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can be described by an effective Hamiltonian ˆ H$_{eff}$ act- ing in the subspace S$_{1}$ defined by Eq. (2). Treating the first line of ˆ H$_{ffb}$ as perturbation ˆ H$_{1}$ , we can expand ˆ H$_{eff}$ according to degenerate perturbation theory. 1 Up to sec- ond order, using ˆ n$_{ib}$ = 1 − ˆ n$_{if}$ in S$_{1}$ , one finds 

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where ˆ P projects on S$_{1}$ and where we have introduced the usual spin operators ˆ S i ≡ 1 2 ∑ s ′ s ˆ f † is $_{′}$σ s ′ s ˆ f$_{is}$ with Pauli matrices σ s ′ $_{s}$. Moreover, 

with δ ν ij ≡ [( U ν V$_{i}$ − $_{V$_{j}$}$) 2 − 1] − $^{1}$, I bb ij ≡ 4 ( t b $_{ij}$) 2 U bb [1 + δ bb $_{ij}$], I fb ij ≡ 2 ( t f $_{ij}$) $^{2}$+( t b $_{ij}$) 2 U fb [1 + δ fb ij ], δI fb ij ≡ ( t f $_{ij}$) $^{2}$− ( t b $_{ij}$) 2 U fb δ fb ij U fb V$_{i}$ − $_{V$_{j}$}$, and adj( i ) containing all sites adjacent to i . In Hamilto- nian (3), t$_{ij}$ describes boson-fermion swaps, i.e. effective fermion tunneling, W$_{i}$ is the effective potential felt by fermions, J$_{ij}$ stands for the usual fermionic antiferromag- netic superexchange coupling, and U nn ij captures boson- mediated nearest-neighbor interaction between fermions that can be attractive, repulsive or zero. 

Hamiltonian (3) is equivalent to a t - J -type model, de- scribing a purely fermionic system when strong repulsion suppresses double occupancy. This can be seen by intro- ducing composite-fermion creation operators 

We define ˜ n$_{is}$ ≡ ˆ c † $_{is}$ˆ c$_{is}$ and ˜ S i ≡ 1 2 ∑ s ′ s ˆ c † is $_{′}$σ s ′ $_{s}$ˆ c$_{is}$ . Using ˆ P ˆ f † is ′ ˆ f$_{is}$ ˆ P = ˆ P ˆ f † is ′ (1+ ˆ b † $_{i}$ˆ b$_{i}$ ) ˆ f$_{is}$ ˆ P = ˆ P ˆ c † is ′ ˆ c$_{is}$ ˆ P , yields both ˜ n$_{is}$ = ˆ n$_{is}$ and ˜ S i = ˆ S i in S$_{1}$ . Bosons transform to empty sites (holes) and S$_{1}$ to the subspace defined by ˜ n$_{i}$ ≤ 1 with ˜ n$_{i}$ ≡ ˜ n$_{i}$$_{↑}$ + ˜ n$_{i}$$_{↓}$ . Thus, we can rewrite ˆ H$_{eff}$ as t - J 

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Hamiltonian in terms of ˆ c † $_{is}$-fermions alone: 

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Transformation (8) (being inverse to a the slave-boson transformation) is illustrated in Fig. 1. nn 

The t - J model (9) with W$_{i}$ = const., U ij = 0, t$_{ij}$ = t , J$_{ij}$ = J ≈ t/ 3, and boson fraction x , is the most simple candidate to explain high-temperature cuprate supercon- ductivity [8–10, 21]. The fermion-boson mixture consid- ered here, allows one to realize these parameters quite accurately: Starting from a cubic lattice, the tunneling amplitudes t f ij and t b ij can be suppressed in one direction by ramping up the lattice in that direction, leading to a stack of uncoupled square lattices layers. From now on, we will only consider the intra-layer physics. Creating the optical lattice by using a rather broad laser beam, within the occupied region of the trap (created by fur- ther beams) the lattice will be practically homogeneous. Thus, t f ij ≃ t f and t b ij ≃ t $^{b}$. Furthermore, potential differ- ences | V$_{i}$ − V$_{j}$ | between neighboring sites are much smaller than U $^{ff}$, U fb and U $^{bb}$, giving δ ν ij ≲ 10 − 3 ( ν = ff, fb, bb ) for the parameters estimated in a previous paragraph. With that the model parameters (4)-(6) are to good ap- proximation homogeneous within the simulator region, t$_{ij}$ ≃ t , J$_{ij}$ ≃ J , and U nn ij ≃ U $^{nn}$. By the very same arguments, also the last three terms contributing to W$_{i}$ [Eq. (7)] have a negligible spatial dependence compared to the effective hopping parameter t , being the relevant energy scale here. Thus, if also the difference between boson and fermion potential δV$_{i}$ is smaller than t , one achieves a practically flat effective potential W$_{i}$ ≃ W , without fermions leaking out of the system, cf. Fig. 1(b). 

In an experimental realization, the hole fraction x is controlled by the boson fraction (also amenable to postselection). Moreover, t/J ≃ τu$_{f}$/ 2 and U $^{nn}$/J ≃ [2 u − 1 b − τ − 2 − 1] u$_{f}$/ 2, with τ ≡ t $^{b}$/tf and u$_{ν}$ ≡ U $^{νν}$/U fb ( ν = f, b ), can be tuned by using Feshbach resonances and by modifying the lattices for bosons and fermions, either in depth or relative position. A candidate system is a mixture of Ytterbium isotopes [25–27]: The total an- gular momentum of Yb is just given by the nuclear spin I , not influencing the interparticle interaction. A mix- ture of two spin states of fermionic $^{173}$Yb ( I = 5 / 2) with bosonic $^{168}$Yb or $^{174}$Yb (both I = 0) is described by three positive s -wave scattering lengths, with a$_{ff}$/a$_{fb}$ = 5 . 2 (1.4), a$_{bb}$/a$_{fb}$ = 6 . 5 (0.8) and a$_{fb}$ = 2.0nm (7.3nm) for $^{168}$Yb ($^{174}$Yb) [26]. Also optical Feshbach resonances are available [27]. Using a far off-resonant optical po- tential allows one to create practically equal traps for bosonic and fermionic isotopes without fine-tuning. Con- sidering the Mott regime, according to band structure calculations, typical Hubbard interaction parameters for Yb will roughly be on the order of 10 kHz. The effec- tive tunneling matrix element t will be about 200 times 

