Non-invertible symmetries and boundary conditions for the transverse-field Ising model

Abstract

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 ${\rm Z}_2$ 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 ${\rm Z}_2$ 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 ${\rm Z}_2$ 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.