For gapped phases, an alternative approach uses module categories in [17].

The input data that determines the Hilbert space is ($\mathcal{C}$, $\mathcal{M}$, $\rho \in \mathcal{C}$), where $\mathcal{C}$ is a fusion category, $\mathcal{M}$ is a $\mathcal{C}$-module category and $\mathcal{S}={\mathcal{C}}_{\mathcal{M}}^{*}$ is the actual symmetry category.

For the expert reader, we choose the categorical input data, $\mathcal{C}={\mathsf{Vec}}_{D_8}$, $\mathcal{M}=\mathsf{Vec}$, $\rho ={\oplus }_{g\in D_8}g,$ so that the symmetry of the model is $\mathcal{S}={\mathcal{C}}_{\mathcal{M}}^{*}=\mathsf{Rep}(D_8)$.

$H_{(F,\beta )}$ is Hermitian as the projectors are and $(L_{\beta }^{\mathsf{f}}{)}^{†}=L_{\beta }^{{\mathsf{f}}^{-1}}$, $(R_{\beta }^{\mathsf{f}}{)}^{†}=R_{\beta }^{{\mathsf{f}}^{-1}}$, and we sum over $f\in F$ where $F$ is a group. One can write equivalent Hamiltonians by summing only over generators of $F$ and their inverses.

The operators $L_g,R_g,Z_{\mathrm{\Gamma }}$ for dihedral groups presented as ${\mathbb{Z}}_N⋊{\mathbb{Z}}_2$ can be found in Appendix B of [48].

The local and string order parameters here are given by ${\mathcal{O}}_{\mathrm{loc}}={\mathrm{\Sigma }}_i⟨{X^{\mathrm{II}}{X}^{\mathrm{III}}⟩}_i/L$ and ${\mathcal{O}}_{\mathrm{str}}=⟨X_i^{\mathrm{II}}{X}_{i+1}^{\mathrm{III}}[{\mathrm{\Pi }}_{k=1}^{l-2}(Z^{\mathrm{II}}{Z}^{\mathrm{III}}{)}_{i+k}]X_{i+l-2}^{\mathrm{II}}{X}_{i+l-1}^{\mathrm{III}}⟩$. In the time-evolving block decimation simulation, we consider the string at the center of the system with a length $l=6$.

There is $\sqrt{2}$ “relative Euler term” between ${\mathsf{Ising}}_0$ and each of ${\mathsf{Ising}}_{\pm }$, since $E$ has quantum dimension 2, whereas the Ising noninvertible symmetry has quantum dimension $\sqrt{2}$.