Chern Numbers Associated with the Periodic Toda Lattice

The periodic Toda lattice is solved by exploiting the spectral properties of the Lax operator, in which the boundary states play an important role. We show that these boundary states have a topological origin similar to that of the edge states in topological insulators, and consequently, that the bulk wave functions of the Lax operator yield nontrivial Chern numbers. This implies that the periodic Toda lattice belongs to the same topological class as the Thouless pump. We demonstrate that the cnoidal wave of the Toda lattice exhibits a Chern number of $-1$ per period.