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arXiv:2605.13363

Invertible Symmetry and Spontaneous Duality Breaking in the Transverse-Field Ising Model

José Dupont, Jasper van Wezel

cond-mat.str-elquant-phSubmitted May 13, 2026• Announced May 14, 2026• 0 citations
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Abstract

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.

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7 pages, 1 figure

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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 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.
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 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.
In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation UKT{U}_{\mathrm{KT}}UKT​, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. In this paper, we find that UKT{U}_{\mathrm{KT}}UKT​ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under UKT{U}_{\mathrm{KT}}UKT​) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when HHH is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model H+UKTHUKTH+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}H+UKT​HUKT​ is exactly equivalent to a gapless spin-1/21/21/2 XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.
Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) S_3 S 3 symmetry and the other with non-invertible \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1d symmetry-topological-order (SymTO) of type JK_4\boxtimes \overline{JK}_4 J K 4 ⊠ J K ¯ 4 . The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual S_3 S 3 -symmetric and \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) -symmetric models.
Spontaneous symmetry breaking is a well-understood mechanism for generating distinct phases of matter. Recently, the notion of symmetry has been broadened to include operations without inverses, leading to the concept of non-invertible symmetries. How do symmetry-breaking phases associated with non-invertible symmetries differ from those arising from conventional invertible symmetries? We address this question using concrete lattice models of the gapped phases with non-invertible Rep(H8H_8H8​) symmetry as an example. We find that, despite the symmetry being non-invertible, the symmetry-breaking phases can still be characterized by the long-range correlation of local order parameters, which obey a more general algebraic structure than in the invertible setting. Furthermore, via generalized gauging, certain non-invertible symmetry-breaking transitions can be mapped to deconfined quantum critical points of invertible symmetries, and vice versa. We establish precise conditions under which this duality holds and illustrate them with several families of examples, providing a systematic route to studying beyond-Landau phase transitions.