Andreas MielkeWe (Leon Haag-Fank and myself) got interesting new results for the bosonic Hubbard model on a three-dimensional flat-band lattice, the line graph of the cubic lattice. Among other results we prove that at a critical particle number, the system has a sub-extensive ground state entropy. This occurs often in highly frustrated spin systems but has, to our knowledge, not been observed in a bosonic lattice model.
The problem is related to the number of 4-cycle decompositions of the cubic lattice with periodic boundary conditions, a question that occurs in graph theory.
Here is the link to arxiv.org:
https://arxiv.org/abs/2604.19703
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#FlatBand
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The bosonic Hubbard model on a three dimensional flat band lattice
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 $N_c$. We prove that at this particle number the ground state entropy is subextensive $\propto N_c^{2/3}$. 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.
Dr. P. M. Secular