A critical lattice model for a Haagerup conformal field theory

Abstract

We use the formalism of strange correlators to construct a critical classical lattice model in two dimensions with the \emph{Haagerup fusion category} $\mathcal{H}_3$ as input data. We present compelling numerical evidence in the form of finite entanglement scaling to support a Haagerup conformal field theory (CFT) with central charge $c=2$. Generalized twisted CFT spectra are numerically obtained through exact diagonalization of the transfer matrix and the conformal towers are separated in the spectra through their identification with the topological sectors. It is further argued that our model can be obtained through an orbifold procedure from a larger lattice model with input $Z(\mathcal{H}_3)$, which is the simplest modular tensor category that does not admit an algebraic construction. This provides a counterexample for the conjecture that all rational CFT can be constructed from standard methods.

Figures (4)

• Figure 1: The partition function of the model on the hexagonal lattice in a maximally occupied configuration, i.e. as many type-$\bf{1}$ plaquettes as allowed by the Fibonacci grading of the fusion rules. There are three maximally occupied configurations as indicated by the three different sublattices. The adjacency rules of neighbouring particles can be shown by the corresponding Dynkin diagram.
• Figure 2: Finite entanglement scaling for the fixed point MPS of the transfer matrix calculated using VUMPS with explicit $\mathcal{H}_3$ anyonic symmetry. The results are consistent with a central charge close to $c=2$.
• Figure 3: The exact diagonalization scheme with anyonic symmetries (shown for $L=6$). The grey lines of the one-row transfer matrix indicate that they have been fixed on the strange correlator $\rho$. The eigenvector is written as an anyonic fusion tree (black lines). Whenever a line crosses the sector label (the black line going up) a half-braiding is required. The anyonic symmetry is ensured by Eq. \ref{['naturality']}.
• Figure 4: Spectra for the transfer matrix, twisted with topological defects $\bf{1}$ (upper left), $\alpha$ (upper middle) and $\rho$ (bottom), numerically obtained with anyonic symmetry-preserving exact diagonalization on $L=15$ sites. The eigenvalues are labeled by their corresponding topological sectors $Z(\mathcal{H}_3)$ according to Table \ref{['centerLabeling']}. For the non-trivial twists, the conformal spins are only integers up to a topological spin correction (Table \ref{['topoSpins']}). Upper right: the identity sector on $L=18$ sites. The first excited states of the vacuum ($\Delta=2, s=-2,2$) are circled in black.