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Web of non-invertible dualities for (2+1) dimensional models with subsystem symmetries

by Avijit Maity, Vikram Tripathi, Andriy H. Nevidomskyy

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Authors (as registered SciPost users):

Avijit Maity

Abstract

We extend non-invertible duality concepts familiar from one-dimensional systems to two spatial dimensions by constructing a web of non-invertible dualities for lattice models with subsystem symmetries. Focusing on $\mathbb{Z}_2\times\mathbb{Z}_2$ subsystem symmetry on the square lattice, we construct two complementary non-invertible dualities: a duality that maps spontaneous subsystem symmetry-broken (SSSB) phases to the trivial phase (often referred to as the Kramers-Wannier (KW) duality in 1+1D models), 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. Crucially, these dualities are boundary-sensitive. On open lattices, both the subsystem KW and KT transformations can be implemented as unitary, invertible operators. In particular, the subsystem KT map not only identifies the bulk Hamiltonians of the dual phases but also carries the spontaneous ground-state degeneracy of the SSSB phase directly onto the protected boundary degeneracy characteristic of the SSPT phase. In contrast, on closed manifolds, the subsystem KW/KT maps become intrinsically non-unitary and non-invertible when restricted to the original Hilbert space. We establish this non-invertibility from three complementary perspectives -- ground state degeneracy matching (applied to two copies of the Xu-Moore/Ising-plaquette model), analysis of the symmetry-twist sector mapping, and the fusion algebra of the duality operator. We further show that enlarging the Hilbert space to include twisted sectors allows the formulation of the subsystem KW duality as a projective unitary which preserves quantum transition probabilities, consistent with the recent formulations of generalized Wigner theorem for non-invertible symmetries. 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. We conclude with field-theoretic consistency checks and discuss implications for the classification and detection of subsystem-protected phases. Our construction provides a concrete lattice realization of non-invertible subsystem dualities, highlighting the central role of symmetry-twist sectors in characterizing generalized symmetries and exotic phases of quantum matter.

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