Abstract Flat-band systems are ideal model systems to study strong correlations. In a large class of one or two dimensional bosonic systems with a lowest flat-band it has been shown that at a critical density the ground states are Wigner crystals. Under very special conditions it has been shown that pair formation occurs if one adds another particle to the system. The present paper extends this result to a much larger class of lattices and to a much broader region in the parameter space. Further, a lower bound for the energy gap between these pair states and the rest of the spectrum is established. The pair states are dominated by a subspace spanned by states containing a compactly localised pair. Overall, this strongly suggests localised pair formation in the ground states of the broad class of flat-band systems and rigorously proves it for some of the graphs in it, including the inhomogeneous chequerboard chain as well as two novel examples of regular two dimensional graphs. Physically, this means that the Wigner crystal remains intact if one adds a particle to it.
Abstract Flat-band systems are ideal model systems to study strong correlations. In a large class of one or two dimensional bosonic systems with a lowest flat-band it has been shown that at a critical density the ground states are Wigner crystals. Under very special conditions it has been shown that pair formation occurs if one adds another particle to the system. The present paper extends this result to a much larger class of lattices and to a much broader region in the parameter space. Further, a lower bound for the energy gap between these pair states and the rest of the spectrum is established. The pair states are dominated by a subspace spanned by states containing a compactly localised pair. Overall, this strongly suggests localised pair formation in the ground states of the broad class of flat-band systems and rigorously proves it for some of the graphs in it, including the inhomogeneous chequerboard chain as well as two novel examples of regular two dimensional graphs. Physically, this means that the Wigner crystal remains intact if one adds a particle to it.
To explore superfluidity in flat-band systems, we consider a Bose-Hubbard model on a cross-linked ladder with $\ensuremath{\pi}$ flux, which has a flat band with a gap between the other band for noninteracting particles, where we study the effect of the on-site repulsion nonperturbatively. For low densities, we find exact degenerate ground states, each of which is a Wigner solid with nonoverlapping Wannier states on the flat band. At higher densities, the many-body system, when projected onto the lower flat band, can be mapped to a spin-chain model. This mapping enables us to reveal the existence of a Tomonaga-Luttinger liquid comprising pairs of bosons. Interestingly, the high- and low-density regions have an overlap, where the two phases coexist.
In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.
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.
We study the Wigner crystallization on partially filled topological flat bands of kagome, honeycomb and checkerboard lattices. We identify the Wigner crystals (WCs) by analyzing the Cartesian and angular Fourier transform of the pair correlation density of the many-body ground state obtained using exact diagonalization. The crystallization strength, measured by the magnitude of the Fourier peaks, increases with decreasing particle density. The Wigner crystallization observed by us is a robust and general phenomenon, existing in all three lattice models for a broad range of filling factors and interaction parameters. The shape of the resulting WCs is determined by the boundary conditions of the chosen plaquette. It is to a large extent independent on the underlying lattice, including its topology, and follows the behavior of classical point particles.
Bosons interacting repulsively on a lattice with a flat lowest band energy dispersion may, at sufficiently small filling factors, enter into a Wigner-crystal-like phase. This phase is a consequence of the dispersionless nature of the system, which in turn implies the occurrence of single-particle localized eigenstates. We investigate one of these systems---the sawtooth lattice---filled with strongly repulsive bosons at filling factors infinitesimally above the critical point where the crystal phase is no longer the ground state. We find, in the hard-core limit, that the crystal retains its structure in all but one of its cells, where it is broken. The broken cell corresponds to an exotic kind of repulsively bound state, which becomes delocalized. We investigate the excitation spectrum of the system analytically and find that the bound state behaves as a single particle hopping on an effective lattice with reduced periodicity, and is therefore gapless. Thus, the addition of a single particle to a flat-band system at critical filling is found to be enough to make kinetic behavior manifest.
The lowest eigenstates of the hopping matrix on the line graph of a cubic lattice with periodic boundary conditions are highly degenerate, they form a lowest flat band. Further, these states are localized. If one considers a repulsive bosonic Hubbard model on this lattice, it is possible to construct exact multi-particle ground states simply by putting particles in the localized single particle ground states such that they avoid each other. This can be done up to a certain critical particle number [Formula: see text]. We prove that at this particle number the ground state entropy is subextensive [Formula: see text]. For lower densities, the entropy is extensive. We further show that the problem is related to the number of 4-cycle decompositions of the cubic lattice with periodic boundary conditions.