On integrable matrix product operators with bond dimension $D=4$

TL;DR

This paper constructs a family of integrable matrix product operators with bond dimension 4, revealing their commutation properties and relation to the Heisenberg chain and Yang-Baxter equation, with special cases linked to valence-bond-solid states.

Contribution

It introduces a two-parameter family of integrable matrix product operators of bond dimension 4 with novel commutation and integrability properties.

Findings

Operators commute with the Heisenberg Hamiltonian on the unit circle.

Operators are mutually commuting when parameters lie on the unit circle.

Special case relates to the reduced density matrix of a valence-bond-solid state.

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 $({\mathbb C}_2)^{\otimes N}$, 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 $y=1/\sqrt{3}$, 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: $x^2 + y^2=1$. 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, $M(x_1,y_1)$ and $M(x_2,y_2)$ are mutually commuting if $x^2_i + y^2_i=1$ for both $i=1$ and $2$. These remarkable properties of $M(x,y)$ are proved as a consequence of the Yang-Baxter equation.