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arXiv:2603.21290

Non-invertible symmetries and boundary conditions for the transverse-field Ising model

Huan-Qiang Zhou, Qian-Qian Shi

Published Mar 22, 2026• 0 citations
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Abstract

Non-invertible Kramers-Wannier (KW) duality symmetries are constructed for the transverse-field Ising model (TFIM) at the self-dual point under various boundary conditions (BCs), as long as the resultant Hamiltonian commutes with the ${\rm Z}_2$ symmetry operator. This is achieved by introducing extra degrees of freedom into the Hilbert space, in order to turn a non-translation-invariant Hamiltonian in the original Hilbert space into a translation-invariant Hamiltonian in the augmented Hilbert space. One may lift the trivial identity operator, the ${\rm Z}_2$ symmetry operator and the non-invertible KW duality symmetry operator to their counterparts in the augmented Hilbert space, valid for each of four types of toroidal BCs. As it turns out, they yield a lattice version of fusion rules, which bears a resemblance to the Tambara-Yamagami ${\rm Z}_2$ fusion category. Our construction is thus consistent with the basic physical requirement that all possible BCs should yield a converging result in the thermodynamic limit. In particular, the lattice versions of fusion rules, constructed by Seiberg, Seifnashri and Shao [SciPost Phys. \textbf{16}, 154 (2024)], are reproduced for periodic and anti-periodic BCs, but a discrepancy is revealed for duality-twisted BCs.

DOI
10.48550/arxiv.2603.21290

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Non-invertible Kramers-Wannier (KW) duality symmetries are constructed for the transverse-field Ising model (TFIM) at the self-dual point under various boundary conditions (BCs), as long as the resultant Hamiltonian commutes with the Z2{\rm Z}_2Z2​ symmetry operator. This is achieved by introducing extra degrees of freedom into the Hilbert space, in order to turn a non-translation-invariant Hamiltonian in the original Hilbert space into a translation-invariant Hamiltonian in the augmented Hilbert space. One may lift the trivial identity operator, the Z2{\rm Z}_2Z2​ symmetry operator and the non-invertible KW duality symmetry operator to their counterparts in the augmented Hilbert space, valid for each of four types of toroidal BCs. As it turns out, they yield a lattice version of fusion rules, which bears a resemblance to the Tambara-Yamagami Z2{\rm Z}_2Z2​ fusion category. Our construction is thus consistent with the basic physical requirement that all possible BCs should yield a converging result in the thermodynamic limit. In particular, the lattice versions of fusion rules, constructed by Seiberg, Seifnashri and Shao [SciPost Phys. \textbf{16}, 154 (2024)], are reproduced for periodic and anti-periodic BCs, but a discrepancy is revealed for duality-twisted BCs.
The Kramers-Wannier transformation of the 1+1d transverse-field Ising model exchanges the paramagnetic and ferromagnetic phases and, at criticality, manifests as a non-invertible symmetry. Extending such self-duality symmetries beyond gauging of abelian groups in tensor-product Hilbert spaces has, however, remained challenging. In this work, we construct a generalized 1+1d Ising model based on a finite-dimensional semisimple Hopf algebra HHH that enjoys an anomaly-free non-invertible symmetry Rep(H)\mathrm{Rep}(H)Rep(H). We provide an intuitive diagrammatic formulation of both the Hamiltonian and the symmetry operators using a non-(co)commutative generalization of ZX-calculus built from Hopf-algebraic data. When HHH is self-dual, we further construct a generalized Kramers-Wannier duality operator that exchanges the paramagnetic and ferromagnetic phases and becomes a non-invertible symmetry at the self-dual point. This enlarged symmetry mixes with lattice translation and, in the infrared, flows to a weakly integral fusion category given by a Z2\mathbb{Z}_2Z2​ extension of Rep(H)\mathrm{Rep}(H)Rep(H). Specializing to the Kac-Paljutkin algebra H8H_8H8​, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six Rep(H8)\mathrm{Rep}(H_8)Rep(H8​)-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six Rep(H8)\mathrm{Rep}(H_8)Rep(H8​)-symmetric gapped phases on the lattice via the HHH-comodule algebra formalism, in agreement with the module-category classification of Rep(H8)\mathrm{Rep}(H_8)Rep(H8​). Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.
A bstract The Kramers-Wannier self-duality of critical quantum chains is examined from the perspective of model wave functions. We demonstrate, using the transverse-field Ising chain and the 3-state Potts chain as examples, that the symmetry operator for the Kramers-Wannier self-duality follows in a simple and direct way from a ‘generalised’ translation symmetry of the model wave function in the anyonic fusion basis. This translation operation, in turn, comprises a sequence of F -moves in the underlying fusion category. The symmetry operator thus obtained naturally admits the form of a matrix product operator and obeys non-invertible fusion rules. The findings reveal an intriguing connection between the (non-invertible) translation symmetry on the lattice and topological aspects of the conformal field theory describing the scaling limit.
The self-duality of the transverse-field Ising model is an archetype for dualities that, alongside symmetry and topology, are used as an organizing principle throughout modern physics. This duality, however, is not exact. The original and dual models have different symmetries and numbers of ground states, and the duality is implemented by a non-invertible operator giving rise to a non-invertible symmetry at the quantum critical point. Here, we show that by adjusting the model to accommodate open rather than periodic boundary conditions, it allows for an exact duality implemented by a unique invertible operator. In the model with exact duality, the symmetry at the quantum critical point is also exact, and hence invertible. Moreover, we find that the exact duality necessitates the presence of an anomalous edge degree of freedom, thus realizing a duality rather than topology based bulk-boundary correspondence. Finally, the exactness of the duality implies that the spontaneous breakdown of a global symmetry in terms of the original model can equivalently be described as spontaneously breaking a local symmetry in the dual system. We show that this seeming contradiction of Elitzur's theorem can be explained by the original and dual models obtaining different sensitivities to spatially local perturbations in any physical implementation of the Hamiltonian. Although the dual partners are mathematically equivalent, their physical implementations therefore are not. In analogy to the spontaneous breakdown of symmetries, we term this emergent distinction due to arbitrarily small environmental influences spontaneous duality breaking.
Non-invertible symmetries extend conventional symmetries beyond group-like transformations and play an important role in quantum field theory and many-body physics. However, a general framework of non-invertible symmetries in open quantum systems is still lacking. Here, we formulate a general Liouville-space duality framework that extends Hilbert-space non-invertible dualities to open quantum systems. The resulting defect relates Liouvillians with dual Hamiltonian and dissipative parameters and becomes an exact symmetry at self-duality. We apply this framework to the periodic transverse-field Ising chain with two collective Lindblad channels exchanged by Kramers--Wannier (KW) duality. A strong Z2\mathbb{Z}_2Z2​ symmetry divides Liouville space into four parity sectors, while the defect acts nontrivially only in the even-even sector. At self-duality, the KW defect imposes a momentum-resolved selection rule on Liouvillian Jordan blocks. Modes with different Liouville momenta or KW-defect eigenvalues cannot form a common exceptional Jordan block. For a three-site chain, a global Rényi-2 KW asymmetry dynamically identifies the self-dual conditions. At the same time, the spectral structure exhibits both a semisimple degeneracy and a self-dual exceptional point with a perturbation-selective response. These results extend non-invertible dualities to Markovian quantum dynamics and establish a symmetry-based selection rule for Liouvillian Jordan structures.
We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2 ℤ 2 symmetry. We introduce a subsystem \mathbb Z_2 ℤ 2 -gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.
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 Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ 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.