Kramers–Wannier Duality Operators

• Kramers–Wannier operators are duality operators that exchange order variables with disorder variables via gauging and Fourier transforms in lattice and field theories.
• They manifest as non-invertible symmetry defects whose fusion rules yield projectors or sums of defects rather than simple inverses, crucial for modeling critical phenomena.
• Recent developments extend their framework to Hopf-algebraic chains, higher-form gauge theories, and explicit matrix-product and circuit implementations, broadening their applicability.

Kramers–Wannier operators are lattice or field-theoretic duality operators that implement the exchange between order variables and disorder variables, originally in the Ising model and now in a wide range of generalized settings. In the modern formulation, they are typically non-invertible symmetry defects rather than ordinary unitary internal symmetries: they act by intertwining local operator algebras, by gauging and Fourier/Hadamard transforms, or by sector-dependent quantum operations, and their fusion commonly yields projectors or sums of symmetry defects instead of an inverse. Recent work extends this structure from the $1+1$d transverse-field Ising model to Hopf-algebraic chains, subsystem-symmetric models, higher-dimensional bosonization dualities, higher-form gauge theories, integrable quantum chains, and explicit shallow quantum circuits (Lu et al., 10 Feb 2026).

1. Canonical Ising construction and the order–disorder exchange

In the standard $1+1$d setting, the transverse-field Ising chain exhibits the basic Kramers–Wannier map between local spin operators and disorder variables. One formulation uses dual variables

$$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$

so that the Ising interaction and transverse field are exchanged, and the Hamiltonian is mapped to its dual with couplings interchanged. In operator language this becomes the familiar order–disorder exchange: local order operators are mapped to bond or string operators, and vice versa (Su et al., 27 Aug 2025).

For critical quantum chains, the Kramers–Wannier operator is naturally realized as a “half-step translation” in an anyonic fusion basis. In the critical transverse-field Ising chain, the resulting operator $\mathsf{D}$ commutes with the Hamiltonian at criticality and satisfies

$$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$

while its square is not the identity but

$$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$

This makes the operator non-invertible and shows that lattice translation mixes with the defect fusion rule (Zhang et al., 2024).

A related and more sector-sensitive formulation appears for the transverse-field Ising chain on a ring, where the duality must be implemented with a proper treatment of charge sectors and twisted boundary conditions. There the Kramers–Wannier map is encoded by a superoperator $U$ on the tensor product of the Ising and dual-Ising Hilbert spaces, satisfying

$$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$

The associated quantum operation has an operator-sum representation

$$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$

and reproduces the well-known non-invertible fusion rules (Khan et al., 2024).

2. Non-invertibility, fusion, and categorical structure

A defining feature of Kramers–Wannier operators is that their composition does not produce an inverse. In the Ising CFT language, the duality defect satisfies the canonical fusion rule

$D \times D = 1 + \psi,$

and higher-dimensional analogs preserve the same structural principle: the square of the defect becomes a sum or projector built from symmetry defects rather than the identity (Kaidi et al., 2021).

On the lattice, this non-invertibility often appears as a projector onto symmetry-neutral sectors. For the critical Ising and $1+1$0-state Potts chains constructed from fusion-category $1+1$1-moves, the corresponding MPOs satisfy

$1+1$2

so the categorical fusion survives on the lattice but is mixed with translation (Zhang et al., 2024).

In the transverse-field Ising chain on a ring, the quantum-operation approach makes the same point in sector language. The duality defect has fusion

$1+1$3

and the checkerboard of twisted and untwisted sectors shows that the duality exchanges charge and twist rather than acting as a single unitary symmetry on a fixed Hilbert space (Khan et al., 2024).

This non-invertible structure persists in many generalized settings. In modulated-symmetry chains, the non-invertible reflection defect satisfies

$1+1$4

so the square is proportional to projectors onto neutral sectors of each modulated symmetry generator (Pace et al., 2024). In integrable constructions, Kramers–Wannier-like operators built from transfer matrices satisfy

$1+1$5

in Ising-type models, and

$1+1$6

in the $1+1$7 parafermionic case (Zhu, 2 Sep 2025).

A common misconception is that Kramers–Wannier operators are merely unitary basis changes. The recent literature instead treats them as non-invertible defects, MPOs, or quantum operations whose algebra is controlled by projectors, higher-form sectors, or defect fusion, not by group inverses (Khan et al., 2024).

3. Generalized constructions beyond the ordinary Ising chain

A substantial generalization replaces the $1+1$8 data of the Ising chain by a finite-dimensional semisimple Hopf $1+1$9-algebra $$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$0. In the Hopf–Ising chain, local operators are “Hopf Pauli” operators

$$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$1

and

$$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$2

When $$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$3 is self-dual, a symmetric Hadamard form $$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$4 defines a unitary $$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$5, and the Kramers–Wannier duality operator is

$$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$6

It exchanges order and disorder operators through

$$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$7

and obeys

$$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$8

At criticality it becomes a non-invertible symmetry defect, and in the infrared flows to a weakly integral $$\mu_{j+1/2}^x=\sigma_j^z\sigma_{j+1}^z,\qquad \mu_{j+1/2}^z=\prod_{k\le j}\sigma_k^x,$$9-graded fusion category with $\mathsf{D}$0 and $\mathsf{D}$1 (Lu et al., 10 Feb 2026).

A parallel generalization appears in symmetry-TFT treatments of generalized Ising models. There, Kramers–Wannier duality is implemented by an interface $\mathsf{D}$2 that combines gauging of a subsymmetry with Fourier transform of local weights. For finite-group symmetry $\mathsf{D}$3, the generalized Ising partition function

$\mathsf{D}$4

is mapped by group Fourier transform to a $\mathsf{D}$5-valued formulation, and fully gauging the symmetry produces the non-abelian Kramers–Wannier dual (Delcamp et al., 2024).

The same theme appears in modulated-symmetry chains. There the canonical gauging map is

$\mathsf{D}$6

and the dual modulation profile is reflected: $\mathsf{D}$7 This yields a Kramers–Wannier duality even when ordinary reflection symmetry is absent, because the composite operator $\mathsf{D}$8 can commute with the Hamiltonian at the self-dual point although $\mathsf{D}$9 itself is not a symmetry (Pace et al., 2024).

A plausible implication is that “Kramers–Wannier operator” is no longer tied to a single abelian gauging recipe. In the current literature it denotes a broader class of duality defects whose microscopic realization may come from Hopf pairings, symmetry TFT boundary changes, modulated-symmetry gauging, or sector-selective projectors, while preserving the order–disorder exchange and non-invertible fusion.

4. Higher dimensions, higher-form symmetries, and subsystem versions

Kramers–Wannier operators now appear well beyond $$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$0 dimensions. In a bosonization framework for parity-gauged Majorana systems, minimal fermionic translations become higher-dimensional Kramers–Wannier operators after projection onto higher-form symmetry sectors. On the square lattice, the bosonized operator takes the form

$$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$1

and acts by conjugation as

$$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$2

Its non-invertibility comes from the higher-form projectors $$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$3 (Su et al., 27 Aug 2025).

Subsystem-symmetric lattice models furnish another extension. In the plaquette Ising model with subsystem $$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$4 symmetry, the subsystem Kramers–Wannier operator $$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$5 gauges the full subsystem symmetry and maps local operators by

$$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$6

$$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$7

Gauging twice yields a “grid” operator rather than an inverse, and the associated duality defects are mobile in both spatial directions (Cao et al., 2023).

Higher-form and gauge-theoretic versions also exist. In lattice higher gauge theory, the Kramers–Wannier operator is the topological operator $$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$8 acting on Wilson-surface operators $$\mathsf{D} X_{l} = Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D}, \qquad \mathsf{D} Z_{l} Z_{l+1\,(\mathrm{mod}\,L)} = X_{l+1\,(\mathrm{mod}\,L)} \,\mathsf{D},$$9 by the intersection-number phase

$$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$0

Under the generalized Kramers–Wannier duality,

$$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$1

the symmetry operator is mapped to a dual Wilson operator, and order/disorder braiding becomes the higher-form analog of the Ising $$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$2–$$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$3 algebra (Kawana, 9 Jul 2025).

In $$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$4d gauge theory, codimension-1 Kramers–Wannier-like defects $$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$5 arise by gauging a $$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$6-form $$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$7 symmetry and dressing the interface with a $$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$8 Chern–Simons theory. Their fusion is

$$\mathsf{D}^{2} = \frac{1}{2}\,T\,\left( 1 + \eta_{(L)} \right), \qquad \eta_{(L)}=\prod_{l=1}^L X_l.$$9

which is explicitly the higher-dimensional analog of $U$0 (Kaidi et al., 2021).

5. MPOs, quantum circuits, and explicit implementations

A major development is the explicit realization of Kramers–Wannier operators as finite-depth or matrix-product operators. In the wave-function construction for critical Ising and Potts chains, the operator is obtained from a sequence of $U$1-moves and admits a translationally invariant MPO form. For the Ising case,

$U$2

with local tensor

$U$3

making the non-invertible defect explicit at the tensor-network level (Zhang et al., 2024).

Sequential-circuit realizations are now also available. For a nonintegrable scarred Ising model, the Kramers–Wannier operator is represented by the finite-depth sequential quantum circuit

$U$4

which implements

$U$5

in the $U$6 sector, with boundary twists encoding the non-invertible sector dependence (Fontana et al., 7 Aug 2025).

Integrable trotterizations of the critical Ising chain reveal an additional doubling in discrete time. The continuous-time duality operator

$U$7

splits into two inequivalent discrete-time operators,

$U$8

with

$U$9

They implement “half-translations” along discrete light-cone directions $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$0 (Sinha et al., 6 Nov 2025).

The most recent purely unitary implementations achieve logarithmic depth with nonlocal connectivity. In $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$1d $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$2, one constructs a unitary $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$3 such that

$$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$4

while in $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$5d the map becomes

$$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$6

These circuits have depth $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$7 and generalize to arbitrary $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$8 dualities (Cheng et al., 2 Jul 2026).

A common misunderstanding is that Kramers–Wannier duality necessarily requires linear-depth local circuits. The shallow-circuit results show that logarithmic-depth, spatially nonlocal unitary circuits can realize the exact duality maps in $$U \left( X_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{Z}_i \hat{Z}_{i+1} \right) U, \qquad U \left( Z_{i-1} Z_i \otimes \mathbb{1} \right) = \left( \mathbb{1} \otimes \hat{X}_i \right) U.$$9d and $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$0d, although purely local connectivity still imposes linear-depth costs (Cheng et al., 2 Jul 2026).

6. Phases, interfaces, and applications

Kramers–Wannier operators organize phase structure and critical behavior. In the Hopf–Ising case based on the self-dual Kac–Paljutkin algebra $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$1, the self-dual Hamiltonian is studied numerically and four of the six $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$2-symmetric gapped phases are identified: A: fully symmetric (GSD 1), B: preserves the invertible $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$3 subgroup (GSD 2), C: preserves the diagonal $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$4 (GSD 2), D: fully SSB (GSD 5). The transitions $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$5 and $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$6 are described by Ising critical lines, and a multicritical point appears at $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$7 (Lu et al., 10 Feb 2026).

In symmetry-enriched topological phases, translation can enforce Kramers–Wannier self-duality. For the Wen–plaquette model on a cylinder, one-site translation along the edge swaps the boundary operators $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$8, which in the effective spin chain become $$\mathcal{E}(\rho) = \sum_{\{s\}} D_s \rho D_s^\dagger, \qquad D_s := \langle s | \Delta | \psi \rangle,$$9. As a result, translation-invariant perturbations force both the edge Hamiltonian and the entanglement Hamiltonian to be Kramers–Wannier self-dual. However, edge–entanglement-spectrum correspondence does not hold generically; it appears only in a finite domain in Hamiltonian space where both effective theories realize the critical Ising model (Ho et al., 2014).

Mixed-state constructions provide another application. In higher-order subsystem SPT phases, tracing out the bulk can produce $D \times D = 1 + \psi,$0d mixed states with a strong non-invertible Kramers–Wannier symmetry $D \times D = 1 + \psi,$1 satisfying

$D \times D = 1 + \psi,$2

Interfaces then diagnose phase distinctions: when no local projector can preserve the relevant Kramers–Wannier symmetry across an interface, the two mixed states are in distinct phases (Mana et al., 3 Mar 2026).

Quantum many-body scarring offers a dynamical application. In a nonintegrable model with exact scar states $D \times D = 1 + \psi,$3, $D \times D = 1 + \psi,$4, and $D \times D = 1 + \psi,$5, the Kramers–Wannier circuit maps

$D \times D = 1 + \psi,$6

where $D \times D = 1 + \psi,$7 lies in a twisted sector with no exact scar. This is used as a diagnostic: scars are stable when the duality preserves the embedding conditions, and fragile when the dual image lies outside the protected sector (Fontana et al., 7 Aug 2025).

Other applications are computational or combinatorial. For the SU(2) principal chiral model, the Kramers–Wannier dualization rewrites constrained worldline fluxes as unconstrained dual plaquette variables and disorder loop variables, producing a second representation with real and positive weights suitable for Monte Carlo simulation (Gattringer et al., 2017). For Boolean satisfiability, a generalized Kramers–Wannier duality maps the #SAT partition function to a dual problem involving non-negative solutions of a Diophantine system of equations (Mitchell et al., 2013). For planar Ising models with arbitrary bond couplings, the exact free energy is expressed as the determinant of ordered and disordered operators defined on vertices and dual vertices, making the duality explicit even in the random-bond case (Song, 2023).

A final limitation concerns universality claims. Several works emphasize that Kramers–Wannier self-duality alone does not force a unique infrared outcome. In the Wen–plaquette setting, tricritical Ising or first-order self-dual transitions remain possible (Ho et al., 2014). In the Hopf case, the Frobenius–Schur indicator of the infrared defect cannot be fixed purely at the lattice level because the defect mixes with translation (Lu et al., 10 Feb 2026). This suggests that Kramers–Wannier operators are best understood as precise algebraic structures whose physical consequences depend on the symmetry sector, boundary conditions, and renormalization-group realization rather than as a single universal mechanism.