Simulating a chiral gauge theory

Laurens Lootens; Hersh Singh

AI summary · arXiv: 2611.06880v1 · hep-lat

Simulating a chiral gauge theory

Laurens Lootens, Hersh Singh

Original paper Submitted to arXiv on 04 Nov. 2026 20 pages, 4 figures License: CC BY 4.0 Also in: fr

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The essentials

The authors construct a finite-dimensional, gauge-invariant Hamiltonian lattice formulation and use tensor networks to simulate the 34-50 chiral gauge theory at N=6.

Question

Can 1+1-dimensional abelian chiral gauge theories be regularized and simulated with exact gauge invariance without sign problems or infinite-dimensional local Hilbert spaces? The paper tests whether the finite-N 34-50 model reproduces the continuum theory’s expected infrared physics.

Method

The authors build a lattice model from Z_N gauge fields and Z_{N^2} clock-shift qudits, retaining the chiral symmetry and anomaly modulo N, then solve the Gauss constraints to obtain an unconstrained chain. They use infinite matrix product states, including VUMPS and finite-entanglement scaling, to study the Z_N Schwinger model and the 34-50 model; for the latter they use N=6, couplings J_p=1, J_x=1/4, J_e=1, κ=0, and bond dimensions up to χ=256.

Findings

Result The lattice construction preserves the chiral symmetry and its anomaly modulo N, and permits exact gauging when the lattice anomaly cancels.
Result For the Z_N Schwinger model, the mass-gap extrapolation agrees closely with the analytic Schwinger-boson value.
See the passage in the paper
A linear fit in $N^{-2}$ to $N=4$–$8$ already extrapolates to $0.1\%$ below the exact value $e/\pi\sqrt{2}=0.22508$, while adding an $N^{-4}$ term results in an agreement within $5\times 10^{-5}$.
Result At N=6, the 34-50 model has central charge c=1.002(1), consistent with the predicted compact-boson infrared theory.
See the passage in the paper
The entanglement entropy grows with the correlation length $\xi$ of the transfer matrix as $S=(c/6)\ln\xi+\mathrm{const}$, with $c=1.002(1)$, in excellent agreement with the analytic prediction that the ir is a free massless Dirac fermion or a compact boson [43].
Authors’ interpretation The measured compactification radius is slightly displaced from the free-fermion value, indicating a weakly interacting massless Dirac fermion.
See the passage in the paper
This answers the question above: at $N=6$, the ir of the $\mathbb{Z}_{N}$ chiral lattice gauge theory is the predicted $c=1$ theory, slightly off the free-fermion radius: a weakly-interacting massless Dirac fermion, as is expected due to the presence of an exactly marginal Thirring interaction.
Result The lattice formulation realizes anomalous fermion-number violation through the gauge dynamics.
See the passage in the paper
A nonzero $\langle X^{N}\rangle$ therefore means that fermion number is not conserved.

Limitations

The demonstrated continuum-like infrared result is for the 34-50 model at N=6 and the specified couplings, so it does not establish equivalent behavior for other models or parameter choices.
The construction is developed for 1+1-dimensional abelian theories; extension to higher dimensions or nonabelian gauge groups remains unresolved.
See the passage in the paper
Extension to higher dimensions or nonabelian gauge groups is not straightforward but certainly conceivable; see Refs. [95, 25, 96] for recent ideas in this vein.

Why it matters

The authors argue that finite-dimensional local Hilbert spaces can support an efficient, gauge-invariant numerical formulation of 1+1-dimensional abelian chiral gauge theories, contrary to the perceived need for infinite-dimensional rotors. They present the 34-50 simulation as a benchmark against which future proposals can be tested, while noting that a practical lattice definition of the Standard Model still requires further ideas.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: chiral gauge theory; lattice field theory; qudit; tensor-network simulation; fermion-number violation

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Comprehensive Summary
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Original abstract

The paper addresses a longstanding practical challenge in lattice field theory: simulating chiral gauge theories while preserving exact gauge invariance. Standard lattice fermion constructions encounter the Nielsen–Ninomiya obstruction, while alternative bosonized approaches may require infinite-dimensional local Hilbert spaces or present difficulties for numerical methods. The authors study abelian chiral gauge theories in one space and one time dimension, focusing on the 34-50 model, whose left- and right-moving fermions have charges (3, 4) and (5, 0), respectively. They begin by bosonizing the fermions and then replace the compact bosonic degrees of freedom with finite-dimensional clock-and-shift qudits. Each local matter register has dimension $N^2$. The construction preserves a $\mathbb{Z}_N$ subgroup of the chiral symmetry and its anomaly modulo $N$. When the continuum gauge anomaly cancels, the corresponding lattice charges commute, so the symmetry can be gauged consistently. The resulting Gauss constraints are imposed and solved exactly. For the 34-50 model, a qudit T-duality simplifies the constraints. Further reduction produces an unconstrained chain with two $\mathbb{Z}_N^2$ registers per site. This formulation is suitable for infinite matrix product state simulations, which avoid the sign problem associated with Monte Carlo methods. The authors first test their construction on a $\mathbb{Z}_N$ Schwinger model. At gauge coupling $e=1$, its mass gap approaches the analytic continuum value of 0.22508. A fit to results for $N=4$ through $8$, including corrections proportional to $N^{-4}$, agrees with this value to within 0.00005. They then simulate the 34-50 model at $N=6$, with couplings $J_p=1$, $J_x=1/4$, and $J_e=1$, and with discrete theta angle zero. Finite-entanglement scaling yields a central charge of 1.002(1), consistent with a massless compact boson. The measured momentum and winding scaling dimensions are approximately 0.967(7) and 0.259(3), respectively. Their product, 0.250(3), agrees with the compact-boson prediction of one quarter. The inferred boson radius is 1.017(3) times the free-fermion radius. These results indicate that the infrared theory is a massless Dirac fermion with a weak Thirring interaction, rather than an exactly free-fermion theory. The paper also demonstrates a lattice version of anomalous fermion-number violation. In the 34-50 model, the anomaly coefficient is 2, so a unit change in Chern–Simons number changes fermion number by two units. The lattice formulation encodes this relation as an exact operator identity. In the simulations, the expectation value of the relevant ’t Hooft operator remains nonzero as the correlation length increases, while a comparison operator charged under an exact symmetry decays with the expected scaling behavior. This supports the interpretation that gauge dynamics violates fermion-number conservation. The authors conclude that finite-dimensional local Hilbert spaces can reproduce the expected infrared physics of this chiral gauge theory and that their tensor-network method is effective at modest bond dimensions. The study is limited to abelian theories in 1+1 dimensions and demonstrates the 34-50 model’s infrared behavior at a single finite truncation, $N=6$. Extending the construction to higher dimensions or nonabelian groups, and developing a practical lattice formulation of the Standard Model, remain open challenges.

Simulating a chiral gauge theory with finite-dimensional quantum systems

Chiral gauge theories describe particles whose left- and right-moving components interact differently with gauge fields. They are central to modern particle physics, but they are notoriously difficult to simulate on a lattice. In “Simulating a chiral gauge theory,” Laurens Lootens and Hersh Singh present a way to tackle this challenge in one space and one time dimension, using finite-dimensional quantum systems and tensor-network methods.

Why chiral gauge theories are hard to simulate

On a lattice, the usual methods for representing fermions run into the Nielsen–Ninomiya obstruction: under standard assumptions, unwanted fermion species appear alongside the ones researchers are trying to study. Alternative approaches can avoid some of these difficulties, but may rely on local systems with infinitely many states or prove challenging for numerical calculations.

The authors take a different route. They first translate the fermionic degrees of freedom into bosonic ones, then represent those bosonic variables using finite-dimensional “clock-and-shift” systems, also known as qudits. This gives each local matter register a manageable dimension of N² while retaining important features of the original theory.

Building a gauge-invariant lattice model

The study focuses on the 34-50 model, an abelian chiral gauge theory with left-moving fermions carrying charges 3 and 4, and right-moving fermions carrying charges 5 and 0. The qudit construction preserves a discrete Zₙ subgroup of the chiral symmetry, along with its anomaly modulo N. Crucially, when the continuum gauge anomaly cancels, the lattice charge operators commute. That makes it possible to impose the gauge symmetry consistently.

The researchers impose and solve the Gauss constraints exactly, rather than treating gauge invariance as an approximate property. For the 34-50 model, a qudit version of T-duality simplifies these constraints. Further transformations yield an unconstrained chain with two Zₙ² registers per site. Removing the constraints in this way makes the model more convenient for numerical work.

The resulting formulation is suited to infinite matrix product state simulations. Tensor-network methods can study the system without the sign problem that obstructs many Monte Carlo approaches, and can often provide useful results even at modest computational resources.

Testing the method

Before studying the chiral model, the authors test their approach on a Zₙ Schwinger model. At gauge coupling e = 1, they calculate its mass gap and compare it with the known continuum value of 0.22508. A fit to results at N values from 4 through 8, including corrections proportional to N⁻⁴, matches the analytic result to within 0.00005.

This agreement is an important check: it suggests that the finite-dimensional qudit system can reproduce the relevant continuum physics as N increases. The team then turns to the 34-50 model, simulating it at N = 6 with couplings Jₚ = 1, Jₓ = 1/4, and Jₑ = 1, and with discrete theta angle zero.

What the infrared theory looks like

To identify the model’s low-energy behavior, the authors use finite-entanglement scaling, a way to infer critical properties from tensor-network data. They obtain a central charge of 1.002(1), consistent with a massless compact boson.

They also measure the scaling dimensions of momentum and winding operators, finding approximately 0.967(7) and 0.259(3). Their product, 0.250(3), agrees with the compact-boson prediction of one quarter. From these results, the authors infer that the boson radius is 1.017(3) times the free-fermion radius.

Together, these measurements point to an infrared theory that behaves like a massless Dirac fermion with a weak Thirring interaction. The interaction matters: the results are close to, but not exactly, those of a free-fermion theory.

Seeing anomalous fermion-number violation

The simulations also probe a defining feature of chiral gauge theories: an anomaly can cause fermion number to change even when classical reasoning would suggest it is conserved. In the 34-50 model, the anomaly coefficient is 2, meaning that a unit change in Chern–Simons number changes fermion number by two units.

The lattice construction captures this relationship as an exact operator identity. In the simulations, the expectation value of the relevant ’t Hooft operator remains nonzero as the correlation length grows. A comparison operator, charged under an exact symmetry, instead decays with the expected scaling behavior. This contrast supports the interpretation that gauge dynamics really does violate fermion-number conservation.

Promise and open questions

The work shows that finite-dimensional local Hilbert spaces can reproduce key features of a chiral gauge theory while preserving exact lattice gauge invariance. It also demonstrates that tensor-network methods can investigate the model at modest bond dimensions. Still, the evidence has limits: the study concerns abelian theories in 1+1 dimensions, and the 34-50 model’s infrared behavior is examined at a single finite truncation, N = 6.

Extending this approach to higher dimensions or nonabelian gauge groups remains a major challenge. So does developing a practical lattice formulation of the Standard Model. For now, “Simulating a chiral gauge theory” offers a promising proof of principle: carefully designed qudit systems can make an otherwise difficult class of quantum field theories more accessible to numerical study.

Created on 09 Nov. 2026

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Simulating a chiral gauge theory

The essentials

Question

Method

Findings

Limitations

Why it matters

Read further

Definitions

Simulating a chiral gauge theory with finite-dimensional quantum systems

Why chiral gauge theories are hard to simulate

Building a gauge-invariant lattice model

Testing the method

What the infrared theory looks like

Seeing anomalous fermion-number violation

Promise and open questions

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