Unconventional superconductivity in a strongly correlated band-insulator without doping

We present a novel route for attaining unconventional superconductivity (SC) in a strongly correlated system without doping. In a simple model of a correlated band insulator (BI) at half-filling we demonstrate, based on a generalization of the projected wavefunctions method, that SC emerges when e-e interactions and the bare band-gap are both much larger than the kinetic energy, provided the system has sufficient frustration against the magnetic order. As the interactions are tuned, SC appears sandwiched between the correlated BI followed by a paramagnetic metal on one side, and a ferrimagnetic metal, antiferromagnetic (AF) half-metal, and AF Mott insulator phases on the other side.

The discovery of unconventional superconductivity in a variety of materials, such as high T c superconductivity in cuprates [1], iron pnictides and chalcogenides [2], in organic superconductors [3], heavy fermions [4] and very recently in magic angle twisted bilayer graphene [5,6], has always ignited worldwide interest owing to their rich phenomenonology, the theoretical challenges they pose, scientific implications and broad application potential. In almost all of these examples, superconductivity appears upon chemically doping a parent compound away from commensurate filling [1,2,[5][6][7][8], though in some cases inducing charge fluctuations by changing pressure also leads to the superconducting phase [3,8]. An important experimental fact is that chemical doping inevitably induces disorder, as is clearly the case in high T c superconductors, which makes these materials very inhomogeneous [9][10][11][12]. It is a theoretical and experimental challenge to come up with new mechanisms and materials for clean high T c superconductors.
Strong e-e correlations are crucial for unconventional superconductivity (SC). In most of the known unconventional superconductors [1][2][3][5][6][7][8] the low temperature phase of the parent compound is either a strongly correlated antiferromagnetic (AF) Mott insulator where charge dynamics is completely frozen, or a AF spindensity-wave phase with at least moderately strong correlations. But the possibility of a SC phase in a strongly correlated band-insulator has been explored very little so far, either theoretically or experimentally.
In this work, we show how an AF spin-exchange mediated SC can be realized without doping in a simple model of a strongly correlated band insulator (BI), where the bare band gap (∆) and the e-e interaction (U ) both dominate over the kinetic energy. As U is increased (but still remains of the order of ∆), the single particle excitation gap in the BI closes, resulting in a metallic phase. Upon further increasing U , SC develops by the formation of a coherent macroscopic quantum condensation of electron pairs, provided the metal has enough low energy quasiparticles and the system has enough frustration against the magnetic order. The SC features tightly bound short Corr. Band 2ρ(ω=0) Para Metal Insulator FIG. 1: Zero temperature phase diagram for the ionic Hubbard model (see text) on a square lattice with e-e interaction U = 10t and t ′ = 0.4t as a function of the interaction to bare band-gap ratio (U/∆). For ∆ ≫ U ≫ t, the system is a correlated BI without any magnetic order. On increasing U/∆, first the gap in the single particle excitation spectrum closes, as shown by the non-zero single particle density of states at the Fermi energy ρ(ω = 0), resulting in a metallic phase. On further increasing U/∆, SC sets in and lasts over a significanly broad range (∆ ∈ [9.3 : 10]t) before ferrimagnetic order with non-zero staggered (ms) and uniform (m f ) magnetization sets in via a first order transition. This is a Ferri metal phase with non-zero, spin asymmetric, spectral density at the Fermi-level: ρ ↑ (ω = 0) = ρ ↓ (ω = 0) > 0. As U/∆ increases further, m f → 0 whence the magnetic order becomes AF and an up-spin spectral gap opens up such that ρ ↑ (ω = 0) = 0 while the down-spin electrons are still conducting, resulting in a sliver of AF half-metal. Eventually the system becomes an AF Mott insulator as U/∆ increases further.
coherence length Cooper pairs with a T c well separated from the energy scale at which the pairing amplitude builds up. The phase diagram, whose section with all model parameters fixed except for U/∆ is shown in Fig.1, presents a plethora of exoctic phases in the vicinity of a broad region of the SC phase.
Our starting point is a variant of the Hubbard model, known as the ionic Hubbard model (IHM), where, on a bipartite lattice with sub-lattices A and B, a staggered ionic potential ∆/2 is present in addition to coulomb repulsion (U ): The amplitude for electrons with spin σ to hop between sites i and j is t ij = t for near-neighbours and t ij = t ′ for second neighbours. The chemical potential µ is chosen to fix the average occupancy at n = 1 (half-filling). The staggered potential doubles the unit cell, and (for t ′ < ∆/4) induces a gap between the two electronic bands that result, making the system a BI for U = 0.
The parameter range of interest for this work is U ∼ ∆ ≫ t, t ′ , where a theoretical solution can be obtained based on a generalization of the projected wavefunctions method [13][14][15][16][17][18][19]. In this limit, at half-filling holons (doublons) are energetically expensive on the A (B) sites and can be projected out of the low energy Hilbert space. Consequently, though all hopping processes connecting the low and high energy sectors of the Hilbert space are eliminated, the system still has charge dynamics through first neighbor hopping processes such as |d A h B ⇔ | ↑ A ↓ B (with d representing a doublon and h a holon) and second neighbour hopping processes which allow doublons (holons) to hop on the A (B) sublattice [20].
The effective low energy Hamiltonian at half-filling, H ef f , is an extended t − t ′ − J − J ′ model acting on a projected Hibert space: Here J = 2t 2 /(U +∆) and J ′ = 4t ′ 2 /U . H 0 is the rescaled Hubbard interaction term in the projected Hilbert space and H d (H tr ) indicates other dimer (trimer) processes. We treat the projection constraint in H ef f using the generalised Gutzwiller approximation [18] and solve it using a renormalized Bogoliubov mean field theory. Gutzwiller approximations [15,16,18] of the sort we use have been well vetted against quantum Monte Carlo calculations [14,19] and dynamical mean field theory [18]. Details of this renormalized mean field theory, Gutzwiller approximation and the various terms in H ef f are given in the Supplementary Material (SM) [21]. Our main findings are summarised in the phase diagram of Fig. 1, which shows a linear section (along the U/∆ axis) of the full phase diagram in Fig. 2[e], for the IHM on a 2d square lattice at t ′ = 0.4t. The correlated BI, stable for ∆ ≫ U ≫ t, is paramagnetic and adiabatically connected to the BI phase of the non-interacting IHM. As ∆ approaches U , the low energy hopping processes (|d A h B ⇔ | ↑ A ↓ B ) become more prominent, increasing charge-fluctuations such that the gap in the single particle excitation spectrum closes, leading to a paramagnetic metallic (PM) phase with finite single particle density of states (DOS) ρ(ω = 0) at the Fermi energy, though for most of the parameter regime the PM phase is a compensated semi-metal with small Fermi pockets ( Fig. 3 and following discussion) [22]. On further increasing U/∆, in the presence of sufficiently large t ′ , robust SC sets in for ∆ ∼ U over a broad range of U/t (as shown in SM) due to the formation of coherent Cooper pairs of quasi-particles which live near the Fermi pockets, and survives for a sizeable range of ∆ (∈ [9.3 : 10]t). The pairing amplitudes ∆ d/s for both the pairing symmetries we have studied, namely, the d-wave and the extended s-wave, increase monotonically with U/∆ and drop to zero via a first order transition at the transition to the ferrimagnetic metal [27].
The ferrimagnetic metal (FM) phase is characterised by non-zero values of the staggered and uniform magnetizations m s,f = (m A ∓m B )/2 with m A/B being the sublattice magnetizations, along with finite spin asymmetric DOS ρ σ (ω = 0) at the Fermi energy. With further increase in U/∆ the FM evolves into an AF half-metal phase in which the system has only staggered magnetization (i.e., m f = 0) and the single particle excitation spectrum for up-spin electrons is gapped while the down-spin electrons are still in a semi-metal phase. Eventually, for a large enough U/∆, both spin spectra become gappedthe system becomes an AF Mott insulator [31].
We next discuss the changes in the behavior of the system with increasing U/∆ for varying values of t ′ , as depicted in Fig. 2. For t ′ = 0, the system shows a direct first order transition from an AF ordered phase to a correlated BI with a sliver of a half-metallic AF phase close to the AF transition point, consistent with most other theoretical work in this limit [32][33][34][35] barring one exception [36]. When t ′ is non-zero but small, due to the breaking of particle-hole symmetry as well as the frustration induced by the second neighbour spin-exchange coupling J ′ , the system first attains ferrimagnetic order for a range of U/∆, beyond which it has pure AF order as shown in panels (a,b) of Fig. 2. The magnetic transition occurs at increasingly larger values of U/∆ with increasing t ′ (except for an initial decrease for small values of t ′ ), which helps in the development of a stable SC phase.
To stabilize the SC phase, a minimum threshold value of t ′ (which is a function of U ) is required, partly in order to frustrate the magnetic order as mentioned above, but more importantly to gain sufficient kinetic energy by intra-sublattice hopping of holons and doublons on their respective sublattices where they are energetically allowed. While a stable d-wave SC phase turns on for t ′ > 0.1t for U = 10t, as shown in Fig. 2, SC in the extended s-wave channel gets stabilized for much larger value of t ′ > 0.35t . In an intermediate regime of U/∆ and t ′ , states with both d-wave and extended s-wave symmetry are viable solutions with energies that are very close (See SM for details). As t ′ increases, the pairing amplitude increases and the range of U/∆ over which the SC phase exists becomes broader for both the pairing symmetries studied [37].
The SC order parameter Φ d/s is defined in terms of the off-diagonal long-range order in the correlation function iγ creates a singlet on the bond (i, i + γ). Fig.  2 shows the SC order parameter, which has been obtained after taking care of renormalization required in the projected wavefunction scheme (see SM). Since the SC order parameter for this system is much smaller than the strength of the pairing amplitude, with increase in temperature the SC will be destroyed at T c by the loss of coherence among the Cooper pairs, leaving behind a pseudo-gap phase with a soft gap in the single particle density of states due to the Cooper pairs which will exist even for T > T c . Thus Φ d/s also provides an estimate of the SC transition temperature T c . The maximum estimated T c for U = 10t on a square lattice is approximately 0.03t for the d-wave SC phase, which for a hopping amplitude comparable to that in cuprates (t ∼ 0.4eV ) gives a T c ∼ 150K, and there is a considerable scope for enhancing T c by tuning U/∆ as well as t ′ .
FIG. 3: The top two rows show the spin resolved low energy spectral functions Aσ(k, ω ∼ 0) (integrated over |ω| ≤ (0.01 − 0.02)t for a 3000 × 3000 system) in the full BZ in the FM phase for t ′ = 0.35t, U = 10t, with up-spin (downspin) spectral functions shown in the first (second) row. At U/∆ = 1.09, the up spin channel has electron pockets while the down spin channel has small hole pockets. As U/∆ decreases, these Fermi pockets become bigger, the down (up) spin spectral function gets additional electron (hole) pockets. The last row shows A(k, ω ∼ 0) (same for up or down spins) for the PM phase. Moving towards the SC phase by increasing U/∆, Fermi pockets in the PM state go on expanding until they almost start touching each other, at which point the SC sets in by formation of Cooper pairs between electrons close to the Fermi energy.
A striking feature of the phase diagram in Fig. 2 is that, though the origin of SC in this model is due to the AF spin-exchange interactions, SC sets in only after the system has evolved to a PM or a FM phase. In order to understand the charge dynamics as the system approaches the SC phase, we have analysed the single particle spectral functions A σ (k, w ∼ 0) which can be directly measured in angle resolved photoemission spectroscopy (ARPES). Fig. 3 shows A σ (k, ω ∼ 0), non-zero values of which determine the energy contours on which low energy quasiparticles live in the Brillouin zone (BZ) (see SM for details). Panels (a-c) show A σ (k, w ∼ 0) in the FM phase for which the up-spin channel has electron pockets around the points K = (±π/2, ±π/2) in the BZ and the down spin channel has small hole pockets around the points K ′ = (±π, 0), (0, ±π) (see SM for details), as shown in panel (a). As U/∆ decreases within the FM phase, and approaches the SC phase, the electron pockets (hole-pockets) in the up-spin (down-spin) spectral function become bigger, and the down-spin channel gets additional electron pockets while the up-spin channel gets additional hole pockets, as shown in panel (c).
In the PM phase, A(k, w ∼ 0) has spin symmetric electron pockets (around K) and hole pockets (around K ′ ). As U/∆ increases through the PM phase, these Fermi pockets slowly expand such that they almost touch each other before the system enters into the SC phase. Similar behaviour is seen with an increase of t ′ in the PM or the FM phases (see SM for details). show the spin resolved single particle density of states (DOS) ρσ(ω) for t ′ = 0.15t and U = 10t. At U/∆ ∼ 1.04, ρ ↓ (ω = 0) is finite where as ρ ↑ (ω = 0) = 0 with a finite spectral gap, corresponding to the AF half-metal phase. At U/∆ = 1.03, the DOS at the Fermi energy is finite in both the spin channels but ρ ↑ (ω) = ρ ↓ (ω) corresponding to the FM phase. Panel (c) shows ρ(ω) for the d-wave and extd-swave SC phases for U = 10t and t ′ = 0.4t. ρ(ω) shows a linear increase with |ω| for ω ∼ 0 for both the SC phases. Panel (d) shows the gap in the DOS, which is the peak to peak distance in ρσ(ω), for both the d-wave and the extended s-wave pairing symmetries. Fig. 4. shows the spin-resolved DOS ρ σ (ω) vs ω which provides additional evidence for the existence of various metallic phases as depicted in the phase diagram in Fig. 2. The para metal, ferri-metal and the AF halfmetal phases are all compensated semi metals, which is reflected in the depletion in the DOS at the Fermi energy and is consistent with the small Fermi pockets shown in Fig. 3. We have also analysed the DOS in the SC phase. As shown in Fig. 4[c], ρ(ω ∼ 0) ∼ |ω| which is a signature of the gapless nodal excitations in the d-wave SC phase. Interestingly, even for the extended s-wave SC phase ρ(ω ∼ 0) ∼ |ω| as the pairing takes place around the small Fermi pockets centered at the K or K ′ points in the BZ, where the pairing amplitude ∆ s (k) = ∆ s (cos(k x ) + cos(k y )) has nodes as well, resulting in gapless excitations. The gap, which is the peak to peak distance in the DOS, is much larger in the d-wave SC phase than in the extended s-wave phase, consistent with the former being the stable phase . Infact for the extended s-wave phase, Gap s is only slightly larger than the SC order parameter Φ s , which indicates that the extended s-wave SC phase will have a narrower pseudogap phase above T c , compared to the d-wave case.
The origin of unconventional SC in most of the materials known today [3,5,7,8] can be understood in terms of the strongly correlated limit of the paradigmatic Hubbard model (single or multi band) but only upon doping the system away from half-filling [7,8,19,[38][39][40]. In the theoretical model we have studied here, SC appears even at half-filling, and therefore without the disorder that inevitably accompanies doping. A remarkable feature is that the SC phase in this model of a correlated BI is sandwiched between paramagnetic metallic and ferrimagnetic metallic phases, which makes the zero temperature phase diagram very different from that of the known unconventional SCs like high T c cuprates [7] or the more recent magic angle twisted bilayer graphene [5]. We expect that the SC phase in this model has transition temperatures comparable to those of cuprates and that it also has a pseudogap phase like in cuprates.
The IHM has been realized for ultracold fermions on an optical honeycomb lattice [41], where the state-of-the art engineering allows the parameters in the Hamiltonian to be tuned with great control. Hence it will be interesting and perhaps the easiest to explore our theoretical proposal in these systems. In the context of the recent developments in layered materials and heterostructures, it is possible to think of many scenarios where the IHM can be used as a minimal model, for example, graphene on h-BN substrate or bilayer graphene in the presence of a transverse electric field [42] which generates the staggered potential. The limit of strong correlation, crucial for realizing the SC phase, can be achieved in these materials by applying a strain or twist. Band insulating systems with two inequivalent strongly correlated atoms per unit cell, frustration in hopping and antiferromagnetic exchange, and lack of particle-hole symmetry, are likely tantalizing candidate materials as well. Our work suggests that the search for such novel materials where superconductivity can be realized at half filling with sufficiently high transition temperatures can perhaps emerge as an exciting, though challenging new research frontier in condensed matter physics. (1) of the main paper in the limit U ∼ ∆ ≫ t, t ′ . In this limit and at half-filling, holons are energetically expensive on the A sites (with onsite potential − ∆ 2 ) and doublons are expensive on the B sites (with onsite potential ∆ 2 ); i.e., in the low energy subspace h A and d B are constrained to be zero. We do a generalized similarity transformation on this Hamiltonian,H = e −iS He iS , such that all first and second neighbour hopping processes connecting the low energy sector to the high energy sector of the Hilbert space are eliminated. The similarity operator of this transfor- where H + t/t ′ represents first or second neighbour hopping processes which involve an increase in h A or d B by one and H − t/t ′ on the other hand represent hopping processes which involve a decrease in h A or d B by one.
Further details can be found in [18]. H ef f acts on a projected Hilbert space which consists of states |Φ = P|Φ 0 where the projection operator P eliminates components with h A ≥ 1 or d B ≥ 1 from |Φ 0 . We use here the Gutzwiller approximation [15,18,19] to handle the projection, by writing the expectation value of an operator Q in a state P|Φ 0 as the product of a Gutzwiller factor g Q times the expectation value in |Φ 0 so that Q ≃ g Q Q 0 . The standard procedure [15] for calculating g Q has been generalised by us for the case where holons are projected out from one sublattice and doublons from the other [18].
We thus obtain the renormalized effective Hamiltonian with the inter-sublattice kinetic energy c † iAσ c jBσ ≈ g tσ c † iAσ c jBσ 0 , and intra-sublattice kinetic energy c † iασ c jασ ≈ g ασ c † iασ c jασ 0 . The intersublattice spin correlation S iA · S jB ≈ g sAB S iA · S jB 0 while the intra-sublattice spin exchange term gets renormalized with a different factor of g sαα . The only other dimer term which does not get rescaled under the Gutzwiller projection is as it consists of only density operators [15,18]. Then we have the important trimer terms: The various Gutzwiller factors involved (see [18] for details) are as follows: g Aσ = 2δ/(1 + δ + σm A ),g Bσ = 2δ/(1 + δ − σm B ), g tσ = √ g Aσ g Bσ , Below we give details about the superconducting order parameter and the spectral functions.
Superconducting order parameter Φ d/s : The SC correlation function is the two particle reduced density matrix defined by i+γB↑ creates a singlet on the bond (i, i+γ) where γ is x or y, considering d-wave pairing symmetry (∆ x AB = −∆ y AB ≡ ∆ d ) and extended s-wave pairing symmetry (∆ x AB = ∆ y AB ≡ ∆ s ) separately. The SC order parameter Φ d/s is defined in terms of the off-diagonal long-range order in this correlation Since F γ1γ2 (r i − r j ) also corresponds to hopping of two electrons from (j, j + γ 2 ) to sites (i, i + γ 1 ), in the projected wavefunction scheme it scales just like the product of two hopping terms such that F γ1γ2 ≈ g A↑ g B↓ F 0 γ1γ2 . Hence the rescaled form of the superconducting order is the order parameter calculated in the unprojected wavefunction of the low energy effective Hamiltonian in Eq. (2).
Spectral Functions and Density of States: In the main paper we also discussed the single particle density of states (DOS) and the spectral functions. In the Gutzwiller projection method, the Green's function is rescaled with the appropriate Gutzwiller factor such that G ασ (k, ω) = g ασ G 0 ασ (k, ω) where G 0 ασ (k, ω) is calculated in the unprojected basis. Here α represents the sublattice A or B and σ is the spin index. The spectral function, A ασ (k, ω) which is imaginary part of the Green's function also get rescaled with the same Gutzwiller factors. The results presented in the paper are for the spectral functions averaged over the two sublattices A σ (k, ω) = 2↓ (k)). The down spin spectral function can be obtained by replacing u i↑k ↔ v i↓k (and vice-versa) and by replacing E iσ (k) by −E iσ (k). Here E 1,2,↑ (k) are the eigenvalues of the BdG equation for a given k in the BZ with eigenvectors (u 1↑k , u 2↑k , −v 1↓k , −v 2↓k ) and (u 3↑k , u 4↑k , −v 3↓k , −v 4↓k ) respectively and −E 1,2↓ are eigenvalues corresponding to eigenvectors obtained by u iσk → v iσk and v iσk → −u iσk . In order to get the low energy spectral functions, presented in the main paper, we integrate A σ (k, ω) over a small ω range such that |ω| ≤ (0.01 − 0.02)t.
The single particle density of states is defined as, ρ ασ (ω) = k A ασ (k, ω). The results presented in the paper are for the single particle density of states (DOS) in the up spin and down spin channels, defined as ρ σ (ω) = (ρ Aσ (ω) + ρ Bσ (ω))/2. The zero temperature momentum distribution function, which helps in identifying whether a Fermi pocket is an electron pocket or a hole pocket (and is presented in section SM F) can also be obtained from the spectral function using n σ (k) = 0 −∞ dωA σ (k, ω).

SM B. Details of the renormalized mean field theory:
We solve the renormalized effective low energy Hamiltonian using three different versions of the renormalized mean field theory (RMFT). To explore the SC phase, we use a generalised spin-symmetric Bogoliubov mean field theory, which basically maps onto a two-site Bogoliubov-deGennes (BdG) mean field theory for each allowed k point in the BZ. We do a mean field decomposition of the various terms in the Hamiltonian, and self-consistently solve for the following mean fields : and extended s-wave pairing symmetry (∆ x AB = ∆ y AB ≡ ∆ s ) separately; (b) density difference between two sublattices, δ = (n A − n B )/2; (c) inter sublattice fock shifts, χ ± y or i ± 2y ± x; and (d) intra sublattice fock shift on A(B) sublattice, with χ αασ = c † iασ c i±2x/2yασ +h.c. , and χ ′ αασ = c † iασ c i±x±yασ +h.c. . To explore the magnetic order and the phase transitions involved, we solve the renormalized Hamiltonian using standard mean field theory allowing non-zero values of the sublattice magnetization m α = n α↑ − n α↓ with α = A, B, from which one gets the staggered magnetization m s = (m A − m B )/2 and the uniform magnetisation m f = (m A + m B )/2, along with all other mean-fields mentioned above except for the SC pairing amplitudes ∆ s/d . The third calculation, where we allow for both the SC pairing amplitudes and the magnetization along with all other mean fields metioned above, uses a standard canonical transformation followed up by the Bogoliubov transformation to diagonalise the mean field Hamiltonian neglecting the inter-band pairing as weak. We solve the RMFT self-consistent equations on the square lattice for various values of U, ∆ and t ′ to obtain the phase diagram reported in the paper. In the parameter regime where solutions with nonzero SC pairing amplitudes and magnetization (from the first two calculations) are both viable, we compare the ground state energy of the two mean-field solutions to determine the stabler ground state. We finally compare the energy of this state with the one obtained in the third calculation to determine the true ground state. Below we give details about the ground state energy comparisons.

SM C. Competing Order-Parameters and Ground State Energy Comparison:
Comparison of the results from the first two calculations shows that there is a significantly broad regime of parameters over which the SC and magnetic orders both exist and compete with each other. In order to determine the true nature of the ground state in this parameter regime, we compare the ground state energies of the different RMFT solutions. As shown in Fig. S1, even for small values of t ′ , the SC pairing amplitudes, in both the pairing channels studied, turn on but the magnetic transition precedes the transition into the SC phase.
Once the magnetic order turns on, the ground state energy of the non-superconducting solution becomes lower than that of both the SC phases studied as shown in the right panels of Fig. S1. Thus for t ′ < 0.1t there is no stable SC phase, as shown in Fig (2e) of the main paper. For larger values of t ′ , as U/∆ increases superconductivity turns on before the magnetic order sets in. There continues to be a solution of the RMFT with pairing amplitudes, in either of the symmetry channels, non zero even in the magnetically ordered regime, but the non-superconducting magnetically ordered solution is lower in energy here. Thus the pure SC phase is a stable phase only before the magnetic transition point.  There is a third scenario possible where one can do a RMFT allowing for non-zero values of both SC and magnetic order parameters along with other mean fields. Before the magnetic order turns on, this theory is consistent with the spin-symmetric Bogoliubov theory described above. After the magnetic order sets in, differences between the two calculations become visible. In the third calculation, the SC order coexists with the ferrimagnetic order for a range of parameters as shown in Fig. S2 though the pairing amplitudes decrease with increasing U/∆. Comparing the energy of this phase with that of the ferrimagnetic metal phase, which was found to be the stabler phase by comparing energies of first two calculations in this regime, we find that the coexistence phase is also a metastable phase and the system actually stabilizes into the ferrimagnetic metallic phase as shown in Fig. 2, of the main paper.

M E T A -S T A B L E S C S T A B L E
SM D. Phase-diagram in U/t − U/∆ plane for a fixed t ′ : In the main paper we have shown phase-diagrams for the IHM on a 2d square lattice for a fixed value of U/t. Fig. 2[e] showed the phase diagram in t ′ /t − U/∆ plane for a fixed U and Fig. 1 showed a section of this phase diagram for t ′ = 0.4t.
In order to understand how the different phases and the phase boundaries between them evolve with varying U , here we have shown the phase diagram in U/t − U/∆ plane for a fixed t ′ /t. As shown in Fig. S3, superconductivity always turns on for U ∼ ∆ irrespective of the value of U/t though with increase in U/t, the range of U/∆ over which both pairing symmetries are almost degenerate solutions shrinks rapidly such that eventually, for large enough values of U/t, the system has only a d-wave SC phase.

SM E. Low Energy Spectral Functions with Varying t ′ :
In order to understand the charge dynamics as the system approaches the SC phase with the tuning of second neighbour hopping, t ′ , we have analysed the single particle spectral functions for a fixed U/∆ in the ferrimagnetic metallic phase. Note that the main text showed how the low energy spectral functions A σ (k, ω ∼ 0) change with the tuning of U/∆ for a fixed t ′ . We can understand why the SC phase does not get stabilized for small values of t ′ by looking at the evolution of A σ (k, ω ∼ 0) for a fixed U/∆ as one tunes t ′ . Fig.  S4 shows A σ (k, ω ∼ 0) close to the magnetic transition point of t ′ = 0, that is, for U/∆ = 1.02. For small values of t ′ , at this value of U/∆ the system is in the ferrimagnetic metal phase. As we increase t ′ inside the ferrimagnetic metal phase, the up spin spectral functions get bigger electron pockets around K = (±π/2, ±π/2) points while the down spin spectral functions get bigger hole pockets around K ′ = (±π, 0), (0, ±π) points. In addition to this, as t ′ increases even the up-spin spectral functions get hole pockets and the down spin spectral functions get electron pockets. As a result of both these effects, an almost connected contour of Fermi pockets is formed, whence superconductivity emerges by the formation of Cooper pairs of the corresponding low energy quasiparticles. The momentum distribution function n σ (k), defined in the section (SM A), is uniformly half in the entire BZ for any insulating phase of the model studied here. When the system goes into a metallic phase, at least one of the bands cross the Fermi level resulting in filled or empty Fermi pockets depending on the curvature of the band. Filled Fermi pockets, also called electron pockets, have n σ (k) > 1/2, while empty Fermi pockets, also called hole pockets, have n σ (k) < 1/2. Fig. S5 shows n σ (k) for t ′ = 0.35t for two values of U/∆. Panel (a) shows the result for the ferrimagnetic metal phase and panel (b) shows the results in the para metal phase. In the ferri-metal phase, n ↑ (k) has filled pockets around the K points while the down-spin component has hole pockets around the K ′ points in the BZ. In the para-metal phase, shown in panel (b), there is a spin symmetry and n σ (k) has electron and hole pockets for both the spin channels. Fig. S6 shows the band dispersion E nσ (k) for both the bands on paths along high symmetry directions in the BZ. In the AF half-metal phase, the down spin channel has small hole pockets around K ′ and tiny electron pockets around K. In the Ferrimagnetic metal phase, the down spin band E 1↓ (k) crosses the Fermi energy around the K ′ points resulting in small hole pockets and E 2↑ (k) crosses the Fermi energy near the K points resulting in small electron pockets. In the paramagnetic metal phase, E 1 (k) crosses the Fermi FIG. S5: Momentum distribution function nσ(k) in the ferrimagnetic metal and the para metal phases for t ′ = 0.35t. In the ferrimagnetic metal phase shown in panel (a) n ↑ (k) > 1/2 on (electron) pockets centered around the K points while n ↓ (k) < 1/2 on (hole) pockets centered around the K ′ points in the BZ. Panel (b) shows the results for the paramagnetic metal phase, where the systen has spin symmetry and nσ(k) < 1/2 around the K ′ points while nσ(k) > 1/2 around the K points for both the spin components. Everywhere else in the BZ nσ(k) = 1/2 in all the panels. energy around the K ′ points resulting in hole pockets and E 2 (k) crosses the Fermi level around K resulting in electron pockets, where, because of the spin symmetry, we have suppressed the spin indices.
A σ (k, ω ∼ 0) for the AF half-metal phase (see Fig. S7), which is fully consistent with the band-dispersions shown above. The up-spin channel is gapped while A ↓ (k, ω ∼ 0) has tiny electron pockets at the K points and hole pockets at the K ′ points in the BZ.