surfaces. Preprint at https://arxiv.org/abs/1810.10653 (2018). 56. Min, H., Bistritzer, R., Su, J.-J. & MacDonald, A. H. Room-temperature Reprints and permission information is available online at http://npg.nature.com/ superfluidity in graphene bilayers. Phys. Rev. B 78, 121401(R) (2008). reprintsandpermissions/ 57. Sobota, J. A. et al. Ultrafast optical excitation of a persistent surface-state Journal peer review information: Nature Communications thanks Ping Zhang and the population in the topological insulator Bi2Se3. Phys. Rev. Lett. 108, 117403 (2012). other anonymous reviewer(s) for their contribution to the peer review of this work. Peer 58. Wang, Y. H. et al. Measurement of intrinsic Dirac fermion cooling on the reviewer reports are available. surface of the topological insulator Bi2Se3 using time-resolved and angle- resolved photoemission spectroscopy. Phys. Rev. Lett. 109, 127401 (2012). Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in 59. Hajlaoui, M. et al. Tuning a Schottky barrier in a photoexcited topological published maps and institutional affiliations. insulator with transient Dirac cone electron-hole asymmetry. Nat. Commun. 5, 3003 (2014). 60. Neupane, M. et al. Gigantic surface lifetime of an intrinsic topological insulator. Phys. Rev. Lett. 115, 116801 (2015). Open Access This article is licensed under a Creative Commons 61. Triola, C., Pertsova, A., Markiewicz, R. S. & Balatsky, A. V. Excitonic gap Attribution 4.0 International License, which permits use, sharing, formation in pumped Dirac materials. Phys. Rev. B 95, 205410 (2017). adaptation, distribution and reproduction in any medium or format, as long as you give 62. Beidenkopf, H. et al. Spatial fluctuations of helical Dirac fermions on the appropriate credit to the original author(s) and the source, provide a link to the Creative surface of topological insulators. Nat. Phys. 7, 939–943 (2011). Commons license, and indicate if changes were made. The images or other third party 63. Sun, K., Yao, H., Fradkin, E. & Kivelson, S. A. Topological insulators and material in this article are included in the article’s Creative Commons license, unless nematic phases from spontaneous symmetry breaking in 2D Fermi systems indicated otherwise in a credit line to the material. If material is not included in the with a quadratic band crossing. Phys. Rev. Lett. 103, 046811 (2009). article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Acknowledgements licenses/by/4.0/. Authors are grateful to A. Burkov, T. Sedrakyan, C.S. Ting, Q.H. Wang, L. Hao, and L.B. Shao for fruitful discussions. This work was supported by 973 Program under Grant No. 2017YFA0303200, and by NSFC (Grants No. 60825402, No. 11574217). O.E. acknowl- © The Author(s) 2019 edges the support from ASU startup grant.

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