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From semimetal to chiral Fulde-Ferrell superfluids Ting Fung Jeffrey Poon and Xiong-Jun Liu ∗ International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China and Collaborative Innovation Center of Quantum Matter, Beijing 100871, China The recent realization of two-dimensional (2D) synthetic spin-orbit (SO) coupling opens a broad avenue to study novel topological states for ultracold atoms. Here, we propose a new scheme to realizeexoticchiralFulde-Ferrellsuperfluidforultracoldfermions,withagenerictheorybeingshown thatthetopologyofsuperfluidpairingphasescanbedeterminedfromthenormalstates. Themain findingsaretwofold. First,asemimetalisdrivenbyanewtypeof2DSOcouplingwhoserealization 7 is even simpler than the recent experiment, and can be tuned into massive Dirac fermion phases 1 withorwithoutinversionsymmetry. Withoutinversionsymmetrythesuperfluidphasewithnonzero 0 pairingmomentumisfavoredunderanattractiveinteraction. Furthermore,weshowafundamental 2 theorem that the topology of a 2D chiral superfluid can be uniquely determined from the unpaired normal states, with which the topological chiral Fulde-Ferrell superfluid with a broad topological n regionispredictedforthepresentsystem. Thisgenerictheoremisalsousefulforcondensedmatter a J physics and material science in search for new topological superconductors. 2 1 Introduction.–The recent experimental realization of First,weproposethatanewtypeof2DSOcoupledDirac ] two-dimensional (2D) spin-orbit (SO) coupling for ultra- semimetal can be realized based on a scheme adopting n cold atoms [1–3], which corresponds to synthetic non- only a single Raman transition and simpler than the re- o Abeliangaugepotentials[4–6],advancesanessentialstep centexperiments[1–3]. Byreadilybreakingtheinversion c - toexplorenoveltopologicalquantumphasesbeyondnat- symmetry of the Dirac system, we find that the FF su- r p ural conditions. Ultracold fermions with SO coupling perfluid phase can be favored under an attractive inter- u canfavortherealizationoftopologicalsuperfluids(TSFs) action. Moreover, we show a generic formalism for the s (similar as topological superconductors in solids [7–13]) Chern number of a 2D superfluid induced in the normal . t based on an s-wave Feshbach resonance [14, 15], which states, with which the topological chiral FF superfluid a m are highly-sought-after quantum phases for their ability with a broad topological region is predicted. tohostnon-AbelianMajoranazeromodesandimplement - d topological quantum computation [22–25]. Note that a n superfluidphasehastoexistin2Dor3Dregime, sohav- o ing a 2D or 3D SO coupling is the basic requirement for c such realization of gapped TSFs. While experimental [ studies of TSFs are yet to be available, different propos- 2 als have been introduced for Rashba and Dirac type SO v coupled systems [16–21], with BCS or FFLO pairing or- 2 9 ders[26,27]. WhentheFermienergycrossesonlyasingle 9 (or odd number of) Fermi surface (FS), the SO coupling 1 forces Cooper pairs into effective p-wave type, rendering 0 a TSF phase [23]. However, in the generic case it is not . 1 known so far whether there is a universal way to pre- 0 cisely determine the topology, e.g. Chern numbers, of a 7 superfluid phase induced in normal bands. On the other 1 hand, toachieveaTSFphaseintegratesseveralessential : v ingredients, which may bring challenges for the experi- i X ment. The minimal schemes of realization are therefore FIG.1: (a)Schematicdiagramofbreakingtheinversionsym- desired to ensure high feasibility of real studies. r metry (process I) and opening gap at Dirac points (II) for a a 2D SO coupled Dirac semimetal. (b) Proposed experimen- In this letter we propose an experimental scheme to tal setting for realization. The standing wave lights formed realize chiral Fulde-Ferrell (FF) superfluids, and show by E generate a blue-detuned square lattice. The inci- 1x,1z a generic theory to compute the Chern number of TSF dent polarization of E is eˆ = αeˆ +iβeˆ , and the λ/4- 1z ⊥ x y phases through the normal states. The findings are of wave plate changes the polarization of the reflected field to significance in both experiment and fundamental theory. eˆ(cid:48)⊥ =αeˆx−iβeˆy. The Raman coupling, illustrated in (c) for 40K fermions, is generated by E and an additional running 1z light E which has tile angle θ with respect to x-z plane. 2 ∗Correspondingauthor: [email protected] 2DDiracSemimetal.–Westartwiththeeffectivetight- 2 binding Hamiltonian H on a square lattice, whose re- we finally get the effective Hamiltonian (1), with the TB alization shall be presented below: parameters being t = t cos[(π/2)cosθ] and t = x 0 xI 2t sin[(π/2)cosθ]. It can be seen that the inversion H = (cid:88)(cid:16)c† ,c† (cid:17)H (cid:18)ck↑(cid:19); sy0mmetry is controlled by the tilt angle θ, and the gap TB k↑ k↓ TB ck↓ opening at Dirac points is controlled by the β, the eˆy- k component of the E field. H = (m −2t cosk −2t cosk )σ 1z TB z x x z z z Theaboverealizationisclearlysimplerthantherecent + 2t sink σ +t sink σ +m σ , (1) so z y xI x 0 x x experiments on 2D SO coupling, for it involves only one Raman transition. We note that a 2D Dirac semimetal where c (c† ) is annihilation (creation) operator with ks ks driven by SO interaction, being different from graphene spin s=↑,↓, t is the hoping constant along x/z direc- x/z whoseDiracpointsareprotectedbysymmetryonlywhen tion, t is the strength of spin-flip hopping, and m so z,x there is no SO coupling, has not been discovered in solid denote the effective Zeeman couplings. As described state materials [29]. The high feasibility and controlla- in Fig. 1(a), the above Hamiltonian describes a Dirac bility ensure that the 2D Dirac semimetal proposed here semimetalifm =0and|m |<2(t +t ),withtwoDirac x z x z can be well observed based on the current experiment. points at Λ = (cid:0)±cos−1[(m −2t )/2t ],0(cid:1). The term ± z x z t sink σ breaks the inversion symmetry and leads xI x 0 to an energy difference between the two Dirac points. 0.4 0.1 Finally, a nonzero m opens a local gap at the two x Dirac points. Without loss of generality, we take that t =t ,t =t cosθ andt =2t sinθ tofacilitatethe -0.1 z 0 x 0 0 xI 0 0 -0.4 further discussion. As we show below, the above Hamil- -1. 1. -1. 1. tonian can be readily realized in experiment. The realization is sketched in Fig. 1(b,c). The ingredients of realization include a blue-detuned spin- 1. 1. independent square lattice and a Raman lattice generated via only a single Raman transition (details can be found in the Supplementary Material [28]). In particu- -1. -1. lar, the two standing-wave lights (red and black lines in - - Fig. 1(b)) of frequency ω form the electric field compo- 1 nents E1x =2E1eˆzcosk0x and E1z =2E1(αeˆxcosk0z+ FIG. 2: (a)-(b) The Hall conductaces σxy versus chemical βeˆ sink z), and generate the square lattice potentials potential for the band structures (c) and (d) at k = 0. In y 0 z V cos2k x and V cos2k z, respectively, with V ∝ (b)aninsulatingregimecanbereached. Theparametersare 0x 0 0z 0 0x |E1|2 and V0z ∝ (α2 − β2)|E1|2. For our purpose we rescaledbytakingtz =1, andwetakemz =3,tso =1,mx = 0.3 and t = 0.92, where t = 0.8t for (a) and (c), and take that α is large compared with β, namely, the field x xI x t =0.1t for (b) and (d). E is mainly polarized in the eˆ direction. The π/2- xI z 1z x phase difference between eˆ and eˆ polarized compo- x y nents is easily achieved by putting a λ/4-wave plate be- Anomalous Hall effect.–When both the inversion and fore mirror M , as shown in Fig. 1(b). All the initial time-reversalsymmetriesarebroken,theHamiltonian(1) 1 phases of the lights are irrelevant and have been ne- leadstoanomalousHalleffect,whichreflectsthenontriv- glected. To generate the Raman lattice another run- ialityof2DSOinteraction[31]. Fig.2showsthehallcon- ning light of frequency ω2 is applied along a tilted di- ductanceσxz versuschemicalpotentialµandthespectra rection so that its wave vector k = k (xˆcosθ+yˆsinθ) in different regimes. The hall conductance at zero tem- 2 0 (blue line in Fig. 1 (b)). E together with the light perature is calculated by (Θ is the step function) 2 E induces the Raman transition between spin-up | ↑ ˆ 1z 2 (cid:88) (cid:105) = |9/2,9/2(cid:105) and spin-down | ↓(cid:105) = |9/2,7/2(cid:105), as illus- σ = dk dk Θ((cid:15) −µ)(1−Θ((cid:15) −µ)) xz h x z m n tratedinFig.1(c). ThegeneratedRamanpotentialtakes n,m the form Meff = 2M0(αcosk0z + βsink0z)eik1x|↑(cid:105)(cid:104)↓|, (cid:16)(cid:68)u (cid:12)(cid:12)∂H(cid:12)(cid:12)u (cid:69)(cid:68)u (cid:12)(cid:12)∂H(cid:12)(cid:12)u (cid:69)(cid:17) with k1 = k0cosθ and M0 ∝ |E1E2|. The former α- ×Im n(cid:12)∂kz(cid:12) m m(cid:12)∂kx(cid:12) n . (2) term in Meff leads to a spin-flip hopping along z di- ((cid:15)n−(cid:15)m)2 rection t(cid:126)sjo,(cid:126)j±1z = ±(−1)jzeiπjxcosθtso, and the latter β- term gives an onsite Zeeman term m(cid:126)jσx, with m(cid:126)j = Itcanbeseenthatσxz isalwayszeroatµ=0orwhenµ (−1)jzeiπjxcosθmx [28]. Note that a small β-term can isinthebandgap[Fig.2(b,d)]. However,thenonzeroσxz inducearelativelylargem comparedwithhoppingcou- for finite µ crossing bulk bands shows the nontrivial 2D x plings, sincem isinducedbyon-sitecoupling. Through SOeffect,implyingthatthesuperfluidphaseofnontrivial x a gauge transformation c(cid:126)j,↓ → c(cid:126)j,↓(−1)jzeiπjxcosθ [20], topology may be obtained. 3 States of superfluidity.– The superfluid phase can be and the normal states at the Fermi energy, which govern induced by considering an attractive Hubbard interac- the phase of ∆(iM,j). (cid:80) k tion. The total Hamiltonian H = H −U n n . TB i i↑ i↓ Due to the existence of multiple FSs corresponding to different Dirac cones, in general we can have intra-cone pairing(FFLO)andinter-cone(BCS)pairingorders,defined by ∆ = (U/N)(cid:80) (cid:104)c† c (cid:105), with q = ±Q 2q k q+k↑ q−k↓ or 0 [21, 32–34]. On the other hand, since the inversion symmetry is broken, the BCS pairing would be typically suppressed. The topology of the superfluid phase can be characterized by the Chern number. However,tocomputetheChernnumberofthepresent system is highly nontrivial. The reason is because the FFLO order breaks translational symmetry can fold up theoriginalBrillouinzoneintomanysub-Brillouinzones. In particular, if Q = pπ/q, with p and q being mutual FIG. 3: (a)-(b) Configurations of FSs of normal bands. The primeintegers,thesystemincludesqsub-Brillouinzones. solid lines with arrows denote the FSs, which are boundaries Moreover, if Q is incommensurate with original lattice, of the vector areas S(cid:126) in the k space. The blue and green iM,j thesystemfoldsupintoinfinitenumberofsub-Brillouiin colors represent the closed FSs, and the red color represents zones. In such generic way, it is not realistic to compute open FSs. numerically the Chern number of the system. To facilitate the presentation, here we show the above Generic theory for Chern number.–Toresolvethediffi- theorem for a multiband system, but with only a single culty,weshowagenerictheoremtodeterminetheChern FS.ThegenericproofcanbefoundintheSupplementary number of a 2D system after opening a gap through Material [28]. We write down the BdG Hamiltonian by havingsuperfluid/superconductivity. Forconveniencewe classify the normal bands of the system without pairing (cid:18) (cid:19) H(k)−µ ∆(k) into three groups: upper (lower) bands which are above HBdG(k)= ∆†(k) −HT(−k)+µ , (4) (below) the Fermi energy, and the middle bands which arecrossedbyFermienergy. Eachmiddlebandmayhave where H is the normal Hamiltonian, ∆(k) is the pairing multiple FSs (loops), and we denote by (i ,j) the j-th order matrix, and k is the local momentum measured M FS loop of the i -th middle band. Let the total Chern from FS center. Note that if FS is not symmetric with M number of the upper (lower) bands be n (n ). We can respect to its center, one can continuously deform it to U L showthattheChernnumberthesuperfluidpairingphase be symmetric without closing the gap. Finally we can induced in the system is given by always write down H in the above form to study BdG the topology. Denote by u(iM) the eigenvector of the (cid:20) k Ch1 = nL−nU +(cid:88) (−1)qiMnF(iM)− npoarirminagl obnanFdSc,rtohsesinegiegFeenrsmtaiteesneorfgyH. Fochuasisngthoenfotrhme iM˛ (cid:21) [α(iM)u(iM),β(iM)u(iM)∗]T. TheChernBnduGmberofthesu- (cid:88)j (−1)qiM,j ∂S(cid:126)iM,j∇kθk(iM,j)·dk . (3) pekrfluidkphasek: Ch1−=k nL−nU +¸ ∇k×Ak(iM)d2k, with A(iM) = i[α(iM)(k)u(iM)]†∇ (cid:2)α(iM)(k)u(iM)(cid:3)+ k k k k Here n(FiM) is the Chern number of the iM-th middle i[β(iM)(k)u(iM)∗]†∇ (cid:2)β(iM)(k)u(iM)∗(cid:3). (5) band and θ(iM,j) =arg[∆(iM,j)] is the phase of the pair- −k k −k k k ing order projected onto the (iM,j)-th FS loop [35]. Let θ(iM) =arg(cid:2)(cid:104)u(iM)|∆(k)|u(iM)∗(cid:105)(cid:3) be the phase of or- k k −k The integral direction is specified by arrows along FS der parameter on FS. Taking that ∆(k) → γ∆(k) with lines in Fig. 3 (a,b), which defines the boundary ∂S(cid:126)iM,j γ →0+ (without closing bulk gap), and with some alge- of the vector area S(cid:126)iM,j in k space. The quantities bra we can reach {q ,q }={0,1}arethendeterminedby“right-hand ˛ im,j iM (cid:20) rule” specified below. The quantity qiM,j = 0 (or 1) if Ch1 = nL−nU +(−1)qiM n(FiM)+2 A(cid:101)(kiM)·dk the energy of normal states within the area S(cid:126)iM,j is pos- ˆ ˛ ∂S(cid:126)iM (cid:21) itive (or negative), while q = 1 (or 0) if the regions (S(cid:126)iM,out) unenclosed by aniyMFS loop have positive (or − 2 S(cid:126)iM ∇k×A(cid:101)(kiM)d2k− ∂S(cid:126)iM ∇kθk(iM)·dk (6) negative) energy (Fig. 3). With this theorem the Chern numberofthesuperfluidphasecanbesimplydetermined with A(cid:101)(kiM) = i[u(kiM)]†∇kuk(iM) being the Berry connec- once we know the properties of lower and upper bands, tion for the normal band states. If we choose a gauge so 4 that A(cid:101)(kiM) is smooth on S(cid:126)iM, the two terms regarding the gap closes at the right hand Dirac point Q˜ = Q, A(cid:101)(iM) in the right hand side of Eq. (6) cancels. We then where the original bulk gap less than 2mx before having k reach the formula (3) for the case with a single FS. superfluid pairing [Fig. 4(e)]. Importantly, for mz =3tz, The proof can be generalized to the case with generic we find that ∆(−c2)Q ∼ 2tz, which is of the order of band multiple FSs which may be closed or open [Fig. 3(b)], width. In this regime Q˜ is away from the right hand with multiple bands crossed by Fermi energy, and with Dirac point, and corresponds to a relatively large bulk the pairing within each FS or between two different FSs, gap before adding ∆ [Fig. 4(f)]. As a result, a large −2Q given that the pairing fully gaps out the bulk [28]. Since ∆ is necessary to drive the phase transition, giving a −2Q the result is not restricted by pairing types, the generic broad topological region, as shown in Fig. 4 (d). theorem shown here is powerful to quantitatively deter- minethetopologyofthesuperfluidphases. Ontheother hand, it should be noted that this theorem is applied to judgethetopologyofthephasegappedoutbysmallpair- ing orders. Starting from the phase governed by Eq. (3), when the magnitude of superfluid order increases, the system may undergo further topological phase transition and enter a new phase with different topology. Monitor- ing such phase transitions can determine the topological region of the phase diagram. Phase Diagram.– With the above theorem we can easily determine the topology of the superfluid phase. Note that for the present Dirac system, a BCS pairing can- not fully gap out the bulk but leads to nodal phases, while the FFLO or FF order can. When there is only one FS, the system with an FF order is topological since ´ 12 n = n = n(iM) = 0 and ∇θ(iM) · dk = ±1, L U F ∂S(cid:126)(iM) k 6. 10 giving the Chern number Ch = ∓1. In contrast, when 1 8 there are two FSs, from the same or different bands, one 5. can readily find that the contributions from both FSs 6 cancels out and the Chern number is zero, rendering a 0. 0.5 1. 0. 0.5 1. trivial phase. This result implies that the system can be topological when it is in an FF phase. From the mean field results shown in Fig. 4 (a,b), we can see that the BCS pairing is greatly suppressed, and ∆ dominates over ∆ , with ±Q ≈ Λ for positive −2Q 2Q ± µ. A rich phase diagram is given in Fig. 4 (c,d), where the topological and trivial FF phases, and FFLO phase are obtained. It is particularly interesting that in Fig. 4 FIG. 4: (a)-(b) The superfluid order ∆ dominates over −2Q (d) a broad topological region is predicted when mz is others for µ > 0. (c) Phase diagram with normal-size away from m = 2t . The broad topological region im- topological region when t = t = t ,t = 0.7t ,m = z z x so z xI z z pliesthattheuppercriticalvalue∆(c),characterizingthe 2tz,mx = 0.3tz. (d) Phase diagram with broad topological 2q region. The values of the parameters in diagrams (a,b,d) are transition from topological FF state to other phases, is m = 2.92t ,t = 0.92t ,t = t ,t = 0.8t ,m = 0.3tz. largely enhanced compared with the case for m = 2t . z z x z so z xI z x z z Schematic diagrams showing the underlying mechanism for This is a novel effect explained below. Note that the su- the normal-size (c) and broad (d) topological regions. perfluid order ∆ also couples the particle-hole states −2Q at Q˜ = (π−Q,0). Increasing ∆−2Q to ∆(−c2)Q closes the Conclusion.–In conclusion, we have proposed an ex- bulk gap at Q˜ momentum, with the critical value being perimentalschemetorealizechiralFulde-Ferrell(FF)su- solved from BdG Hamiltonian as perfluids,andshowedagenerictheoremtodeterminethe topology of TSF phases through the normal states. 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TheincidentpolarizationofE iseˆ =αeˆ +iβeˆ ,andtheλ/4-waveplatechangesthepolarizationofthere- 1z ⊥ x y flectedfieldtoeˆ(cid:48) =αeˆ −iβeˆ . (b,c)Asillustratedfor40Kfermions,thelatticepotentialforthestate|F =9/2,m =9/2(7/2)(cid:105) ⊥ x y F isgeneratedbyE . (d)TheRamancouplingisgeneratedbyE andE ,whichhasatiltangleθ withrespecttox-axisin 1x,1z 1z 2 the x-y plane. B. Light fields The electric field of standing-wave lights for the present realization can be written as: (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17) E = E ei(k0x+φ1)+ei(−k0x+φ2) eˆ +E α ei(k0z+φ3)+ei(−k0z+φ4) eˆ +β ei(k0z+φ3)+ei(−k0z+φ4+π) eˆ 1x z 1z x y +E ei(k1y+φ5)eˆ 2 x (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) = 2E1xei(φ1+2φ2)cos k0x+ φ1−2 φ2 eˆz+2E1z αei(φ3+2φ4)cos k0z+ φ3−2 φ4 eˆx (cid:18) (cid:19) (cid:19) +iβei(φ3+2φ4)sin k0z+ φ3−2 φ4 eˆy +E2ei(k1y+φ5)eˆx = 2eiφBE cosk xeˆ +2E (αcosk zeˆ +iβsink zeˆ )+E ei(k1y+φA)eˆ , (S1) 1x 0 z 1z 0 x 0 y 2 x where φA =−kk01φ1−2φ2 − φ3+2φ4 +φ5, φB = φ1+2φ2 − φ3+2φ4. In the last line of the above equation we have made the changeofparametersx→x−(φ −φ )/(2k )andz →z−(φ −φ )/(2k )andamultiplicationofanoverallphase 1 2 0 3 4 0 factor e−i(φ3+φ4)/2. The Rabi-frequencies of the transitions described in Fig. S1(b-d) can be derived. The light denoted by black line has the Rabi-frequency Ω = 2Ω(cid:48) E cosk . Those lights depicted as red lines and connecting existing states π π 1x x Ω(cid:48) with ∆F = ±1 have the Rabi-frequency Ω± = √±2E1z(∓αcosk0x+βsink0z), and the one depicted as blue line 2 is Ωp = Ω(cid:48)pE2eik0xcosθ = Ω(cid:48)pE2eik1x. The phase φA,B are irrelevant and so are omitted here. The quantities Ω(cid:48)i is the dipole matrix element (cid:104)g|r |e(cid:105), where g and e is the corresponding ground state and excited state, and r is the q q spherical component of vector r. C. Lattice and Raman potentials The lattice potential is generated by the two-photon processes which intermediate states are in the manifold 2S 1/2 (described in the diagram Fig. S1(b,c) by the process generated by black and red lines). Each process gives a contribution (cid:80) |Ω |2/∆ to the lattice potential, where Ω is the corresponding Rabi-frequency, j runs through all j i j i possible atomic states and ∆ =∆ or ∆+∆ if the intermediate state considered is in the manifold 2P or 2P . j s 1/2 3/2 Accordingtoexperimentaldata,forreference,see [30])alldifferent∆or∆+∆ correspondingtointermediatestates s 7 1 → 3;9 q=−1 q=0 q=1 1 → 3;7 q=−1 q=0 q=1 2 2 2 (cid:113) (cid:113) (cid:113) 2 2 2 (cid:113) (cid:113) (cid:113) F(cid:48) = 11 1 − 1 1 F(cid:48) = 11 9 − 9 3 2 2 11 110 2 22 55 110 (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) F(cid:48) = 9 8 − 16 F(cid:48) = 9 16 392 − 256 2 33 297 2 297 2673 2673 (cid:113) (cid:113) (cid:113) F(cid:48) = 7 − 14 F(cid:48) = 7 28 98 2 135 2 1215 1215 1 → 1;9 q=−1 q=0 q=1 1 → 1;7 q=−1 q=0 q=1 2 2 2 (cid:113) (cid:113) 2 2 2 (cid:113) (cid:113) (cid:113) F(cid:48) = 11 1 − 2 F(cid:48) = 11 2 49 − 32 2 3 27 2 27 243 243 (cid:113) (cid:113) (cid:113) F(cid:48) = 9 16 F(cid:48) = 9 32 112 2 27 2 243 243 TABLE I: The value of the ratio of (cid:104)g|r |F(cid:48),m(cid:48) =m −q(cid:105) and |(cid:104)J =1/2||er||J(cid:48)(cid:105)| = α . The top-left corners of each table q F F i containsthreenumbersdenotingJ,J(cid:48) andm ,respectively,whereJ andJ(cid:48) isthequantumnumberofJcorrespondingtothe F ground and excited states respectively and |g(cid:105)=(cid:12)(cid:12)F = 29,mF(cid:11). ofdifferentm areofthesameorderofmagnitude(orderofTHz)andthedifferencesamongeachgrouparenegligible F (order of 100MHz to 1GHz). The lattice potential of both spin is V (x,z)= 4(cid:18)2α22 + α12 (cid:19)(cid:0)E2 cos2k x+(cid:0)α2−β2(cid:1)E2 cos2k z(cid:1), (S2) 3 ∆ ∆+∆ 1x 0 1z 0 s wherewehaveomittedtheconstantterm, andαi =(cid:12)(cid:12)(cid:10)J = 12||er||J =i− 21(cid:11)(cid:12)(cid:12)(i=1,2)isthereducedmatrixelement betweenthetotalangularmomentumJ =1/2andJ =3/2. (ThecoefficientsinV canbecalculatedwiththehelpof 0 Table I in a straightforward way.) Notice that the coupling between spin-up |9/2,9/2(cid:105) and spin-down |9/2,7/2(cid:105) state (cid:12) (cid:12) throughtheseprocessesarenegligiblesinceEg↑−Eg↓ (cid:29)(cid:12)Ω2/∆(cid:12), whereEgs istheenergyofthespin-uporspin-down state. The Raman lattice is generated via only one Raman process (described in the diagram Fig. S1(d)). The Raman potential generated is given by the formula (cid:80) Ω∗Ω /∆ . In this case, the generated potential is j − p j M =M (αcosk z+βsink z)eik1x|↑(cid:105)(cid:104)↓|, (S3) eff 0 0 0 whereM0 = 91(cid:16)α∆22 − ∆+α21∆s(cid:17)√2E2E1z. TheZeemantermmzσz isgeneratedbytheasmalloff-resonantintheRaman process, where m =δ˜/2=(cid:0)E −E −ω +ω (cid:1)/2. z g↑ g↓ 1 2 D. Tight-binding Model We derive the tight-binding model by considering the hopping contributed from lattice and Raman potentials, respectively. The lattice potential contributes the spin-conserved hopping terms as   HT(1B) =−t(cid:48)x (cid:88) Cj†x,jz,↑Cjx+1,jz,↑+Cj†x,jz,↓Cjx+1,jz,↓−tz (cid:88) Cj†x,jz,↑Cjx,jz+1,↑+Cj†x,jz,↓Cjx,jz+1,↓+h.c., (S4) jx,jz jx,jz ´ ´ where t(cid:48) = −V E2 dxdzφ∗ (x,z)cos2k xφ (x,z), t = −V (cid:0)α2−β2(cid:1)E2 dxdzφ∗ (x,z)cos2k zφ (x,z) x 0 1x 0,0 0 1,0 z 0 1z 0,0 0 0,1 and k a = π, and φ is the wavefunction at centered at (x,z) = (j ,j )a. On the other hand, for the spin-flip 0 jx,jz x z term induced by Raman coupling, we have that the first term of (S3) provides spin-flip hopping along the z-direction of strength ˆ αM dxdzφ∗ (x,z)cosk zeik1xφ (x,z) 0 jx,jz 0 jx,jz±1 ˆ = αM dxdzφ∗ (x,z)cosk (z+j a)eik1(x+jxa)φ (x,z) 0 0,0 0 z 0,±1 ˆ = (−1)jzeikk01πjxαM0 dxdzφ∗0,0(x,z)cosk0zeik1xφ0,±1(x,z) = ∓(−1)jzeikk10πjxtso, 8 ´ where tso =−αM0 dxdzφ∗0,0(x,z)cosk0zeik1xφ0,1(x,z). The same term gives no contribution to the hopping along x-direction since cosk z is antisymmetric in the x-direction at the local minimum of lattice potential. Likewise, the 0 second term of (S3) gives rise to an onsite Zeeman term ˆ βM dxdzφ∗ (x,z)sink zeik1xφ (x,z) 0 jx,jz 0 jx,jz ˆ = βM dxdzφ∗ (x,z)sink (z+j a)eik1(x+jxa)φ (x,z) 0 0,0 0 z 0,0 = (−1)jzeikk10πjxmx, ´ where mx =βM0 dxdzφ∗0,0(x,z)sink0zeik1xφ0,0(x,z). This term gives negligible contribution to the hopping term since β (cid:28)1. Therefore, in the tight-binding model, the Raman potential contributes HT(2B) = (cid:88) (−1)jzeikk01πjx(cid:104)tso(cid:16)c†jx,jz,↑cjx,jz+1,↓−c†jx,jz,↑cjx,jz−1,↓(cid:17)+mxc†jx,jz,↑cjx,jz,↓(cid:105)+h.c. (S5) jx,jz The total tight-binding Hamiltonian reads H = H(1) +H(2), which can be simplified by applying the gauge TB TB TB transformation cjx,jz,↑/↓ →(−i)jz+jxe±i(cid:16)k21k0πjx+π2jz(cid:17)cjx,jz,↑/↓. (S6) With the Fourier transformation cjx,jzσ → √1Nckσeik·(jx,jz)a, we can finally get the Bloch Hamiltonian of the tight- binding model  (cid:16) (cid:17)  HTB = (mz−2tzcoskz)σz+2tsosinkzσy+mxσx+2t(cid:48)xcos kx+0π2kk01 − π2 cos(cid:16)kx−0π2kk01 − π2(cid:17) = (m −2t cosk −2t cosk )σ +2t sink σ +t sink σ +m σ , (S7) z x x z z z so z y xI x 0 x x where t = t(cid:48) sin(πcosθ/2) and t = 2t(cid:48) cos(πcosθ/2). All the parameters are independently tunable except that x x xI x t and t are related by t2 +t2 /4=t2. x xI x xI 0 It can be seen that the inversion symmetry is controlled by the tilt angle θ, and the gap opening at Dirac points is controlled by the β, the eˆ -component of the E field. Note that m is induced by onsite spin-flip transition. Thus y 1z x a small β-term in Eq. (S3) can generate a relatively large m . For our purpose, we shall consider a small β-term x compared with α-term. Thus the E field is mainly polarized in the eˆ direction. 1z x S-2. BDG HAMILTONIAN (cid:80) An attractive Hubbard interaction can be described effectively as H = −U n n . We introduce three order U i i↑ i↓ (cid:68) (cid:69) parameters ∆ and ∆ when considering superconducting pairing, where ∆ = (−U/N)(cid:80) c† c† and ±2Q 0 2q k q+k q−k q = ±Q or 0. If only one of them is non-zero, the BdG Hamiltonian can be written as H = (cid:80) Ψ†H Ψ /2, BdG k k BdG k where (cid:18) (cid:19) H (k) ∆ TB 2q H = (S8) BdG ∆† −HT (2q−k) 2q TB (cid:16) (cid:17)T and the basis of Ψ is c ,c ,c† ,c† . The pairing matrix ∆ is assumed to be real valued, which can be k k↑ k↓ 2q−k↑ 2q−k↓ 2q set via a change of overall phase factor of c. Here, q is the center-of-mass momentum of the non-zero pairing. If there are more than one of them is non-zero, we need to fold the Brillouin zone. Let q = Q = mπ/n, where m 9 and n are coprime integers. Then the density of the BdG Hamiltonian can be written as   H0 0 0 ··· 0 ∆(cid:101)0 ∆(cid:101)−Q 0 ··· ∆(cid:101)2Q  0 H2Q 0 ··· 0 ∆(cid:101)Q ∆(cid:101)0 ∆(cid:101)−2Q ··· 0   0 0 H4Q ··· 0 0 ∆(cid:101)Q ∆(cid:101)0 ··· 0   ... ... ... ... ... ... ... ... ... ...     0 0 0 ··· H−2Q ∆(cid:101)−2Q 0 0 ··· ∆(cid:101)0  HBdG = ∆(cid:101)† ∆(cid:101)† 0 ··· ∆(cid:101)† H(cid:48) 0 0 ··· 0 , (S9)  0 Q −2Q 0  ∆(cid:101)† ∆(cid:101)† ∆(cid:101)† ··· 0 0 H(cid:48) 0 ··· 0   −2Q 0 Q −2Q   0 ∆(cid:101)†−2Q ∆(cid:101)†0 ··· 0 0 0 H−(cid:48) 4Q ··· 0   ... ... ... ... ... ... ... ... ... ...  ∆(cid:101)† 0 0 ··· ∆(cid:101)† 0 0 0 ··· H(cid:48) 2Q 0 2Q (cid:18) (cid:19) where Hr(Q(cid:48)) =±H(T)(±kx+rQ,±kz), ∆(cid:101)2pQ = −∆0 ∆20pQ , r =0,1,2,··· ,n−1 and p=0 or±1. 2pQ The Nambu basis in the folded Brillouin zone, where k ∈[0,2π/n], is y (cid:16) (cid:17)T c c ··· c c† c† ··· c† . kx,ky kx+2Q,kz kx−2Q,kz −kx,−kz −kx−2Q,−kz −kx+2Q,−kz In this basis, the gap equation can be derived through diagonalizing (S8) or (S9) and then calculate the expectation values of the order parameters. Therefore, from a particular value of U the order parameters can be solved self- consistently. S-3. BKT TRANSITION The BKT Temperature can be calculated through the approximate BKT criterion π π k T = ρ (T )≈ ρ (T =0), (S10) B BKT 2 s BKT 2 s where ρ is the superfluid density, which is can be calculated by ρ = j /δq. j is the supercurrent density and −Q s s s s is the center-of-mass momentum of two paired fermions when the system cut only one FS, which is of center-of-mass momentum−Q. j istobecalculatedbythevariationofthezero-temperatureenergy,i.e. j =δE /δq. Thetotal s s Total energy E can be calculated through diagonalization of the total Hamiltonian in the Nambu basis, by assuming Total the center-of-mass momentum of the pairing to be −Q+δq. The numerical result is shown in Fig. S2 0.4 0.2 5 10 FIG. S2: The BKT temperature when m =2.92t ,t =0.92t ,t =t ,t =0.8t ,m =0.3t . z z x z so z xI z x z S-4. GENERIC THEORY FOR CHIRAL TOPOLOGICAL SUPERFLUID After presenting the generic result, we shall show step by step the generic theory that the topology of 2D chiral superfluid/superconductor can be precisely determined by knowing only the properties of the Fermi surfaces and the 10 topology of other normal bands away from the Fermi surfaces. A. Generic formalism For simplicity, we do not consider the case that the normal bands barely touch the Fermi energy. We classify the normal bands into three groups: upper (lower) bands, which are those with energy above (below) Fermi energy, and middlebandswhicharecrossedbyFermienergy. Letn (n )bethetotalChernnumberoftheupper(lower)bandand L U n(iM) be the Chern number of i -th middle band. Each middle band may form multiples Fermi surfaces. We denote F M thepairingofthe(i ,j)-thFermisurface,projectedoni -thband,tobe∆(iM,j) anditsphaseθ(iM,j) =arg∆(iM,j). M M k k k We denote S(cid:126) to be the area enclosed by the Fermi surface ∂S(cid:126) and S(cid:126) to be the unenclosed area. Notice iM,j iM,j iM,out thatamongeachoftheregionsS(cid:126) orS(cid:126) , theenergyofthei -thbandisofthesamesign. Weshallshowthat iM,j iM,out M in general, the formula for the Chern number of the negative energy states, after considering the superconducting pairing, can be written as  ˆ  Ch1 =nL−nU +(cid:88)(−1)qiMn(FiM)+(cid:88)(−1)qi(cid:48)M,j ∇kθk(iM,j)·dk, (S11) iM j ∂S(cid:126)iM,j wheretheintegralinthebracketisthewindingnumberofθ(iM,j). Theintegralisevaluatedinthedirectionsatisfying k right hand rule. The phase factor (−1)qi(cid:48)M,j and (−1)qiM is 1 if the energy of the iM-th band is negative in the region S(cid:126) and S(cid:126) respectively, and they are −1 if the corresponding energy is positive. We have chosen a gauge here iM,j iM,out such that the phase of the Berry connection of the i -th band is continuous throughout the regions S(cid:126) . The case M iM,j with an alternative gauge can be found below. This quantity is non-zero if the system is topological. B. One band, One FS First, we consider the case where there is only one band in the system and there is only one FS. So the Berry curvature, and thus the Chern number, of the normal states is zero since we can choose a gauge of real eigenvectors where the Berry connection is identically zero. We can calculate the Berry connection, after considering the superconducting pairing, by direct calculation over each region S(cid:126). The density of the BdG Hamiltonian and the Berry connection can be written as (cid:18) (cid:19) (cid:15) −µ ∆ H (k) = k k ; BdG ∆∗ −(cid:15) +µ k −k (cid:18) (cid:19) A = i(cid:0)α∗(k) β∗(k)(cid:1)∇ α±(k) , k± ± ± k β (k) ± where the ± sign denotes upper (lower) band. α and β can be found by diagonalizing the H : BdG (cid:114) (cid:16) (cid:17)2 (cid:15)k+(cid:15)−k −µ± |∆ |2+ (cid:15)k+(cid:15)−k −µ 2 k 2 α (k) = ; ± N (k) ± ∆∗ β (k) = k ; ± N (k) ± (cid:113) N (k) = |N (k)α (k)|2+|N (k)β (k)|2. ± ± ± ± ± So the Berry connection of the lower band is: ∆ ∇ ∆∗ −∆∗∇ ∆ A =i k k k k k k . k (cid:114) (cid:32)(cid:114) (cid:33) (cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17) 4 (cid:15)k+(cid:15)−k −µ +|∆ |2 (cid:15)k+(cid:15)−k −µ +|∆ |2− (cid:15)k+(cid:15)−k −µ 2 k 2 k 2