The Kramers-Wannier (KW) duality operator and the Kennedy-Tasaki (KT) transformation of one-dimensional spin chains - the latter mapping symmetry-protected topological (SPT) phases to spontaneous symmetry breaking (SSB) phases - were generalized by Mana, Li, Sukeno, and Wei [Phys. Rev. B 109, 245129 (2024)] to subsystem-symmetric lattice models in two and higher dimensions, where the KW operator is realized as a sequential circuit plus projection onto symmetry eigenspaces and the KT transformation maps subsystem SPT (SSPT) phases to subsystem SSB (SSSB) phases for Z2\mathbb{Z}_2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 subsystem symmetries. That paper's conclusion poses an openly stated generalization program: (i) KW transformations obtained by gauging higher-form symmetries, with explicit circuit-plus-projector realizations; (ii) a higher-form KT transformation - e.g., mapping 3D cluster SPTs protected by 1-form symmetries to copies of higher-form SSB phases - and its algebra; (iii) classification of SSPT phases beyond two dimensions, with KT transformations for every class, for symmetry groups beyond Z2\mathbb{Z}_2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2; (iv) the physical interpretation and anomaly characterization of the fermionic exchange symmetry of Majorana plaquette and hypercube models; and (v) Lieb-Schultz-Mattis-type constraints imposed by non-invertible duality symmetries in higher-dimensional subsystem-symmetric lattice models. Substantial progress exists on the higher-form KW/KT axis - a realized non-invertible KT map with its fusion algebra for the 3D Raussendorf-Bravyi-Harrington cluster model (Cao-Watanabe 2026), systematic higher-dimensional KW dualities from fermionic bosonization (Su-Martin 2025), and 2+1d lattice gauge-theory KT relations (Choi et al. 2025) - but no comprehensive SSPT classification beyond two dimensions, no anomaly characterization of the Majorana exchange symmetry, and no non-invertible LSM-type theorem in higher-dimensional subsystem models has been established. The problem remains open.

Background

A symmetry-protected topological (SPT) phase is a gapped quantum many-body phase that is adiabatically connected to a trivial product phase only if a symmetry is allowed to be broken along the path; a spontaneous symmetry breaking (SSB) phase instead has an order parameter and degenerate ground states related by the symmetry. The Kramers-Wannier (KW) duality of the one-dimensional transverse-field Ising model exchanges its ordered and disordered phases. Promoted from a change of variables to an operator D\mathbf{D} on the Hilbert space, this map is not invertible: its fusion with itself reproduces a sum over symmetry operators (a Tambara-Yamagami-type fusion algebra, the algebraic structure of fusion categories in which a non-invertible element squares to a direct sum of invertible symmetry elements). At the self-dual point (where the Hamiltonian is mapped to itself by the duality), D\mathbf{D} therefore acts as a non-invertible symmetry - a symmetry operator without an inverse. The Kennedy-Tasaki (KT) transformation (Kennedy and Tasaki 1992) composes gauging of the symmetry (summing over the orbits of the symmetry group, which amounts to coupling the system to a dynamical discrete gauge field) with stacking a cluster-state SPT layer; it maps an SPT phase to an SSB phase, converting nonlocal string order parameters into local ones and exposing the hidden symmetry breaking of the Haldane phase, the prototypical SPT phase of the spin-1 chain. Li, Oshikawa, and Zheng (2023) recast the 1D KT transformation as KW gauging followed by stacking an SPT, and Seiberg and Shao (2024) showed that the 1D KW operator emerges, after gauging the fermion parity, from the anomalous translation symmetry of a Majorana chain (a system of Hermitian fermionic operators γ=γ†\gamma=\gamma^\dagger satisfying {γi,γj}=2δij\{\gamma_i,\gamma_j\}=2\delta_{ij}) - tying non-invertible duality to anomalies of ordinary symmetries, an anomalous symmetry being one whose action cannot be realized by onsite operators in the same number of dimensions.

Subsystem symmetries act on rigid lower-dimensional subspaces (for example, all spins on a lattice row or column) rather than on the whole system. Their spontaneous breaking (SSSB) yields extensively degenerate symmetry-broken manifolds, and SPT phases protected by them (SSPT phases) include the two-dimensional cluster-type models, with strong and weak SSPT classes classified for Z2\mathbb{Z}_2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 planar subsystem symmetries by You, Devakul, Burnell, and Sondhi (2018) and Devakul, Williamson, and You (2018). The Xu-Moore (Ising plaquette) model and the Wen-plaquette model are the canonical SSSB and topologically ordered members of this family. Higher-form symmetries, in turn, act on extended operators (loops or membranes) rather than on local charged operators; the spontaneous breaking of a 1-form symmetry is a topological order, e.g., the deconfined Z2\mathbb{Z}_2 gauge theory (3D toric code), and the 3D cluster state of Raussendorf, Bravyi, and Harrington is an SPT protected by a Z2\mathbb{Z}_2 1-form symmetry (Roberts, Yoshida, Kubica, and Bartlett 2017). Lieb-Schultz-Mattis (LSM) theorems state that a lattice model whose symmetry representation and unit-cell filling are incompatible with a trivial symmetric gapped phase must be gapless or degenerate; Seiberg, Seifnashri, and Shao (2024) proved a 1D version enforced by a non-invertible symmetry (gaplessness or ground-state degeneracy a multiple of three).

The source paper (Mana, Li, Sukeno, and Wei 2024, arXiv:2402.09520) supplied, for dd-dimensional hypercubic Ising models with Z2\mathbb{Z}_2 subsystem symmetry, an explicit realization of the KW duality operator D(d)\mathbf{D}^{(d)} as a sequential quantum circuit (a circuit whose gate layers sweep across the lattice in a fixed spatial order, so that its depth scales with the linear extent of the system rather than being constant) followed by a projection onto subsystem-symmetry eigenspaces, and used it to construct KT transformations that map the Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 SSPT (cluster) models to two copies of SSSB transverse-field Ising models, and the Z2\mathbb{Z}_2 SSPT to one copy of the SSSB double hypercubic Ising model. Its Appendix A identifies, in Majorana plaquette and hypercube models, a fermionic exchange symmetry that exchanges the plaquette term with the onsite fermion-parity term, does not commute with the subsystem fermion parity (hence is anomalous), and becomes the non-invertible KW symmetry after gauging. Its conclusion states the generalization program this task pursues.

Problem Statement

Complete the operator-level theory of non-invertible Kramers-Wannier (KW) and Kennedy-Tasaki (KT) transformations for lattice models beyond one dimension and beyond the Z2\mathbb{Z}_2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 subsystem-symmetric examples treated in Mana, Li, Sukeno, and Wei (2024). The common objective is that each generalized duality be realized at the operator level on the lattice - as in the source's constructions, by a quantum circuit followed by a projection onto the appropriate (higher-form or subsystem) symmetry eigenspaces - and that the associated phase correspondences, classifications, and constraints be derived. The source's conclusion asks for the following linked components.

  1. Higher-form KW: construct KW-type duality operators obtained by gauging higher-form symmetries, and explore the KW transformation more broadly beyond the known example in which gauging a Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 1-form symmetry maps the 3D cluster state to itself and the product state to two copies of the 3D toric code.

  2. Higher-form KT: construct the higher-form generalization of the KT transformation, for example by composing the higher-form KW map with the 3D cluster-state entangler (the finite-depth circuit that prepares the cluster state from the product state) so that a 3D cluster SPT protected by a 1-form symmetry maps to copies of higher-form SSB phases; realize it as a circuit with a projector and formulate its algebra. (The canonical Z2\mathbb{Z}_2 example now has a partial realization by Cao-Watanabe 2026; the general construction for other groups and dimensions, including the multiple-copies structure and complete fusion algebra, remains to be given.)

  3. SSPT classification beyond two dimensions: extend the classification of subsystem symmetry-protected topological phases to higher spatial dimensions, and construct KT transformations that map every SSPT class - for subsystem symmetry groups beyond Z2\mathbb{Z}_2 and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 - to corresponding SSSB phases, including the question of whether imposing the non-invertible KW duality symmetry refines 2+1d subsystem cluster states into distinct non-invertible SSPT phases.

  4. Fermionic origin: clarify the physical interpretation of the fermionic exchange symmetry of the Majorana plaquette and hypercube models - the anomalous symmetry that exchanges the plaquette term with the onsite fermion-parity term, does not commute with the subsystem fermion parity, and becomes the non-invertible KW symmetry after gauging - including its characterization as an anomaly.

  5. Non-invertible LSM constraints: determine the Lieb-Schultz-Mattis-type constraints imposed by non-invertible duality symmetries in higher-dimensional subsystem-symmetric lattice models; that is, for models invariant under both the subsystem symmetries and the non-invertible duality operator D(d)\mathbf{D}^{(d)} at the self-dual point, derive precise constraints on ground-state degeneracy or gaplessness, the 1D analogue being the theorem of Seiberg, Seifnashri, and Shao (gaplessness or degeneracy a multiple of three).

These components jointly support one objective: a general operator-level theory of non-invertible duality transformations in lattice models beyond one dimension, in which the constructions (components 1-2) underlie the phase correspondences and classification (component 3), the fermionic dual explains the origin of the same operators (component 4), and the LSM-type constraints are dynamical consequences of the same non-invertible symmetries (component 5). A complete resolution requires all five components; each is independently answerable or refutable, and a partial result on any one component does not resolve the task.

References

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