Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective

TL;DR

This paper explores the Kramers-Wannier self-duality in quantum chains through wave functions, revealing a connection between non-invertible translation symmetry, fusion categories, and topological features in conformal field theory.

Contribution

It demonstrates how the self-duality symmetry operator arises from a generalized translation symmetry in the wave function, linking lattice symmetries to topological quantum field theory.

Findings

Self-duality operator derived from wave function translation symmetry.

Operator admits a matrix product operator form with non-invertible fusion rules.

Connection established between lattice translation symmetry and topological CFT aspects.

Abstract

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.