Supplementary Figures.

Supplementary Figures. E -µ (e V ) 1 0-1 - (π,0 ) (0,π) (0,0 ) (π,0 ) (π,π) (0,0 ) a b c E -µ (e V ) 1 0-1 k y /π -0.5 - -1.0 (π,0 ) (0,π) (0,0 ) (π,0 ) (π,π) (0,0 ) -1.0-0.5 0.0 k x /π 0.5 1.0 1.0 0.5 0.0 Supplementary Figure S1. The band dispersions of a K 1 x Fe y Se and b Tl 1 x Fe y Se, along the high-symmetry directions of the extended Brillouin zone (one iron/unit cell). The band dispersions have been obtained by diagonalizing the appropriate five-orbital tight binding model. Panel c shows the Fermi surfaces of Tl 1 x Fe y Se with electron doping δ = 0.15, which consist of only electron pockets near zone boundaries M. 1

0.3 0. a 0.1 0 b A 1 g (s x y ) A 1 g (s x + y ) B 1 g (d x -y ) J /D 0.1 I II IV III P. A. 0.0 5 x z /y z o rb ita l J /D = 0.1 0.0 0.0 0.1 0. 0.3 /D 0.0 0 0.0 0.1 0. 0.3 /D Supplementary Figure S. Pairing phase diagram and amplitudes of Tl 1 x Fe y Se. The phase diagram and the strength of the pairing amplitudes are similar to those of K 1 x Fe y Se described in the main text. Panel a shows the zero temperature phase diagram of Tl 1 x Fe y Se for an electron doping δ = 0.15. The regions I, II, III, and IV respectively correspond to an A 1g state with s x y as the dominant pairing channel, a time reversal symmetry breaking A 1g + ib 1g state with s x y and d x y as the dominant A 1g and B 1g pairing channels, a likewise A 1g + ib 1g state with s x +y and d x y as the dominant A 1g and B 1g pairing channels, and a pure B 1g state with d x y pairing channel. Panel b shows the competing dominant pairing amplitudes A 1g s x y, A 1g s x +y, B 1g s x y for xz/yz orbitals of Tl 1 xfe y Se with an electron doping δ = 0.15, and J /D = 0.1.

3 a 3 b J / t1 1 I II III J / t1 1 I II III 0 0 1 3 / t 1 0 0 1 3 / t 1 0.15 c 0.15 d P. A. 0.10 0.05 A 1g s x y A 1g s x +y B 1g d x -y J / t 1 =0.7 P. A. 0.10 0.05 A 1g s x y A 1g s x +y B 1g d x -y J /t 1 =0.7 0.00 0.0 0.5 1.0 1.5.0 / t 1 0.00 0.0 0.5 1.0 1.5.0 /t 1 Supplementary Figure S 3. Qualitatively similar zero temperature phase diagrams of a two orbital t J model obtained a by considering a doping dependent renormalization of the band dispersions due to the local constraint of zero double occupancies and b without imposing δ the zero double occupancy constraint. In a the effective hopping parameters t i = t i. The regions I, II, and III respectively correspond to an A 1g state with s x y as the dominant pairing channel, an A 1g + ib 1g state with s x y and d x y as the dominant A 1g and B 1g channels, and a likewise A 1g + ib 1g state with s x +y and d x y as the dominant A 1g and B 1g channels. 3

b c d P. A. P. A. a Supplementary Figure S4. The pairing amplitudes for xz/yz orbitals of a K1 x Fe y Se, b iron pnictides with electron doping δ = 0.15, c Tl1 x Fe y Se, and d single-layer FeSe. Compared to K1 x Fe y Se and Tl1 x Fe y Se, the iron pnictides have stronger A1g sx +y pairing. 4

U/W 1 Ordered vacancies MI 45 insulating Superconducting OSMP "Semiconducting" Metal 0 n=6 Electron doping Supplementary Figure S5. Sketch of the overall phase diagram for K 1 x Fe y Se. The vertical axis stands for the strength of iron vacancy order, with 1 being fully 5 5 vacancy ordered and 0 being completely vacancy disordered. The iron vacancy order affects the system in a similar way as U/W. The red line refers to the Mott insulator (MI), the yellow shading illustrates the orbitalselective Mott phase (OSMP), and the orange dashed line shows an OSMP-to-metal transition. The diamond symbols indicate the approximate positions where the various samples are located. 5

E -µ (e V ) 1 0-1 - a k y /π (π,0 ) (0,π) (0,0 ) (π,0 ) (π,π) (0,0 ) 1.0 0.5 0.0-0.5 b -1.0-1.0-0.5 0.0 0.5 1.0 k x /π Supplementary Figure S6. Panel a shows the band dispersion of the single-layer FeSe along the high-symmetry directions of the extended Brillouin zone (one iron/unit cell). The band dispersion has been obtained by diagonalizing the appropriate five-orbital tight binding model. Panel b shows the Fermi surfaces of the single-layer FeSe with electron doping δ = 0.1, which consist of only electron pockets near zone boundaries M. 6

J /D 0.3 0. 0.1 a I IV 0.0 0.0 0.1 0. 0.3 /D II III P. A. P. A. 0. 0.1 0.0 0.1 /D 0.0 0.1 0. 0.3 b c x y o rb ita l x z /y z o rb ita l A 1 g (s x y ) A 1 g (s x + y ) B 1 g (d x -y ) 0.0 0.0 0.1 0. 0.3 /D Supplementary Figure S7. Panel a shows the zero temperature phase diagram of the singlelayer FeSe for an electron doping δ = 0.1. The phase diagram is similar to those of K 1 x Fe y Se (Fig. 3a of the main text) and Tl 1 x Fe y Se (Supplementary Figure Sa). The regions I, II, III, and IV respectively correspond to an A 1g state with s x y as the dominant pairing channel, a time reversal symmetry breaking A 1g + ib 1g state with s x y and d x y as the dominant A 1g and B 1g pairing channels, a likewise A 1g + ib 1g state with s x +y and d x y as the dominant A 1g and B 1g pairing channels, and a pure B 1g state with d x y pairing channel. Panels b and c respectively show the competing dominant pairing amplitudes A 1g s x y, A 1g s x +y, B 1g s x y for xy and xz/yz orbitals of the single-layer FeSe with electron doping δ = 0.1, and J /D = 0.1. 7

Supplementary Tables. α = 1 α = α = 3 α = 4 α = 5 ɛ α -0.36559-0.36559-0.56466-0.00096-0.91583 t αα µ µ = x µ = y µ = xy µ = xx µ = xxy µ = xyy µ = xxyy α = 1-0.05475-0.40868 0.0881 0.01557-0.00866-0.03143 0.01899 α = 3 0.353-0.09783-0.00537 α = 4 0.0633 0.0658-0.0355-0.0189 0.0043 α = 5-0.0470 0.01117 0.00177-0.01349 t αβ µ µ = x µ = xy µ = xxy µ = xxyy αβ = 1 0.10161-0.0017 0.0373 αβ = 13-0.31447 0.065 0.01030 αβ = 14 0.13785 0.00105 0.01040 αβ = 15-0.0485-0.10096-0.0104 αβ = 34-0.04795 αβ = 35-0.30966-0.01498 αβ = 45-0.08359-0.00766 Supplementary Table S1. K 1 x Fe y Se. Here we use the same notation as in Ref. Tight-binding parameters of the five-orbital model for 7 of the main text. The orbital index α =1,,3,4,5 correspond to d xz, d yz, d x y, d xy, and d 3z r orbitals, respectively. The units of the parameters are ev. 8

α = 1 α = α = 3 α = 4 α = 5 ɛ α -0.35956-0.35956-1.11574 0.0934-0.74545 t αα µ µ = x µ = y µ = xy µ = xx µ = xxy µ = xyy µ = xxyy α = 1-0.4198-0.34713 0.389 0.1014-0.05889 0.06785 0.0078 α = 3 0.38401-0.0547 0.01306 α = 4 0.38169 0.15837-0.0308-0.04663-0.0685 α = 5 0.03688-0.056-0.0318 0.010 t αβ µ µ = x µ = xy µ = xxy µ = xxyy αβ = 1 0.4408 0.00436 0.01173 αβ = 13-0.36704-0.01-0.067 αβ = 14 0.3177-0.0560 0.03873 αβ = 15-0.1317-0.0608 0.0931 αβ = 34-0.05093 αβ = 35-0.3085 0.0386 αβ = 45-0.17070-0.070 Supplementary Table S. Tl 1 x Fe y Se. Tight-binding parameters of the five-orbital model for 9

α = 1 α = α = 3 α = 4 α = 5 ɛ α -0.0313-0.0313-0.51304 0.4379-0.374 t αα µ µ = x µ = y µ = xy µ = xx µ = xxy µ = xyy µ = xxyy α = 1-0.15497-0.34438 0.3647 0.0363-0.04761 0.00148 0.03479 α = 3 0.38181-0.04947-0.05064 α = 4 0.855 0.1173-0.0079-0.040-0.079 α = 5 0.01989 0.00197-0.03545 0.084 t αβ µ µ = x µ = xy µ = xxy µ = xxyy αβ = 1 0.14939-0.00943 0.0397 αβ = 13-0.3508 0.0987 0.06043 αβ = 14 0.397-0.08589 0.0547 αβ = 15-0.08056-0.06893-0.01788 αβ = 34-0.00317 αβ = 35-0.36967-0.03581 αβ = 45-0.3453 0.0949 Supplementary Table S3. single-layer FeSe. Tight-binding parameters of the five-orbital model for the 10

Supplementary Note 1. Two orbital model with and without zero doubleoccupancy constraints To demonstrate the robustness of our pairing phase diagram against the doping dependent band renormalization effects, we consider a two orbital model involving only xz and yz orbitals. For simplicity we consider hole doping; the electron doping case can be treated in a similar manner after performing a particle-hole transformation. The model and the detail of the theoretical method are described in the Supplementary Methods. We have compared both the pairing phase diagrams and pairing amplitudes obtained using the two methods described in the Supplementary Methods. One method explicitly imposes the no double-occupancy constraint, and the other accounts for it through an effective band renormalization. As shown in Supplementary Figure S3, the pairing phase diagrams and the pairing amplitudes obtained in these two methods are qualitatively similar, when the exchange interactions are scaled by the renormalized bandwidth D. Supplementary Note. Orbital character of the Fermi surface and orbital dependence of pairing amplitudes. We have respectively shown in Fig. c and Fig. d of the main text the orbital weights on the electron pockets for both K 1 x Fe y Se and pnictides, for electron doping δ = 0.15. For pnictides, the xz/yz orbitals have the dominant orbital weights on the two hole pockets (not shown), and also contribute significantly on the electron pockets. Whereas for K 1 x Fe y Se (and also for Tl 1 x Fe y Se ), there are no hole pockets, and the contribution from xy orbital is considerably enhanced on the electron pockets. This indicates that the xy orbital plays a more important role in building the electron pocket of (K,Tl) 1 x Fe y Se than in pnictides. The enhanced xy orbital character on Fermi surface of K 1 x Fe y Se affects the electron pairing. To see this we compare the pairing amplitudes at J /D = 0.1 of xy orbital with xz/yz orbital for both K 1 x Fe y Se and pnictides at electron doping δ = 0.15 in Fig. 4 of the main text. We see that for pnictides, the contribution to the three competing dominant pairings from the xz/yz orbital is comparable with the one from xy orbital. But for K 1 x Fe y Se the contribution from xy orbital can be stronger. This is especially true in the regime /D 0.07, where only A 1g pairing is present. Overall, however, the pairing amplitudes for the xy orbital are similar between (K,Tl) 1 x Fe y Se and iron pnictides. This is seen in the three-dimensional plots given 11

in Figs. 3c,d of the main text and in the Supplementary Figure Sb, which are similar to their counterparts for the xz/yz orbitals (Supplementary Figure S4). Supplementary Note 3. Mott phase with vacancy order and the orbital selective Mott phase in alkaline iron selenides. Experiments reveal that the insulating alkaline iron selenide samples contain 5 5 ordered iron vacancies. Many measurements also suggest that the superconducting region of the superconducting samples include either no iron vacancy, or vacancies that are disordered. This raises the question of what role the vacancy ordered insulating phase (in the so called 45 compound) plays in the superconductivity of the alkaline iron selenides, and how it can be connected to the superconducting phase. In this note we describe this linkage within a phase diagram (Supplementary Figure S5) in the parameter space spanned by U/W associated with a varying degree of vacancy order and carrier doping, derived from the considerations of electron correlations in a multiorbital model for the alkline iron selenides (Ref. 33 of the main text). This phase diagram is supported by ARPES measurements (Ref. 34 of the main text), as well as transport measurements at high pressures (Ref. 35 of the main text). The horizontal and vertical axes respectively denote the electron doping and the strength of the iron vacancy order. We label the degree of vacancy order from 0 (being completely vacancy disordered) to 1 (for the fully vacancy ordered case). Since the vacancy order causes the effective reduction of kinetic energies, from the perspective of electron correlations, it affects the system in a similar way as tuning the ratio U/W. At commensurate filling n = 6, we find the system is in a Mott insulator (MI). Away from this filling, the system is in either an orbital-selective Mott phase (OSMP) or a metallic state, depending on the degree of vacancy order and the electron filling. In the OSMP, the 3d xy orbital is Mott localized, while other 3d orbitals are still delocalized. In this phase diagram, the insulating 45 compounds is located in the MI with full vacancy order. A recent ARPES study (Ref. 34 of the main text) suggests that the normal state of the superconducting sample and another sample which shows semiconducting behavior in its resistivity are respectively located inside the metallic phase and very close to the boundary of the OSMP-to-metal transition, both without vacancy order but at different doping concentrations. This suggests the following effects of chemical doping the 45 insulating 1

sample: On the one hand, doping injects extra carriers which increase the electron filling; on the other hand, the dopants change the chemical environments around the iron ions, and disturb the iron vacancy order. As a result of the combined effects, the system may evolve from the vacancy ordered 45 insulating phase to the vacancy disordered metallic one via the partially vacancy ordered OSMP. In this way, the vacancy ordered 45 Mott insulating phase connects to the superconducting material, with the link provided by the OSMP (Ref. 35 of the main text). Supplementary Methods Here we describe the method used to study the singlet pairing in the two-orbital model. The Hamiltonian of interest is H = i<j,α,β,σ t αβ ij c iασ c jβσ + h.c. µ c iασ c iασ + J αβ ij i,α,σ i<j,α,β with the double occupancy prohibiting constraint σ c iασ c iασ ( Siα S jβ 14 n iαn jβ ) (S1) 1 for each orbital. The constraint is imposed by introducing slave boson operator b i and fermionic spinon operator f i for each orbital (Refs. 1 and 45) and the Hamiltonian is transformed to H = i<j,α,β,σ t αβ ij f iασ b iαb jβ f jβσ + h.c. µ f iασ f iβσ J αβ ij B ij,αβ B ij,αβ i,α,σ i<j,α,β ) + i,α λ iα ( σ f iασ f iασ + b iα b iα 1 (S) where B ij,αβ = (f iα f jβ f iα f jβ ) is the spin-singlet pairing operator for the fermionic spinons, and λ i,α s are Lagrange multipliers which enforce the occupancy constraints. At zero temperature the slave bosons are Bose condensed and the boson operators have finite expectation values. In the tetragonal symmetry breaking particle-hole channel order, we have b i,1 = b i, = δ/, where δ is the hole doping and λ i,1 = λ i, = λ, which is absorbed into the chemical potential. The expectation value of the boson operators renormalizes the kinetic energy term by a factor of δ/. As in the main text we choose the interactions to be diagonal in the orbital space. After mean-field decoupling, the free energy density f = ( x,αα + y,αα ) + J ( x+y,αα + x y,αα ) α α d k 4π (E k+ + E k δ E k+ δ E k + µ) 13 (S3)

is minimized with respect to pairing amplitudes with the constraint n 1 = n = 1 δ. The quasiparticle dispersions E k± in the paired state are calculated from the 4 4 Nambu matrix δ ˆξ k ĥ k = µ1 k k δ ˆξ k + (S4) µ1 where k,αα = ( x,αα cos(k x ) + y,αα cos(k y )) + J ( x+y,αα cos(k x + k y ) + x y,αα cos(k x k y )) (S5) For the band structure we choose the two orbital tight-binding model of Ref. 49. The tight-binding matrix ˆξ k = ˆξ k = ξ k+ 1 + ξ k τ z + ξ kxy τ x, where Pauli matrices τ i ) operate on the orbital indices, and ξ k+ = (t 1 + t )(cos k x + cos k y ) 4t 3 cos k x cos k y, ξ k = (t 1 t )(cos k x cos k y ), ξ kxy = 4t 4 sin k x sin k y are respectively A 1g, B 1g and B g functions. The band dispersion relations E k± = ξ k+ ± ξk + ξ kxy, give rise to two electron pockets at k = (π, 0) and (0, π), and two hole pockets at k = (0, 0) and (π, π). The following values of the hopping parameters, t 1 = t, t = 1.3t, t 3 = t 4 = 0.85t were obtained in Ref. 49, by a fitting of the LDA bands. The quasiparticle dispersions in the paired state are given by [ ( (δξk+ ) E k± = µ + δ ( ξ 4 k + ξkxy) k,11 + + ) { ( ( k, δξk+ ± δξ k + k,11 ) ( k, (δξk+ ) + δ ξkxy µ + 1 4 k,11 k, ) µ ) } 1 ] 1 We then compare the pairing phase diagrams obtained from two means: one way is to explicitly account for the occupancy constraints and associated band renormalization; and the other way is to calculate the pairing without imposing the no double-occupancy constraint. Supplementary References (S6) 49. Raghu, S., Qi, X.-L., Liu, C.-X., Scalapino, D. J. & Zhang, S.-C. Minimal two-band model of the superconducting iron oxypnictides. Phys. Rev. B 77, 0503 (008). 14