Effective extended-Hubbard description and scale-up of exact configuration-interaction benchmarks for ultracold fermions in optical plaquettes

Exact configuration-interaction (CI) solutions for few fermionic atoms in continuum multi-well optical potentials provide microscopic benchmarks for the building blocks of unconventional superconductivity, but they are currently limited to a single and a double square plaquette (up to 8 sites and 6 particles). The open problem is threefold and anchored in the original benchmark study: (1) determine the precise form of the effective extended (non-standard) Hubbard Hamiltonian — including off-site nearest-neighbor repulsion and, if needed, density-dependent (correlated) tunneling and higher-band terms — together with its parameter regime of validity, such that it faithfully reproduces the exact CI spectra, hole-pairing gaps, and density profiles for ultracold fermions in optical plaquettes; (2) extend the exact-benchmark ladder to larger plaquette arrays and higher particle numbers using approximate but controlled methods calibrated against these benchmarks; and (3) realize and detect the predicted entangled charge- and spin-density-wave states of the double plaquette. Items (1) and (2) are the mandatory theory-and-computation core, and completing them resolves the objective; item (3), the experimental realization, lies outside the reach of analytical and numerical work and is not required for that resolution.

Background

A square optical plaquette is a four-well microscopic trapping potential for ultracold atoms, realizable with optical superlattices or programmable tweezer arrays; coupled plaquettes are the finite building blocks from which two-dimensional lattice physics can be assembled 'bottom-up'. The resonating-valence-bond (RVB) idea — a ground state that superposes many singlet-pair coverings of the lattice, proposed by Anderson as a route to high-temperature superconductivity in cuprates — motivates studying such building blocks with repulsively interacting fermions, where hole pair binding (a negative pair-binding energy, i.e., the energy gained when two doped holes occupy the same plaquette rather than two separate ones) is the elementary pairing signature. Configuration interaction (CI) is an exact many-body method that diagonalizes the microscopic Hamiltonian — here the continuum Schrödinger equation for ${}^{6}\mathrm{Li}$ atoms in a realistic multi-well potential with short-range repulsion — in a systematically enlargeable basis of antisymmetrized products, without imposing any effective lattice model. Brandt, Yannouleas, and Landman [Phys. Rev. A 95, 043617 (2017)] used CI to compute exact solutions for one and two square plaquettes: they found entangled d- and s-wave RVB states coexisting with partial double occupancy (so the results map onto the Hubbard model, not onto the double-occupancy-excluding t-J model), a maximum hole-pairing gap $\Delta_p \approx 0.045t$ at $U/t \approx 2.6$, and, for two weakly coupled plaquettes, ground-state manifolds of entangled mirror-reflected charge-density-wave (CDW) and spin-density-wave (SDW) configurations — spatial modulations of the charge or spin density that become visible when the exact parity eigenstates are combined into symmetry-broken combinations. The Hubbard model places fermions on lattice sites with tunneling $t$ and on-site repulsion $U$; an extended Hubbard model adds further terms such as the off-site nearest-neighbor repulsion $V$, and 'non-standard' Hubbard models further add density-dependent (correlated) tunneling and higher-band occupation terms that arise when the continuum Wannier-like orbitals of a real optical potential deform with filling (Dutta et al., Rep. Prog. Phys. 78, 066001 (2015)). Two CI findings exceeded the standard Hubbard model: the pairing gap retains a dependence on the inter-well barrier height $\varepsilon_b$ even after rescaling by $t$ and $U$, and the off-site repulsion $V$ — negligible for high barriers but of order $\Delta_p$ for $\varepsilon_b \lesssim 0.3$ — suppresses hole pairing. Ultracold-atom emulations frequently require such non-standard effective Hamiltonians for their analysis, which is why the authors posed exact microscopic benchmarks as the calibration reference for constructing, assessing, and improving effective models, progressing from one plaquette to two and to systems of hierarchically increasing complexity, with few-atom detection techniques (trap lowering, as in Murmann et al., Phys. Rev. Lett. 115, 215301 (2015); site-resolved microscopy) as the readout path. The full CI treatment is limited by the exponential growth of the many-body Hilbert space to roughly 8 sites and 6 particles, so the extension to larger arrays (e.g., 4×4 and beyond) requires approximate but controlled methods — such as tensor-network or renormalization-group techniques, or entangled-plaquette variational states (Mezzacapo et al., Phys. Rev. B 94, 155120 (2016)) — whose reliability must be calibrated against the exact few-plaquette benchmarks.

Problem Statement

For repulsively interacting ultracold fermions (specifically ${}^{6}\mathrm{Li}$ atoms) in microscopic multi-well optical potentials forming square plaquettes, as treated exactly by configuration interaction in Brandt, Yannouleas, and Landman [Phys. Rev. A 95, 043617 (2017)], three linked questions remain open, addressing one objective — a validated microscopic-to-effective-model bridge for plaquette-based emulation of unconventional-pairing physics. Questions (1) and (2) are the theory-and-computation core of this objective, resolvable by analytical and numerical work against the exact benchmarks; question (3) is the corresponding experimental deliverable: (1) Determine the precise form of the effective extended (non-standard) Hubbard Hamiltonian — beyond the standard on-site-$U$ and tunneling-$t$ terms, including the off-site nearest-neighbor repulsion $V$ and, if required by the exact data, density-dependent (correlated) tunneling and higher-band occupation terms — together with its parameter regime of validity in $U/t$, inter-well barrier $\varepsilon_b$, and geometry, such that it reproduces the exact CI benchmarks (low-energy spectra, hole-pairing gaps $\Delta_p$, single-particle and spin densities, and the RVB/double-occupancy content of the eigenstates) for the single and double plaquette. (2) Building on these benchmarks, extend the calculations to larger plaquette arrays (at least a 4×4 array) and to particle numbers beyond the 6 particles covered by the exact benchmarks, using approximate but controlled methods, quantifying their accuracy against the exact few-plaquette results and characterizing how the hole-pairing and entangled CDW/SDW physics evolves with system size. (3) Realize the predicted entangled charge-density-wave and spin-density-wave states of the fermionic double plaquette experimentally and detect them, testing the microscopic CI predictions (including the $\varepsilon_b$-dependence of the pairing gap and the suppression of pairing by $V$).

References

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