On integrable matrix product operators with bond dimension D = 4

Abstract

We construct and study a two-parameter family of matrix product operators of bond dimension D = 4. The operators M(x, y) act on , i.e. the space of states of a spin-1/2 chain of length N. For the particular values of the parameters: x = 1/3 and , the operator turns out to be proportional to the square root of the reduced density matrix of the valence-bond-solid state on a hexagonal ladder. We show that M(x, y) has several interesting properties when (x, y) lies on the unit circle centered at the origin: x2 + y2 = 1. In this case, we find that M(x, y) commutes with the Hamiltonian and all the conserved charges of the isotropic spin-1/2 Heisenberg chain. Moreover, M(x1, y1) and M(x2, y2) are mutually commuting if for both i = 1 and 2. These remarkable properties of M(x, y) are proved as a consequence of the Yang–Baxter equation.