A Simple Mechanism for Unconventional Superconductivity in a Repulsive Fermion Model

Motivated by a scarcity of simple and analytically tractable models of superconductivity from strong repulsive interactions, we introduce a simple tight-binding lattice model of fermions with repulsive interactions that exhibits unconventional superconductivity (beyond BCS theory). The model resembles an idealized conductor-dielectric-conductor trilayer. The Cooper pair consists of electrons on opposite sides of the dielectric, which mediates the attraction. In the strong coupling limit, we use degenerate perturbation theory to show that the model reduces to a superconducting hard-core Bose-Hubbard model. Above the superconducting critical temperature, an analog of pseudo-gap physics results where the fermions remain Cooper paired with a large single-particle energy gap.

In general, the dominant contributions to an electron Hamiltonian can be modeled on a lattice by a generalized Hubbard model: where I and J include position, orbital, and spin degrees of freedom. The electron hopping t IJ describes the kinetic energy contribution, while the Hubbard interaction U IJ ≥ 0 describes the repulsive Coulomb force. In order to superconduct, pairs of electrons must form a bosonic bound state, the so-called Cooper pair, which must then condense 1 . It may seem unnatural for electrons to form a bound state due to repulsive interactions. Nevertheless, various mechanisms for how this could occur have been proposed in the 1 That is, the Hamiltonian must have ground states with b(x) = 0, where b(x) ∼ IJ ψ IJ (x) c I c J is a Cooper pair annihilation operator and ψ IJ (x) is the superconducting order parameter. This occurs when the charge conservation symmetry (c I → e iθ c I ) is spontaneously broken, as per Ginzburg-Landau theory. (As is commonly done, we are approximating the electrodynamic gauge field as a classical background field, rather than a dynamical field [23] which would be integrated over in the partition function.) literature, such as Fermi surface instabilities [24][25][26][27][28][29][30][31][32] and inhomogeneity-induced superconductivity [33][34][35][36][37].
In this work, we present a simple (and perhaps minimal) toy model of fermions with a repulsive interaction, which gives rise to a superconducting state with short coherence-length Cooper pairs in the strong coupling limit. The mechanism that we are proposing is based on the local physics of a 4-site spinless fermion model with a repulsive interaction; hence it is very different from a Fermi surface instability, while somewhat related to inhomogeneity-induced superconductivity.
We emphasize that our motivation is not to describe any current material. Rather, the motivation is to find a minimal model of unconventional superconductivity to gain physical intuition, which may be useful for designing or discovering new classes of superconducting materials.

I. SUPERCONDUCTIVITY MECHANISM
A simple mechanism for the emergence of an effective attraction due to repulsive interactions can be understood from the following 4-site spinless fermion model: c α are four spinless fermion annihilation operators with site/orbital index α = 1, 2, 3, 4; n α = c † α c α is the fermion number operator. s is a fermion hopping strength; U is a nearest-neighbor Hubbard repulsion; and µ is the chemical potential.

II. MINIMAL MODEL
By using H (4) to generate an effective attractive interaction, we can write down a simple repulsive fermion model with a superconducting ground state in the limit of strong interactions. The model is simply a grid of coupled H (4) models: Here, we are considering two coupled two-dimensional square-lattice layers (black).
ij sums over all pairs of nearest-neighbor unit cells i and j. Each unit cell i is composed of an H (4) model, which includes four spinless fermions c i,α indexed by α = 1, 2, 3, 4. H adds a hopping term t and additional chemical potential µ to the α = 1, 2 fermions. We will focus on the following limit: In this limit, each H to remain in its ground state, H must act twice in order to move two fermions (a Cooper pair) from one site to another. Thus, we can use degenerate perturbation theory [40,41] to obtain a low-energy effective Hamiltonian at order t 2 /s. (See Appendix B for details.) The resulting model can be written in the form of the following hard-core boson model: A phase diagram of the hard-core boson model [Eq. (7)] extracted from Refs. [42,43]. A large chemical potential µ results in a Mott insulating phase with boson number η = 0 or η = 1 on every site. A large boson repulsion V eff induces a charge density wave where half of the hard-core boson states are filled in a checkerboard pattern. In between these phases is a superconducting phase. The fermion model [Eq. (5)] results in V eff /t eff = 2. (There is a hidden SU (2) symmetry when µ = 0 and V eff /t eff = 2.) where V eff = 2t eff and t eff = t 2 /s.
The hardcore constraint implies that the boson number operator (H eff can also be transformed into an XXZ spin model in a magnetic field 3 .) Physically, the boson is a Cooper pair of the fermions: b i ∼ c i,1 c i,2 . The boson hopping term t eff b † i b j results from a virtual process (t c † i,2 c j,2 )(t c † i,1 c j,1 ) that hops two fermions from site j to i. The nearest-neighbor repulsion V eff (η i − 1 2 )(η j − 1 2 ) results from a virtual process (t c † j,α c i,α )(t c † i,α c j,α ) where a fermion hops from site j to i and then back to j. In both virtual processes, the intermediate state has a large energy s t, which penalizes the virtual processes and results in the energy scaling t 2 /s for t eff and V eff .
The phase diagram of this effective boson model is shown in Fig. 1. The ground state is in a superfluid phase when 0 = |µ | < 4t eff (and V eff = 2t eff ). Since the effective boson carries the same charge as two fermions, the superfluid in the effective model corresponds to 3 H eff in Eq. (7) can be viewed as an XXZ spin-1/2 model by replacing the hard-core boson operator b j → 1 2 (σ x j + i σ y j ) with Pauli operators σ µ i . Since V eff = 2t eff , H eff can also be transformed into an SU (2) anti-ferromagnetic Heisenberg model in an applied field H eff = 1 2 t eff ij σ i · σ j − µ i σ z i by rotating the spins on the A sublattice by the unitary operator a superconductor in the original fermion model H [Eq. (5)]. Therefore, the ground state of H is a superconductor in the limit of interest [Eq. (6)] when 0 = |µ | < 4t 2 /s.

III. EXTENSIONS
Finding a material that realizes the model [Eq. (5)] in the previous section may be difficult. In this section, we will exemplify possible ways that the model can be extended in order to make it more realistic. In Appendix A, we will also explore an alternative lattice geometry. An actual material may realize more than one of these extensions.

A. Missing Hopping
The absence of a fermion hopping between sites 3 and 1 (and between 3 and 2) may seem peculiar. Let us then consider the effect of extending the 4-site model [Eq. (2)] with such a hopping: If the hopping energy t is much less than the energy gap s/2, then the low-energy eigenstates and energies in Eq. (3) will not change. But if t s/2, then the low-energy states and energy spectrum will change significantly, which will likely destroy the superconducting ground state when the 4-site clusters are coupled together.
However, we should also consider the effect of a repulsive interaction between sites 3 and 4, which we will write as: Since U is large, it is natural to also consider the possibility that U is also large. In this case, U energetically forbids states that do not have a total of one fermion on sites 3 and 4. Therefore, if we consider extending the 4-site model by these terms, (2), (8), (9)], then this extended model will have the same low-energy eigenstates and energies in Eq. (3) as the original 4-site model as long as t U (and µ = s/2 and s U as before). Therefore, when the 4-site clusters are coupled together, the additional hopping t will not hamper superconductivity as long as t U .

B. Covalent Bonds
The geometry of the 4-site cluster [Eq.
(2)] may seem unnatural in a material realization due to site 4, which only couples to a single site. However, the cluster can be expanded such that the middle section models a covalent bond between two or more ions (or nuclei).
We will now discuss the example involving a covalent bond between two ions, which can be modelled by the following 5-site cluster: c α are five spinless fermion annihilation operators with site/orbital index α = 1, 2, 3, 4, 5. Sites 3, 4, and 5 model a covalent bond between a pair of ions at sites 4 and 5. Site 3 is (very coarsely) modelling the electron states between the two ions. One could think of sites 4 and 5 as s orbitals, while site 3 can be thought of as the superposition of p x orbitals of ions 3 and 4 that constructively interferes in the area between the ions. However, other combinations of orbitals are also possible.
When µ = s/ √ 2 and s U , the two lowest energy levels are: , the ground states are two-fold degenerate and act as hard-core boson states. When an electron is at site 1 or 2, the covalent bond is damaged since the repulsive interaction U prevents fermions from hopping onto site 3. In the covalent bond picture, the covalent bond mediates an effective attractive interaction [of the form of Eq. (4)] between the fermions on sites 1 and 2, and a filled hard-boson corresponds to a damaged covalent bond.
If many 5-site clusters are weakly coupled together in a grid [similar to Eq. (5)], then the low-energy physics can be effectively described by a hard-core boson model [Eq. (7) with V eff = 2t eff and t eff = t 2 / √ 2s]. A superconducting ground state results when 0 = |µ | < 4t eff = 2 √ 2t 2 /s, where µ is a chemical potential for the fermions at sites 1 and 2 [just as in Eq. (5)].

C. Spinful Fermions
The previous models have all involved spinless fermions. But electrons are spin-half particles. In this section, we will show that if a spin degree of freedom is added to the fermions in the 4-site model, then a superconducting ground state may still result as long as a large on-site Hubbard repulsion is included.
Alternatively, an applied magnetic field could gap out the spin degree of freedom and effectively result in the spinless fermion model in Sec. II.
A spinful generalization of the 4-site model is: where n α = σ n α,σ is the total fermion number on site α and σ =↑, ↓ denotes to two electron spin states. When µ 12 = 2µ 34 = s, s U 0 , and s U 1 , the lowest energy levels are: The ground states are five-fold degenerate. |1 σσ denotes four different spin states indexed by σ, σ =↑, ↓. Therefore, the low-energy states behave like a hard-core boson with four different spin states.
If many spinful 4-site clusters are coupled together in a grid [similar to Eq. (5)], then the low-energy physics can be effectively described by a hard-core boson model [similar to Eq. (7)] where the boson has four spin states. It is feasible that this hard-core boson model has a superconducting ground state in some regions of its phase diagram. However, additional perturbations should be added to the model since they will generically split the degeneracy between the spin singlet and triplet states of the hard-core boson.

IV. DISCUSSION
We have considered a simple two-dimensional lattice model [Eq. (5)] with a superconducting ground state. At temperatures below the single-particle fermion gap s/2 [from Eq. (3)], the model is well-approximated by a hard-core boson model [Eq. (7)] with hopping strength t eff ∼ t 2 /s. If we consider a 3D stack of the 2D model with a weak fermion hopping between the stacks, then the resulting three-dimensional model can be expected 4 4 Before coupling the stacks, each layer is in a state with quasilong-range superconducting order below a Berezinskii-Kosterlitz-Thouless transition [44] critical temperature T KT ∼ t eff .
Since the correlation length of each layer is infinite, a weak fermion coupling between the layers will result in a longrange superconducting order with roughly the same critical temperature Tc ∼ T KT .
to exhibit superconductivity at temperatures below T c ∼ t eff ∼ t 2 /s. Although we only considered a single corner [Eq. (6)] of the phase diagram, any sufficientlysmall local perturbation can be added without destroying the superconductivity. The Cooper pairing is ultimately a result of the local Coulomb repulsion physics in the 4-site fermion model [Eq. (2)], and the size of the Cooper pair is just a single unit cell. Because the Cooper pairing results from charge interactions, it is interesting to note that the superconducting phase neighbors a charge density wave order (Fig. 1).
At temperatures above the superconducting critical temperature T c but below T ∼ s, the fermions are Cooper paired with a gap ∆ ≈ s/2 to singlefermion excitations.
Ref. [45] showed that the DC (zero frequency) resistivity of the hard-core boson model [Eqs. (7)] with µ = V eff = 0 is 5 at high temperatures, which could apply to our fermion model H [Eq. (5)] in the temperature range t eff T s. This regime of a large single-particle gap and large resistivity (linear in temperature) therefore appears to be an s-wave analog of the pseudo-gap physics [46][47][48][49] seen in the cuprate and iron-based superconductors. However, Ref. [45] only considered a hard-core boson model at half filling and without a nearest-neighbor Hubbard repulsion (µ = V eff = 0). Future work is required to determine the robustness of the large linear resistivity (ρ ∝ T ) to these perturbations, which are present in our low-energy boson models.
As discussed in Sec. III, the model can be extended in various ways that could help facilitate a material realization. For example, it is possible to view the fermions on sites 3 and 4 as an effective covalent bond between ions (which could be modeled using more than just two sites). In this picture, the presence of a fermion at site 1 or 2 damages the covalent bond by imposing a large Coulombic energy repulsion, which ultimately leads to Cooper pairing. Furthermore, the effective spinless fermion model [Eq. (5)] could result from an applied (or induced) in-plane magnetic field, which can gap out the spin degree of freedom. Such an example is of practical interest since it can result in a superconductor that is more robust to strong magnetic fields.
Another possibility is to think of the 4-site model [Eq. (2)] as a minimal model for a molecule. If a molecule with similar physics can be discovered, then a liquid or crystal of such molecules could exhibit superconductivity. In particular, the lowest-energy states of the molecule should have fermion occupation numbers that differ by even integers (with at least one of them non-zero), as in Eq. (3). In fact, this kind of physics has already been shown to occur in doped buckminsterfullerene C 60 molecules [60][61][62].
Hard-core boson models have been shown to emerge in certain repulsive fermions models in previous work [35,36]. These works considered inhomogeneous spinhalf Hubbard models, which were motivated by existing materials. Our motivation differed in that our objective was to find the simplest possible theoretical model of unconventional superconductivity without a bias towards existing materials. The advantage of this approach is that it helps to elucidate the simplest possible mechanism for Cooper pairing, and resulted in a unique and novel model. Indeed, the physics of our new model is very different from Hubbard model physics since spin plays an important role in the Hubbard model, but not in our model.
We hope that our model will help improve our understanding of unconventional superconductivity and inspire new candidate superconducting materials. In particular, we hope that it will be possible to construct layered materials that are engineered to mimic models similar to the models discussed in this work. This may be feasible since the superconducting critical temperature T c of our models is relatively easy to predict, which could lead to material guidelines for a higher T c in certain cases.
We thank Arun Paramekanti, Alex Thomson, Jong Yeon Lee, and Leonid Isaev for helpful discussions. This work was supported by the NSERC of Canada and the Center for Quantum Materials at the University of Toronto. KS also acknowledges support from the Walter Burke Institute for Theoretical Physics at Caltech.
i and j index different 3-site clusters, while α = 1, 2, 3 indexes the three sites within a cluster. H (6) ij couples two 3-site clusters (i and j) together.
When V = µ = s/2 and s U , the two lowest energy levels of H (6) ij are: The lowest energy level is now triply degenerate. But since the three low-energy states each differ by an even number of fermions, we can still think of them as hardcore boson states | η i η j with fillings η i , η j = 0, 1 but where the | 11 state is gapped out due to a large effective bosonic repulsive interaction. If this effective boson condenses, then a superconducting state will result. To achieve this, we will embed H (6) ij into a layered triangular lattice: ij sums over all nearest-neighbor 3-site clusters (along the solid gray and black lines), while ij only sums over the neighboring 3-site clusters with a red line between them. We will focus on the following limit: Again, we can use degenerate perturbation theory to derive a low-energy effective hard-core boson model (see Appendix B 2 for details): The last term in H eff is a four-boson repulsion term that results from a virtual process (t c † j,α c i,α )(t c † i,α c j,α ) where a fermion hops across a black link from site j to i and then back to j. The projection operators (1−ηî)(1− η) result due to the η i ηî = η j η = 0 constraint, which prevents the virtual process from occurring whenî or is occupied by a boson. At a mean-field level, we can think of the projection operators as effectively weakening the repulsive interaction (η i − 1 2 )(η j − 1 2 ) and shifting the chemical potential µ .
Given its similarity to H eff [Eq. (7)] in the previous section, H eff is likely to also have a superfluid ground state for certain µ . However, this will have to be checked numerically, which could be done using sign-free 6 quantum Monte Carlo [63]. This implies that the original model H [Eq. (A3)] is also likely to have a superconducting ground state in the limit considered in Eq. (A4) for some range of µ . and H are defined in Eq. (5). We will work using the limit in Eq. (6), and derive H eff up to corrections of order O(t 4 /s 3 ) and O(s 2 /U ).
First, we will define hard-core boson annihilation and number operators that act on the unperturbed ground states [Eq. (3)] as follows: The boson operators can be written in terms of the fermions as Within the ground state space of H 0 , into which P projects, the above can be simplified to The √ 2 appears in order to cancel the 1 √ 2 in |0 [Eq. (3)]. The first non-constant term of H eff in Eq. (B2) is Eq. (B9) results because the grounds states of H 0 always have η i = n i,1 = n i,2 , The next term is given by 6 H eff does not have a sign problem since −H eff , which appears in the Boltzmann factor e −βH eff , has positive off-diagonal elements when viewed as a matrix in the boson number basis.
In Eq. (B11), we are ignoring higher order O(t 2 /U ) terms. Adding together Eqs. (B9) and (B13) reproduces H eff in Eq. (7) up to constant terms, which we ignore in the main text.

Triangular Model
To derive H eff in Eq. (A5), we define where H (6) ij and H are defined in Eq. (A3). We will work using the limit in Eq. (A4), and derive H eff up to corrections of order O(t 4 /s 3 ) and O(s 2 /U ).
We will define hard-core boson annihilation and number operators that act on the unperturbed ground states [Eq. (A2)] as follows: If is the 3-site cluster across a red link [shown in Eq. (A3)] from j, then b acts similarly but on the first digit in the ket; e.g. b †  | 00 j = | 10 j . Note that within the above Hilbert space, the following constraint is obeyed: ηη j = 0. The boson operators can be written in terms of the fermions as b j = c j,1 c j,2 1 √ 2 (c † j,3 + c † ,3 )(1 − n j,3 )c ,3 η j = b † j b j = n j,1 n j,2 (1 − n j,3 )n ,3 Within the ground state space of H 0 , into which P projects, the above can be simplified to P b j P = √ 2 P c j,1 c j,2 P P η j P = P n j,1 n j,2 P (B17) The first non-constant term of H eff in Eq. (B2) is Eq. (B19) results because the grounds states of H 0 always have η i = n i,1 = n i,2 . The next term is given by