---
title: "Lattice gauge fields and string breaking"
description: "Learn local Gauss constraints, electric strings and quantum screening through eight chapters, worked examples and three interactive experiments."
canonical_url: "https://sandboxphysics.com/stories/lattice-gauge/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/lattice-gauge/"
---

FR04

# Lattice gauge fields and string breaking

Sandbox Physics · 2026-10-27

A local charge determines how electric flux changes from one link to the next. Follow that rule from four sites to a quantum chain: build the allowed states, release a string, and compare the result with a published model calculation.

## Start with four sites

A particle moves one step and the middle electric link disappears. Following that simple change leads to Gauss’s law, the allowed quantum states and a calculation of string release. Each chapter adds one tool when the next question needs it.

Fixed sources at the endpoints · zero boundary flux

$q_0^{\mathrm{ext}}=+1$ $q_3^{\mathrm{ext}}=-1$

Intact string

$0$

$1$

$1$

$1$

$0$

$1$

$1$

One link screened

$0$

$1$

$0$

$0$

$1$

$1$

$1$

Original configuration diagram. Circles show matter occupation; links show rightward electric flux. Removing the particle from odd site one and placing it on even site two changes the middle flux from one to zero.

Chapters 1–3 build the local rule and energy picture. Chapters 4–6 solve and test the finite quantum model. Chapters 7–8 map a published quantum-simulation measurement and reproduce a fixed ideal model curve. Start at the chapter whose prerequisites you meet.

[Shared textbook foundations](https://github.com/yangzichao/sandboxphysics/issues/27) · [Local phase and gauge links](https://sandboxphysics.com/stories/gauge-effective-theory/local-phase-links/) · [Research route and acceptance](https://github.com/yangzichao/sandboxphysics/issues/34)

1. L1 · 1

   ## [Building an electric-flux chain with Gauss’s law](https://sandboxphysics.com/stories/lattice-gauge/local-gauss/)

   Calculate electric flux site by site, distinguish staggered charge from occupation, and test why total charge alone misses local violations.

2. L2 · 2

   ## [Turning allowed configurations into a quantum model](https://sandboxphysics.com/stories/lattice-gauge/hamiltonian-basis/)

   Build the finite Hamiltonian, derive why matter hopping must change link flux, and enumerate the physical basis before evolving a state.

3. L2 · 3

   ## [The energy cost of stretching an electric-flux string](https://sandboxphysics.com/stories/lattice-gauge/flux-energy/)

   Derive the static string energy, calculate the field energy a matter pair can remove, and separate an energetic crossing from a breaking time.

4. L3 · 4

   ## [Following a released electric-flux string in time](https://sandboxphysics.com/stories/lattice-gauge/string-dynamics/)

   Solve finite-chain evolution in the energy basis, derive a two-state resonance, and read electric screening together with matter and conservation tests.

5. L3 · 5

   ## [What electric cutoffs and short chains change](https://sandboxphysics.com/stories/lattice-gauge/cutoff-finite-size/)

   Separate numerical precision, electric truncation and finite size using edge weights, matched local observables and boundary-return comparisons.

6. L3 · 6

   ## [Projecting out violations or penalizing them](https://sandboxphysics.com/stories/lattice-gauge/physical-subspace/)

   Compare an exact physical basis with Gauss-energy protection under a defined gauge-breaking error, using leakage and physical-observable differences.

7. L4 · 7

   ## [Reading an electric string from atomic occupations](https://sandboxphysics.com/stories/lattice-gauge/optical-lattice-mapping/)

   Follow an optical-lattice paper from atomic occupations to gauge observables, then calculate the strong-mass configuration-energy crossing.

8. L4 · 8

   ## [Independently recalculating a four-site pair-production curve](https://sandboxphysics.com/stories/lattice-gauge/reproducible-comparison/)

   Fix the ideal four-site model from Martinez et al. (2016), rebuild its spin Hamiltonian, and compare every sampled observable with the browser.

## Three experiments along the way

- [Build a chain and locate each Gauss violation](https://sandboxphysics.com/labs/gauss-law-builder/)
- [Release a string and follow its electric field](https://sandboxphysics.com/labs/string-breaking/)
- [Compare physical projection with energy protection](https://sandboxphysics.com/labs/gauge-subspace/)

The browser solves short one-dimensional Abelian chains. The course explains their boundary conditions, electric cutoff and observable definitions before comparing results. All plotted trajectories are deterministic model calculations.

## Sources and calculation scope

1. [Sandbox Physics FR04 finite Hamiltonian, conventions and calculation scope](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr04/model-specification.md)

   Defines the course’s zero-based staggered charge, open boundaries, integer-flux hard cutoff, source geometry and deliberate gauge-breaking error. The accompanying paper-reproduction and numerical-validation notes specify which finite-model calculations were checked independently. This is our model definition and calculation record.

2. [David Tong, Gauge Theory (2018 lecture notes)](https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf)

   Sections 2.5.1 and 2.7 discuss static probes and dynamical matter; sections 4.2.1 and 4.3.1 introduce lattice links and two-dimensional staggered fermions; section 7.1 gives the electrostatic energy in one spatial dimension. Our finite Hamiltonian, indexing and hard cutoff are defined explicitly in this course.

3. [Ying Liu et al., String breaking mechanism in a lattice Schwinger model simulator, arXiv:2411.15443v1 (23 November 2024)](https://arxiv.org/html/2411.15443v1)

   Equation (1), Figures 1–3 and 5, and Appendix A fix the spin-one-half quantum-link Hamiltonian, optical-lattice mapping, static sources and measured observables. The reading guide recalculates the zero-hopping configuration-energy relation behind Figure 5 for atomic chain lengths 6, 8 and 10; it does not reproduce the full quantum-link Hamiltonian. Figure 2 supplies the experimental protocol context. The arXiv page displays a perpetual non-exclusive license. No measured points, source figures, fitted slopes or experimental ramp dynamics are reproduced.

4. [E. A. Martinez et al., Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature 534, 516–519 (2016); arXiv:1605.04570v1 (15 May 2016)](https://arxiv.org/html/1605.04570v1)

   Methods equation (3), Figure 2a and the main-text density definition fix the four-site ideal continuous-Hamiltonian calculation: electric energy equal to hopping, mass equal to half the hopping, zero boundary field and the bare spin vacuum. The associated vacuum-persistence observable is defined in the main text and used in Figure 3; no Figure 3 scan is reproduced. DOI: 10.1038/nature18318. The arXiv page displays a perpetual non-exclusive license; no source figures, author code or measured datasets are redistributed.

5. [Sandbox Physics FR04 original numerical references and reproduction scripts](https://github.com/yangzichao/sandboxphysics/tree/main/docs/frontier/fr04)

   benchmarks-browser.json and benchmarks-independent.json retain the separately calculated raw observables; benchmarks-summary.json records their differences, numerical environment, cutoff and finite-size checks. scripts/frontier/fr04/reproduce-benchmarks.mjs and independent-reference.py regenerate the original deterministic records. No experimental data are included.

6. [ITensorMPS.jl, DMRG and MPS Time Evolution official tutorials](https://docs.itensor.org/ITensorMPS/dev/tutorials/DMRG.html)

   The DMRG tutorial defines ground-state optimization and its bond-dimension and discarded-weight controls. The linked MPS Time Evolution tutorial treats a different task. They provide the next numerical method after small-chain exact diagonalization; no large-system tensor-network result is claimed for this course.