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arXiv:2604.19703 · Reviews in Mathematical Physics

The bosonic Hubbard model on a three-dimensional flat band lattice

Leon Haag-Fank, Andreas Mielke

Published Sep 21, 2026• 0 citations
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Abstract

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.

DOI
10.1142/s0129055x26610027

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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.
We consider the repulsive Hubbard model on a class of lattices or graphs for which there is a large degeneracy of the single particle ground states and where the projector onto the space of single particle ground states is highly reducible. This means that one can find a basis in the space of the single particle ground states such that the support of each single particle ground state belongs to some small cluster and these clusters do not overlap. We show how such lattices can be constructed in arbitrary dimensions. We construct all multi-particle ground states of these models for electron numbers not larger than the number of localized single particle eigenstates. We derive some of the ground state properties, esp. the residual entropy, i.e. the finite entropy density at zero temperature.
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.
We consider the repulsive Hubbard model on three highly frustrated one-dimensional lattices---sawtooth chain and two kagom\'e chains---with completely dispersionless (flat) lowest single-electron bands. We construct the complete manifold of exact many-electron ground states at low electron fillings and calculate the degeneracy of these states. As a result, we obtain closed-form expressions for low-temperature thermodynamic quantities around a particular value of the chemical potential ${\ensuremath{\mu}}_{0}$. We discuss specific features of thermodynamic quantities of these ground-state ensembles such as residual entropy, an extra low-temperature peak in the specific heat, and the existence of ferromagnetism and paramagnetism. We confirm our analytical results by comparison with exact-diagonalization data for finite systems.
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.
Spinless fermions on highly frustrated lattices are characterized by the lowest single-particle band which is completely flat.Concrete realizations are provided by the sawtooth chain and the kagom é lattice.For these models a real-space picture is given in terms of localized states.Furthermore, we find a finite zero-temperature entropy for a suitable choice of the chemical potential.The entropy is computed numerically at finite temperature and one observes a strong cooling effect during adiabatic changes of the chemical potential.We argue that the localized states, the associated zerotemperature entropy as well as the large temperature variations carry over to the repulsive Hubbard model.The relation to flat-band ferromagnetism is also discussed briefly.
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.