Numerical evidence for a Haagerup conformal field theory

Abstract

We numerically study an anyon chain based on the Haagerup fusion category, and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is $\sim 2$. Fusion categories generalize the concept of finite group symmetries to non-invertible symmetry operations, and the Haagerup fusion category is the simplest one which comes neither from finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries.

Figures (6)

• Figure 1: Finite group symmetries as realized by walls.
• Figure 3: Ground state energy $E_0$ and gap $\Delta E$ as functions of chain length $L$, juxtaposed with fits to \ref{['GSFit']}. Black dots: DMRG; Red squares: exact diagonalization. The errors in the parameters shown here only reflect the fitting errors.
• Figure 4: Scaling of the gap with the system size. Fits of $\log \Delta E$ over $\log L$ show that the dynamical exponent is $z \sim 1$.
• Figure 5: Entanglement entropy curve of the ground state and fit to \ref{['entanglement']}. The color scheme shows the evolution over DMRG sweeps.
• Figure 6: Periodic chain spectra for $L = 12, 15, 18$ obtained by exact diagonalization, assuming that the lowest state with $p=1$ is the descendant of the first excited state with $p=0$. The filled dots and the hollow dots are for states with $\rho_+=\frac{3+\sqrt{13}}{2}$ and $\rho_-=\frac{3-\sqrt{13}}{2}$, respectively. For $L = 18$ we do not yet have the $\rho$ measurement.