Parameterized families of 2+1d $G$-cluster states

TL;DR

The paper constructs a 2+1d $G$-cluster state with full $G\times2\mathrm{Rep}(G)$ symmetry, realized through a condensation surface in $\mathrm{Rep}(G)$ that yields a non-invertible $2\mathrm{Rep}(G)$ symmetry. Using tensor-network techniques, it defines line-like action tensors and module/comodule MPS to capture the symmetry actions on interfaces, and it extends the symmetry interpolation method to generate $S^{1}$- and $S^{2}$-parameterized families of $G$- and $2\mathrm{Rep}(G)$-symmetric SRE states. For each family, an adiabatic, textured Hamiltonian reveals pumped interface modes that carry nontrivial charges under the corresponding interface symmetry, proving the nontriviality of the families. The work also connects these pumping phenomena to categorical structures such as fiber 2-functors and automorphisms of the interface algebra, providing multiple lenses on the invariant content of these parameterized SRE states. Overall, the results extend the 1+1d G-cluster pump picture to 2+1d and demonstrate that non-invertible symmetry data can be robustly captured and classified via tensor-network constructions and generalized pumping arguments.

Abstract

We construct a $G$-cluster Hamiltonian in 2+1 dimensions and analyze its properties. This model exhibits a $G\times2\mathrm{Rep}(G)$ symmetry, where the $2\mathrm{Rep}(G)$ sector realizes a non-invertible symmetry obtained by condensing appropriate algebra objects in $\mathrm{Rep}(G)$. Using the symmetry interpolation method, we construct $S^1$- and $S^2$-parameterized families of short-range-entangled (SRE) states by interpolating an either invertible $0$-form or $1$-form symmetry contained in $G\times2\mathrm{Rep}(G)$. Applying an adiabatic evolution argument to this family, we analyze the pumped interface mode generated by this adiabatic process. We then explicitly construct the symmetry operator acting on the interface and show that the interface mode carries a nontrivial charge under this symmetry, thereby demonstrating the nontriviality of the parameterized family.

Figures (14)

• Figure 1: A two-dimensional oriented square lattice with periodic boundary conditions. The arrows on the links represent the orientation of the lattice. We place the group ring $\mathbb{C}[G]$ on each vertex and link of the lattice, denoted by the black dots.
• Figure 3: The PEPO representation of the condensation surface. Each white dot represents a generalized Pauli operator $Z_{A}$ associated with the representation $A$ and each black dot represents a 4-valent tensor $V_{A}$ consisting of the multiplication and comultiplication maps.
• Figure 5: The decomposition of an invertible condensation surface associated with the algebra object $A^{G}_{G,\omega}$. To visualize the connectivity of the lines, we use orange and blue colors for the same unitary operators $u^{(p)}_{\omega}$.
• Figure : (a)
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