Exactly solvable Hamiltonians commuting with not-on-site chiral symmetry in 3+1D

Construct an exactly solvable lattice Hamiltonian in 3+1 dimensions, defined on a tensor‑product Hilbert space, that commutes with the prescribed not‑on‑site symmetry G = U(1)_V × U(1)_A acting on the lattice degrees of freedom, thereby enabling the chiral symmetry to be implemented without relying on auxiliary symmetry‑protected topological slabs.

Background

In the 3+1D part of the paper, the authors develop a strategy to realize chiral symmetries on the lattice by employing symmetry disentanglers. However, they note a key limitation: they have not found exactly solvable Hamiltonians that commute with the not-on-site symmetry G = U(1)_V × U(1)_A on the lattice Hilbert space. To proceed, they introduce an alternative construction that couples the D-dimensional model to boundaries of (D+1)-dimensional SPT slabs.

Addressing this limitation directly would streamline the approach by removing the need for domain-wall/SFT slab machinery, and would simplify the path to gauging the symmetry in the microscopic Hamiltonian. Hence, constructing such exactly solvable models is an explicit unresolved task identified by the authors.