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arXiv:2510.17969

Non-invertible bosonic chiral symmetry on the lattice

Lukasz Fidkowski, Cenke Xu, Carolyn Zhang

Published Oct 20, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2510.17969

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In this work we realize the 3 + 1 dimensional non-invertible ZN{\mathbb{Z}}_NZN​ 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)U(1)U(1) rotor site Hilbert spaces. Specifically, our Hilbert space is that of a U(1)U(1)U(1) lattice gauge theory coupled to a charge 111 scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ZN{\mathbb{Z}}_NZN​ one-form symmetry Zm(1)Z^{(1)}_mZm(1)​ , at the lattice Hamiltonian level. We construct the generator of the ZN{\mathbb{Z}}_NZN​ chiral symmetry as as a unitary operator in the subspace of Zm(1)Z^{(1)}_mZm(1)​-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 Zm(1)Z^{(1)}_mZm(1)​, we find a dual description of this subspace, as the subspace of a charge 1/N1/N1/N gauge theory invariant under an electric one-form symmetry Ze(1)Z^{(1)}_eZe(1)​. 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 Ze(1)Z^{(1)}_eZe(1)​ symmetry.
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We show that the 3450 U(1) chiral fermion theory can appear as the low energy effective field theory of a 1+1D local lattice model, with an on-site U(1) symmetry and finite-range interactions. The on-site U(1) symmetry means that the U(1) symmetry can be gauged (gaugeable for both background probe and dynamical fields), which leads to a non-perturbative definition of chiral gauge theory --- a chiral fermion theory coupled to U(1) gauge theory. Our construction can be generalized to regularize any U(1)-anomaly-free 1+1D gauged chiral fermion theory with a zero chiral central charge (thus no gravitational anomaly) by a lattice, thanks to the recently proven Poincare dual equivalence between the U(1) 't Hooft anomaly free condition and the U(1) symmetric interaction gapping rule, via a bosonization-fermionization technique.
A bstract We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers : constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for U(1) symmetries with mixed ’t Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional “3450” chiral gauge theory, and we argue that a related construction applies to the U(1) hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a Zp\mathbb{Z}_pZp​ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality SSS, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT TTT. In the continuum, they form a central extension of PSL(2,Z4)PSL(2,\mathbb{Z}_4)PSL(2,Z4​) for p=2p=2p=2, and of SL(2,Zp)SL(2,\mathbb{Z}_p)SL(2,Zp​) for odd primes ppp, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For p=2p=2p=2 the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a Zp\mathbb{Z}_pZp​ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For p=2p=2p=2 they are generated by the non-trivial semion QCA, and for odd prime ppp they are generated by the non-trivial Zp\mathbb{Z}_pZp​ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
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A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry GGG in 1+1D spacetime is proposed. More precisely, we ask whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry GGG defined on a 1D spatial lattice with continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry GGG? In particular, we show that the 3L_LL​-5R_RR​-4L_LL​-0R_RR​ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffered from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. As an additional remark, we show a topological non-perturbative proof on the equivalence relation between the 't Hooft anomaly matching conditions and the boundary fully gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) of U(1) symmetry. Our proof holds universally independent from Hamiltonian or Lagrangian/path integral formulation of quantum theory.