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

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.Comment: 13 pages, 3 figures, submitted to a special issue of JSTAT on "Quantum Entanglement in Condensed Matter Physics"; Conjectures presented in version 1 have been proved in version 2; typos correcte

Interacting Fibonacci anyons in a Rydberg gas

A defining property of particles is their behavior under exchange. In two dimensions anyons can exist which, opposed to fermions and bosons, gain arbitrary relative phase factors or even undergo a change of their type. In the latter case one speaks of non-Abelian anyons - a particularly simple and aesthetic example of which are Fibonacci anyons. They have been studied in the context of fractional quantum Hall physics where they occur as quasiparticles in the $k=3$ Read-Rezayi state, which is conjectured to describe a fractional quantum Hall state at filling fraction $\nu=12/5$. Here we show that the physics of interacting Fibonacci anyons can be studied with strongly interacting Rydberg atoms in a lattice, when due to the dipole blockade the simultaneous laser excitation of adjacent atoms is forbidden. The Hilbert space maps then directly on the fusion space of Fibonacci anyons and a proper tuning of the laser parameters renders the system into an interacting topological liquid of non-Abelian anyons. We discuss the low-energy properties of this system and show how to experimentally measure anyonic observables