The Kramers-Wannier transformation for spin systems

Al.B. Zamolodchikov

Michael Monastyrsky

Masataka Koide

Victor Matveevich Buchstaber, Michael Monastyrsky

In 1941 H Kramers and G Wannier discovered a special symmetry which relates low-temperature and high-temperature phases in the planar Ising model. The corresponding transformation, the Kramers–Wannier transform, is a special non-local substitution in the partition function. The existence of such transformations is a general property of lattice spin systems. Generalization of the KW transform to spin systems with non-Abelian symmetry is essential for many problems in statistical physics and field theory. This problem is very difficult and cannot be carried out by classical methods (like Fourier transform in the commutative case). We present new results which solve this problem for finite non-Abelian groups.

Lei Su, Ivar Martin

It is well known that the noninteracting Majorana chain is dual to the one-dimensional transverse-field Ising model, either through the Jordan–Wigner transformation or by gauging fermion parity. In this correspondence, the minimal translation of the Majorana chain maps to the celebrated Kramers–Wannier (KW) duality of the spin model, with the critical point mapped to the self-dual point. In this work, we generalize this mapping to two and higher dimensions by constructing a unitary equivalence between the parity-gauged fermionic system and a spin system defined on arbitrary polyhedral decompositions of space. Imposing the flatness condition on the gauge field yields a bosonization duality between the original (ungauged) fermionic system and a gauged spin system obeying a Gauss law. The dependence of the Gauss law in the spin system on the Kasteleyn orientation (and the discrete spin structure) of the fermionic system is made explicit. Applying this bosonization to one or two copies of Majorana fermions on translationally invariant lattices, we derive higher-dimensional analogs of KW (self-) dualities in spin systems arising from fermionic minimal translations. The KW (self-) dualities are non-invertible due to projections onto eigenspaces of higher-form symmetries in the associated symmetry operators. The bosonization framework we present is intuitive, general, and systematic, encompassing other known exact bosonization methods while offering a novel approach to establish new connections between fermionic and spin systems in arbitrary dimensions.

Stanislav Alekseevich Molchanov, Nguyen The Long