Chiral Lattice Gauge Theories from Symmetry Disentanglers

Abstract: 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.

• The paper presents a novel method using symmetry disentanglers to convert non-on-site symmetries into gaugeable on-site forms for chiral lattice systems.
• It develops explicit rotor lattice models in 1+1D and extends the approach to 3+1D via constant-depth circuits and ancilla rotors.
• The framework resolves anomaly matching challenges and enables nonperturbative Hamiltonian formulations with direct implications for the Standard Model.

Symmetry Disentanglers in Hamiltonian Lattice Models for Chiral Gauge Theories

Introduction and Motivation

Chiral gauge theories underlie fundamental interactions in high-energy physics, most notably exemplified by the Standard Model. A long-standing challenge has been the construction of strictly local, nonperturbative lattice Hamiltonians for chiral gauge theories in dimensions greater than $1{+}1$, circumventing the computational and conceptual obstructions imposed by the Nielsen-Ninomiya theorem and fermion doubling. Existing strategies, such as overlap fermion approaches and symmetric mass generation (SMG), either do not yield manifest locality or heavily depend on analyzing strongly-coupled mirror sectors. This work introduces a new paradigm based on symmetry disentanglers—constant-depth circuits of local unitaries that map non-on-site symmetries to on-site ones—thereby rendering certain chiral symmetries gaugeable using standard lattice methods.

Figure 1: Construction of a chiral theory by using a symmetry disentangler to map not-on-site symmetry to an on-site subgroup, allowing local Hamiltonian realization and enabling standard lattice gauging methods.

Lattice Rotor Models and Anomaly Structure in 1+1D

The authors formulate explicitly solvable rotor lattice models in $1{+}1$D with a compact boson field $\phi_j$ and Villain integer-valued gauge fields $n_{j-1,j}$. The $U(1)_V$ and $U(1)_A$ symmetry generators in this setting are constructed to exhibit the characteristic mixed 't Hooft anomaly:

$$Q_V = \frac{i}{2\pi} \sum_j \frac{d}{d\phi_j}, \quad Q_A = \sum_j (\phi_j - \phi_{j-1} - n_{j-1,j})$$

A $2\pi$ $U(1)_A$ transformation over an interval induces endpoint operators charged under $U(1)_V$, signifying anomaly inflow and spectral flow analogous to continuum results. The key advance is the explicit construction of a Villain disentangler, a spatially local operation that transforms the Villain Hilbert space to a tensor product structure, facilitating local implementation of symmetries and precise control of their anomaly content.

Numerical results and exact solvability are exhibited for the "3450 theory" in $1{+}1$D: two left-moving Weyl fermions of charge $3$ and $4$, and two right-moving of charge $5$ and $0$. The construction delivers, for the first time, a manifestly local Hamiltonian solution to this chiral gauge theory.

Disentangling Symmetries via Ancilla Rotors and Constant-Depth Circuits

For both bosonic and fermionic models, the paper develops a general symmetry disentangling scheme:

• Ancilla Rotors: Introduced at each lattice site to serve as local gauge degrees of freedom.
• Disentangler Operators: Explicit constant-depth circuits $W$ are constructed that conjugate a non-on-site symmetry to an on-site form,
• Anomaly-Free Subgroups: Disentangling is only possible for subgroups satisfying appropriate anomaly cancellation conditions: $\sum_\alpha q_V^\alpha q_A^\alpha = 0$ for $1{+}1$D and $\sum_\alpha q_A^\alpha q_V^\alpha q_V^\alpha = 0$ for $3{+}1$D.

This framework is extended to stacks of rotor systems, enabling construction of arbitrary chiral $U(1)$ theories with anomaly-free charge matrices.

Chiral Symmetry Disentangling in 3+1D: Bosonic and Fermionic Models

Moving to $3{+}1$D, the construction is realized on branched simplicial lattices. The $U(1)_A$ symmetry generator is engineered as a constant-depth circuit using cup products and cochain variables, and its anomaly structure is encoded by boundary SPT entanglement terms proportional to $A_A \wedge dA_V \wedge dA_V$.

Figure 2: Lattice model embedding rotors and Majorana fermions, with resolved dual lattice facilitating parity and anomaly content via Kasteleyn orientations.

The fermionic generalization incorporates Majorana degrees of freedom corresponding to faces of the simplicial lattice, and leverages results on Kasteleyn-oriented Kitaev chain decorations for odd vorticity loops (see Figure 3). The $U(1)_A$ generator includes topological fermion-pairing terms controlled by the rotor state, and the resultant lattice model reproduces the correct anomaly—on closed manifolds, $\exp(i\pi Q_A^{\text{f}}) = (-1)^F$ implements fermion parity.

Figure 3: Short Kasteleyn-oriented loop in the resolved dual lattice surrounding an edge, illustrating assignment of Majorana pairing and parity.

The symmetry disentangler for the fermionic system is, again, formulated as a local dressing of the bosonic disentangler by a fermionic component, such that conjugation maps the chiral combination to an on-site symmetry provided the anomaly-free condition holds.

Anomaly In-flow and $D+1$ Dimensional SPT Coupling

A novel variant of the construction is presented from the anomaly in-flow perspective. Here, chiral boundary modes of SPT slabs are gapped against $D$-dimensional anomalous models by constructing interfaces between free-fermion and commuting-projector SPT Hamiltonians. The symmetry disentangler is deployed in the bulk near the lower boundary, ensuring the on-site-ability of the anomaly-free subgroup and enabling the trivial gapping of boundary modes.

Figure 4: Anomaly in-flow construction: stack of $(D{+}1)$-dimensional free-fermion SPTs adjacent to commuting-projector SPTs. Disentangler acts near lower boundary to render a subgroup on-site, enabling gapping.

This formalism renders the SMG scenario tractable by providing controlled mechanisms for interface gapping, decoupling strong interactions from anomaly matching, and offering a promising route to local nonperturbative formulations of chiral lattice gauge theories.

Implications for the Standard Model

The framework is shown to accommodate the $U(1)_Y$ hypercharge assignments for one fermion generation in the Standard Model (including a sterile neutrino): the charge vectors of both the quark and lepton sectors are precisely groupable into anomaly-free stacks, and symmetry disentanglers exist to map the not-on-site hypercharge action to an on-site symmetry. The construction does not immediately generalize to arbitrary chiral charge sectors—groupability is crucial for the existence of a symmetry disentangler, and certain charge configurations with vanishing anomalies are excluded.

Practical and Theoretical Implications

The symmetry disentangler formalism resolves non-on-site symmetry obstructions—previous limitations on the gaugeability of discrete and continuous symmetry-protected topological phases in lattice models. The explicit construction of commuting-projector Hamiltonians for $U(1)$ SPT phases avoids the no-go theorems for finite-dimensional site Hilbert spaces, relying instead on infinite-dimensional rotor spaces.

Applications include new exactly solvable Hamiltonians for chiral gauge theories in $1{+}1$D and analytical progress for $3{+}1$D, with direct relevance for simulating chiral fermion sectors beyond perturbation theory and studying anomaly-constrained topological phases. The approach establishes a concrete, analytically controlled connection between anomaly inflow, lattice chiral symmetries, and the construction of symmetric trivial interfaces.

Conclusion

The work provides a Hamiltonian-level route to formulating chiral gauge theories on the lattice by utilizing symmetry disentanglers, combining techniques from cohomological anomaly matching, tensor product Hilbert spaces, and constant-depth circuit theory. By reducing the problem of gaugeability to on-site-ability via locality-preserving unitaries, it unlocks both practical constructions for explicit chiral lattice systems and conceptual clarity regarding the interplay between symmetry, locality, and anomaly structure in many-body quantum systems. Future research may focus on explicit realization and classification of gapped interfaces in higher dimensions, potential extensions to non-abelian symmetry settings, and rigorous mathematical analysis of infinite-dimensional operator domains in quantum lattice models.