We consider the three-body problem in a generic multiband lattice, and analyze the dispersion of the trimer states that are made of two spin-$\ensuremath{\uparrow}$ fermions and a spin-$\ensuremath{\downarrow}$ fermion due to an on-site attraction in between. Based on a variational approach, we first obtain the exact solution in the form of a set of coupled integral equations, and then reduce it to an eigenvalue problem. As an illustration we apply our theory to the sawtooth lattice, and numerically show that energetically stable trimers are allowed in a two-band setting, which is in sharp contrast with the single-band linear-chain model. In particular, we also reveal that the trimers have a nearly flat dispersion when formed in a flat band, which is unlike the highly dispersive spectrum of its dimers.
We consider the three-body problem in a generic multiband lattice, and analyze the dispersion of the trimer states that are made of two spin-\ensuremath↑ fermions and a spin-\ensuremath↓ fermion due to an on-site attraction in between. Based on a variational approach, we first obtain the exact solution in the form of a set of coupled integral equations, and then reduce it to an eigenvalue problem. As an illustration we apply our theory to the sawtooth lattice, and numerically show that energetically stable trimers are allowed in a two-band setting, which is in sharp contrast with the single-band linear-chain model. In particular, we also reveal that the trimers have a nearly flat dispersion when formed in a flat band, which is unlike the highly dispersive spectrum of its dimers.
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.
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 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 study the bound states of N 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.
In a multiband Hubbard model the self-consistency relations for the two-body bound-state bands are in the form of a nonlinear eigenvalue problem. Assuming that the resultant eigenvectors form an orthonormal set, e.g., in the strong-binding regime, here we reformulate their Berry curvatures and the associated Chern numbers. As an illustration we solve the two-body problem in a Haldane-Hubbard model with attractive on-site interactions and analyze its topological phase diagrams from weak to strong couplings, i.e., by keeping track of the gap closings in between the low-lying two-body bands. The resultant Chern numbers are consistent with the lobe structure of the phase diagrams in the strong-coupling regime.
On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.