Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory

We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of ($2+1$)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.

Figure 1

Topological sector labeling of finite-size CFT spectra (scaling dimension $\mathrm{\Delta }$ versus momentum) on a cylinder with a circumference of 18 sites with no defect line on the left and a $\tau$ defect line on the right. The exact topological correction to the conformal spin is denoted next to the first appearance of the respective idempotents.

Figure 2

Definition of the transfer matrices with $A$ as two blocked string-net tensors on the left and $\mathrm{YL}\otimes \mathrm{YL}$ topological sector labeling of the critical transfer matrix $T_2$ on a cylinder of nine sites on the right.