Special cycles in independence complexes and superfrustration in some lattices

TL;DR

This paper demonstrates that certain grid independence complexes have exponentially many Betti numbers, indicating a high degeneracy of ground states in the corresponding statistical physics models with fermions.

Contribution

It establishes the exponential growth of Betti numbers in independence complexes of specific grids, linking topological properties to physical ground state degeneracy.

Findings

Independence complexes of some grids have exponential Betti numbers.

This exponential growth correlates with superfrustration in the hard-core model.

The work connects topological invariants with physical phenomena in lattice models.

Abstract

We prove that the independence complexes of some grids have exponential Betti numbers. This corresponds to the number of ground states in the hard-core model in statistical physics with fermions in the vertices of the grid.