Kramers-Wannier Duality

Definition

Kramers–Wannier duality is the exact map of the 2d Ising model onto itself that exchanges high and low temperature. High-temperature graphs of one lattice are the low-temperature domain walls of the dual lattice; the self-dual point $\sinh (2{\beta }_c)=1$ locates the critical temperature ${\beta }_c=\frac{1}{2}\ln (1+\sqrt{2})$ without solving the model.

The duality is cleanest in operator language:

• The order variable ${\sigma }_x$ (site spin) has $⟨\sigma \sigma ⟩$ long-ranged at low $T$.
• The disorder variable ${\mu }_{\tilde{x}}$ (Kadanoff–Ceva), living on dual sites, creates a domain-wall seam; $⟨\mu \mu ⟩$ is long-ranged at high $T$.
• They obey a mutual non-locality: ${\sigma }_x{\mu }_{\tilde{y}}=\pm {\mu }_{\tilde{y}}{\sigma }_x$, the sign set by whether the seam crosses $x$. The composite $\sigma \mu$ is a lattice fermion.

Role in Research

Kramers–Wannier is the first appearance in the generalized-symmetries course of the thread “a duality is a statement about a wall.” In the modern reading it is gauging a ${\mathbb{Z}}_2$ symmetry: summing over ${\mathbb{Z}}_2$ backgrounds maps the theory to its orbifold, and at the self-dual point the duality is implemented by a topological defect line whose fusion is a projector — a non-invertible symmetry. This is the ancestor of the condensation-defects the group’s Julia–Toulouse construction produces.

• The disorder operator $\mu$ is the direct 2d ancestor of the ‘t Hooft/monopole disorder operators of lattice-gauge-theory.
• Reading the duality as gauging connects it to the general ${\mathbb{Z}}_N^{(p)}\to {\mathbb{Z}}_N^{(d-p-2)}$ quantum-symmetry rule of higher-form-symmetries.

Relations

• villain-action — the same Poisson-resummation engine executes both the Ising and the XY duality
• higher-form-symmetries — Kramers–Wannier as gauging a 0-form symmetry, producing the dual (quantum) symmetry
• condensation-defects — the self-dual duality defect is the simplest non-invertible / condensation defect
• lattice-gauge-theory — Wegner’s ${\mathbb{Z}}_2$ gauge theory is the gauged version, self-dual in 3d

Papers

Kramers–Wannier, Phys. Rev. 60 (1941) 252; Kadanoff–Ceva, Phys. Rev. B 3 (1971) 3918; Aasen–Mong–Fendley, arXiv:1601.07185 (the duality defect on the lattice). See the course bibliography.

Notes

• Tracking boundary conditions honestly (periodic vs antiperiodic, spin structures) is where the subtleties live; the naive self-dual statement hides how the duality acts on the different sectors.
• The order/disorder algebra is the 2d shadow of a general fact: symmetry operators and the objects they act on are mutually non-local extended operators.