Web of Non-invertible Dualities for (2+1) Dimensional Models with Subsystem Symmetries

TL;DR

This work extends non-invertible dualities to (2+1)D lattice models with subsystem $\\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry, constructing a web of Kramers–Wannier- and Kennedy–Tasaki-type dualities. It shows that subsystem KW duality is unitary on open lattices but non-unitary on closed manifolds, with ground-state and twist-sector analyses revealing non-invertibility; enlarging the Hilbert space to include twisted sectors recovers unitarity in line with generalized Wigner-type theorems. Building on this, the generalized KT transformation maps spontaneous subsystem symmetry breaking (SSSB) phases to subsystem symmetry-protected topological (SSPT) phases, including a weak case (stacks of 1D SPTs) and a strong case (square-lattice cluster state) that dualizes to two copies of the Xu–Moore model; boundary conditions crucially determine invertibility. The work also connects bulk and edge diagnostics through invariants $eta(g)$ and $\,\phi_{ ext{top}}$, showing these quantities are preserved under KT duality even as local SSPT repair operations map to nonlocal dual objects. Overall, the results provide a unified lattice framework for dualities between trivial, SSPT, and SSSB phases, with explicit operator constructions, symmetry–twist sector mappings, and a detailed treatment of open vs closed geometries and their physical consequences.

Abstract

We extend non-invertible duality concepts from one-dimensional systems to two spatial dimensions by constructing a web of non-invertible dualities for lattice models with subsystem symmetries. For the $\mathbb{Z}_2 \times \mathbb{Z}_2$ subsystem symmetry on the square lattice, we build two complementary dualities: a map that sends spontaneous subsystem symmetry-broken (SSSB) phases to the trivial phase (the analogue of the Kramers-Wannier (KW) duality in 1+1D), and a generalized subsystem Kennedy-Tasaki (KT) transformation that maps SSSB phases to subsystem symmetry-protected topological (SSPT) phases while leaving the trivial phase invariant. These dualities are boundary-sensitive. On open lattices, both subsystem KW and KT transformations act as unitary, invertible operators. In particular, the KT map not only matches the bulk Hamiltonians of the dual phases but also carries the spontaneous ground-state degeneracy of the SSSB phase onto the protected boundary degeneracy of the SSPT phase. On closed manifolds, however, both maps become intrinsically non-unitary and non-invertible when restricted to the original Hilbert space. We demonstrate this non-invertibility via ground-state degeneracy matching (in two copies of the Xu-Moore/Ising-plaquette model), analysis of symmetry-twist sectors mapping, and the fusion algebra of the duality operator. Enlarging the Hilbert space to include twisted sectors allows the subsystem KW map to be formulated as a projective unitary preserving quantum transition probabilities, consistent with generalized Wigner-theorem-based constructions. We also show that the KT map faithfully transmits the algebraic content of bulk and edge invariants diagnosing strong SSPT order: although strictly local SSPT repair operators map to highly nonlocal objects in the dual SSSB phase, the essential commutation algebra and the bulk-edge correspondence remain intact.

Figures (7)

• Figure 1: Schematic illustration of the Kramers--Wannier (KW), domain-wall decoration (DW) and Kennedy--Tasaki (KT) dualities in a model with subsystem symmetries such as the one studied in this work. The dualities interrelate the phase with spontaneously broken subsystem symmetry (SSSB), the subsystem symmetry-protected topological (SSPT) and the trivial phase.
• Figure 2: Depiction of the Xu--Moore model on the square lattice, with the red sites hosting the original Ising spins $\sigma_{i,j}$, while the green sites correspond to the dual degrees of freedom $\widehat{\sigma}_{i-\frac{1}{2},j-\frac{1}{2}}$ at the centers of the plaquettes. The mapping of the plaquette operator and the transverse field term through the KW transformation is illustrated, following Eq. \ref{['eq:XM-dual-variables']}.
• Figure 3: (a) The mapping of the Ising interaction terms through the $1$d KW transformation is illustrated; (b) Ground-state wave function showing the domain-wall decoration pattern; (c) Preparation of the $1$d cluster state $\ket{\Psi_{\mathrm{cluster}}}$ from the trivial state $\ket{\Psi_{\mathrm{trivial}}}$ using $U_{\text{DW}}$; (d) Illustration of the sequence of transformations that construct the 1D KT duality, which maps a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-broken state to a $\mathbb{Z}_2 \times \mathbb{Z}_2$ SPT cluster state.
• Figure 4: (a) The Pauli spin $\sigma_{i,j}$ resides on the red sites, while the Pauli spin $\tau_{i-\frac{1}{2},j-\frac{1}{2}}$ is placed on the blue sites. It is important to emphasize that the blue $\tau_{i-\frac{1}{2},j-\frac{1}{2}}$ spins are part of the Hilbert space and are distinct from the dual space degrees of freedom $\widehat{\sigma}_{i-\frac{1}{2},j-\frac{1}{2}}$ in the context of the subsystem KW transformation, shown in green in Fig. \ref{['fig:KW_duality']}. (b) A representative decorated domain wall configuration is illustrated. Solid yellow lines indicate domain walls between sites with opposite $\sigma^z$ values, while dashed yellow squares mark the corners where the domain wall is decorated $\tau^x=-1$Zhou_Detecting_SSPT_PRB2022.
• Figure 5: Sequence of transformations illustrating the construction of the subsystem KT duality $\mathcal{N}_{\text{KT}} = \mathcal{N}_{\text{KW}}^\dagger U_{\text{DW}} \mathcal{N}_{\text{KW}}$: the KW duality $\mathcal{N}_{\text{KW}}$ maps the $H_\text{SSSB}$ to a trivial one in terms of dual spins; the domain-wall decoration operator $U_{\text{DW}}$ then promotes it to the $\widehat{H}_\text{SSPT}$ in the dual basis; and finally $\mathcal{N}_{\text{KW}}^\dagger$ returns the SSPT Hamiltonian $H_\text{SSPT}$ in terms of the original spins.