Geometric Criterion for Solvability of Lattice Spin Systems

Understand

We present a simple criterion for solvability of lattice spin systems on the basis of the graph theory and the simplicial homology.

• The lattice systems satisfy algebras with graphical representations.
• It is shown that the null spaces of adjacency matrices of the graphs provide conserved quantities of the systems.
• Furthermore, when the graphs belong to a class of simplicial complexes, the Hamiltonians are found to be mapped to bilinear forms of Majorana fermions, from which the full spectra of the systems are obtained.