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arXiv:2204.10003 · Physical Review A

Stability of (N+1) -body fermion clusters in a multiband Hubbard model

M. Iskin, Ahmet Keleş

Published Sep 6, 2022• 4 citations
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Abstract

We start with a variational approach and derive a set of coupled integral equations for the bound states of $N$ identical spin-$\ensuremath{\uparrow}$ fermions and a single spin-$\ensuremath{\downarrow}$ fermion in a generic multiband Hubbard Hamiltonian with an attractive on-site interaction. As an illustration, we apply our integral equations to the one-dimensional sawtooth lattice up to $N\ensuremath{\le}3$, i.e., to the $(3+1)$-body problem, and we reveal not only the presence of tetramer states in this two-band model but also their quasiflat dispersion when formed in a flat band. Furthermore, for $N={4,5,\ensuremath{\cdots},10}$, our density-matrix renormalization-group simulations and exact diagonalization suggest the presence of larger and larger multimers with lower and lower binding energies, conceivably without an upper bound on $N$. These peculiar $(N+1)$-body clusters are in sharp contrast with the exact results on the single-band linear-chain model where none of the $N\ensuremath{\ge}2$ multimers appear. Hence their presence must be taken into account for a proper description of the many-body phenomena in flat-band systems, e.g., they may suppress superconductivity especially when there exists a large spin imbalance.

DOI
10.1103/physreva.106.033304

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We start with a variational approach and derive a set of coupled integral equations for the bound states of NNN identical spin-\ensuremath↑\ensuremath{\uparrow}\ensuremath↑ fermions and a single spin-\ensuremath↓\ensuremath{\downarrow}\ensuremath↓ fermion in a generic multiband Hubbard Hamiltonian with an attractive on-site interaction. As an illustration, we apply our integral equations to the one-dimensional sawtooth lattice up to N\ensuremath≤3N\ensuremath{\le}3N\ensuremath≤3, i.e., to the (3+1)(3+1)(3+1)-body problem, and we reveal not only the presence of tetramer states in this two-band model but also their quasiflat dispersion when formed in a flat band. Furthermore, for N=4,5,\ensuremath⋯,10N={4,5,\ensuremath{\cdots},10}N=4,5,\ensuremath⋯,10, our density-matrix renormalization-group simulations and exact diagonalization suggest the presence of larger and larger multimers with lower and lower binding energies, conceivably without an upper bound on NNN. These peculiar (N+1)(N+1)(N+1)-body clusters are in sharp contrast with the exact results on the single-band linear-chain model where none of the N\ensuremath≥2N\ensuremath{\ge}2N\ensuremath≥2 multimers appear. Hence their presence must be taken into account for a proper description of the many-body phenomena in flat-band systems, e.g., they may suppress superconductivity especially when there exists a large spin imbalance.
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We study the bound states of NNN identical bosons that are described by a multiband Bose-Hubbard model with generic hoppings and an attractive on-site interaction. Using a variational approach, we first derive exact integral equations for the dimers, trimers, tetramers, and other multimers, and then apply them to a one-dimensional sawtooth model that features two bands. In particular we reveal the presence of not only the off-site dimer states which consist of two monomers on different sites even in the strong-coupling limit but also the off-site trimer states which consist of either a dimer on one site and a monomer on another site or three monomers on three different sites. Our variational calculations for the ground states of on-site dimers, on-site trimers and off-site trimers benchmark perfectly well with the density matrix renormalization group (DMRG) simulations. We also present DMRG results for the ground states of on-site tetramers, off-site tetramers, on-site pentamers, off-site pentamers, and other multimers.
We consider the two-body problem in a periodic potential, and study the bound-state dispersion of a spin-$\ensuremath{\uparrow}$ fermion that is interacting with a spin-$\ensuremath{\downarrow}$ fermion through a short-range attractive interaction. Based on a variational approach, we obtain the exact solution of the dispersion in the form of a set of self-consistency equations, and apply it to tight-binding Hamiltonians with on-site interactions. We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversal symmetry, and show that the lowest-energy bound states disperse quadratically with momentum, whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlying Bloch states. In particular, we apply our theory to the Mielke checkerboard lattice, and study the special role played by the interband processes in producing a finite effective mass for the bound states in a nonisolated flat band.
We investigate pairing and superconductivity in the attractive Fermi-Hubbard model on the one-dimensional sawtooth lattice, which exhibits a flat band by fine-tuning the hopping rates. We first solve the two-body problem, both analytically and numerically, to extract the binding energy and the effective mass of the pairs. Based on the density matrix renormalization group method, we address the ground-state properties of the many-body system, assuming equal spin populations. We compare our results with those available for a linear chain, where the model is integrable by the Bethe ansatz, and we show that the multiband nature of the system substantially modifies the physics of the BCS-BEC crossover. We find that near a flat band, the chemical potential remains always close to its zero-density limit predicted by two-body physics. In contrast, the pairing gap exhibits a remarkably strong density dependence and, differently from the pair binding energy, it is no longer peaked at the flat-band point. We show that these results can be interpreted in terms of polarization screening effects, due to an anomalous attraction between pairs in the medium and single fermions. Importantly, we unveil that three-body bound states (trimers) exist in the sawtooth lattice, in sharp contrast with the linear chain geometry, and we compute their binding energy. The nature of these states is investigated via a strong-coupling variational approach, revealing that they originate from tunneling-induced exchange processes.
We study the pairing and superconducting properties of the attractive Hubbard model in two quasi-one-dimensional topological lattices---the Creutz and sawtooth lattices---which share two peculiar properties: each of their band structures exhibits a flat band with a nontrivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, due to a chiral (sublattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use a multiband mean field and exact density matrix renormalization group in our work. Our three main results are as follows: (a) For both lattice systems, the superconducting weight, Ds{D}_{s}Ds​, is linear in the coupling strength, UUU, for low values of UUU. (b) For small U,DsU, {D}_{s}U,Ds​ is proportional to the quantum metric for the Creutz system but not for the sawtooth system because its sublattices are not equivalent. We have therefore extended the approach to this more complex situation and found an excellent agreement with the numerical results. (c) At moderate and large UUU, the conventional BCS mean field is no longer appropriate for such systems with inequivalent sublattices. We show that, for a wide range of densities and coupling strengths, these systems are very well described by a full multiband mean-field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean-field parameters.
The peculiar effect of electron correlation in multiband systems is illustrated for the recently proposed ferromagnetism in a Hubbard model having flat bands. Depsite the flatness of the band, an acoustic spin-wave mode with a finite stiffness exists along with optical modes below a Stoner gap. The relevant energy (gap and stiffness) transmutes from the Hubbard U in the weak-coupling limit to t (transfer integral) in the strong-coupling limit. This implies a stable itinerant ferromagnetism with finite band widths.