Non-invertible bosonic chiral symmetry on the lattice

TL;DR

This work constructs a 3+1D lattice realization of a non-invertible ${\mathbb{Z}}_N$ chiral symmetry by employing an infinite-dimensional rotor lattice coupled to a $U(1)$ gauge field in the Villain formalism. A non-onsite axial symmetry on the rotor matter is gauged to produce a gauge-invariant current ${*j_A}$ whose presence, together with a projected magnetic ${Z_m^{(1)}}$ sector, yields a non-invertible ${\mathbb{Z}}_N$ symmetry operator $U_{1/N}(M^3)$ on ${\cal H}^m$; gauging the magnetic one-form symmetry leads to a dual description on ${\cal H}^e$ where the operator extends to a locality-preserving circuit but carries a mixed anomaly with the electric one-form symmetry ${Z_e^{(1)}}$. The authors connect this lattice construction to a target 3+1D field theory with a bosonic analogue of the Adler-Bell-Jackiw anomaly, proposing a bosonic QCD-like theory whose axial rotation pumps a root 2+1D bosonic ${U(1)}$ SPT state at the boundary, and outline how gauging $U(1)_V$ in this field theory yields the non-invertible symmetry seen on the lattice. The framework thus links lattice rotor realizations, one-form gauging, and SPT entanglers into a coherent picture of non-invertible chiral symmetries, with potential extensions to gravitational anomalies and fermionic generalizations. This work provides a concrete operator-theoretic realization of non-invertible chiral symmetries on the lattice and an explicit dual description that clarifies the interplay between higher-form symmetries, anomalies, and quantum cellular automata.

Abstract

In this work we realize the 3 + 1 dimensional non-invertible ${\mathbb{Z}}_N$ chiral symmetry generator as an operator in a many body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite dimensional $U(1)$ rotor site Hilbert spaces. Specifically, our Hilbert space is that of a $U(1)$ lattice gauge theory coupled to a charge $1$ scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ${\mathbb{Z}}_N$ one-form symmetry $Z^{(1)}_m$ , at the lattice Hamiltonian level. We construct the generator of the ${\mathbb{Z}}_N$ chiral symmetry as as a unitary operator in the subspace of $Z^{(1)}_m$-invariant states, and show that it cannot be extended to the entire Hilbert space while preserving locality and unitarity. Using a lattice-level duality based on gauging $Z^{(1)}_m$, we find a dual description of this subspace, as the subspace of a charge $1/N$ gauge theory invariant under an electric one-form symmetry $Z^{(1)}_e$. We show that in this dual formulation, the chiral symmetry generator does extend unitarily to the entire Hilbert space, but has a mixed anomaly with the $Z^{(1)}_e$ symmetry.

Figures (1)

• Figure 1: An example of vortex self-linking. The vortex loop is in fact a ribbon, and Eq. \ref{['chargedaction']} gives each linking of the ribbon a phase.