Temperley-Lieb integrable models and fusion categories

Abstract

We show that every fusion category containing a non-invertible, self-dual object $a$ gives rise to an integrable anyonic chain whose Hamiltonian density satisfies the Temperley-Lieb algebra. This spin chain arises by considering the projection onto the identity channel in the fusion process $a\otimes a$. We relate these models to Pasquier's construction of ADE lattice models. We then exploit the underlying Temperley-Lieb structure to discuss the spectrum of these models and argue that these models are gapped when the quantum dimension of $a$ is greater than 2. We show that for fusion categories where the dimension is close to 2, such as the Fib$\times$Fib and Haagerup fusion categories, the finite size effects are large and they can obscure the numerical analysis of the gap.

Figures (3)

• Figure 1: Spectra of the Hamiltonian for the Fib$\times$Fib (left) and Haagerup (right) chains for $L=4,6,8$. We have indicated in blue (resp. yellow, green) the energy levels that can be recovered from the XXZ chain at zero magnetization and twists $\varphi_0$ (resp. $\varphi_1$, $\varphi_2$) as defined in the main text, see eqs \ref{['twistsFF']} and \ref{['twistsH']}. The remaining energy levels, marked in gray, are recovered by XXZ sectors with nonzero values of the magnetization.
• Figure 2: Low-lying spectra of the Fib$\times$Fib (top) and Haagerup chains (bottom) for increasing system sizes: numerical data (obtained form exact diagonalization for $L$ up to $8$, and from DMRG for $L$ between $10$ and $20$) is shown as black dots, and the blue/orange dotted lines show the lowest energies $E_0(L)$ and $E_1(L)$ in the sectors $(0,\varphi_0)$ and $(0,\varphi_1)$, computed by Bethe ansatz.
• Figure 3: The rescaled gap $- \frac{\log(E_1(L)-E_0(L))}{L/\xi}$ between the ground states of the $(0,\varphi_0)$ and $(0,\varphi_1)$ sectors, as a function of $\Delta$. The red to yellow lines show data from Bethe ansatz for increasing system sizes, where we used larger plot marks to evidence the Fib$\times$Fib and Haagerup points. The blue line gives the value of the correlation length $\xi$ as a function of $\Delta$.