FR04
Lattice gauge fields and string breaking
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
Intact string
One link screened
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 · Local phase and gauge links · Research route and acceptance
L1 · 1
Building an electric-flux chain with Gauss’s law
Calculate electric flux site by site, distinguish staggered charge from occupation, and test why total charge alone misses local violations.
L2 · 2
Turning allowed configurations into a quantum model
Build the finite Hamiltonian, derive why matter hopping must change link flux, and enumerate the physical basis before evolving a state.
L2 · 3
The energy cost of stretching an electric-flux string
Derive the static string energy, calculate the field energy a matter pair can remove, and separate an energetic crossing from a breaking time.
L3 · 4
Following a released electric-flux string in time
Solve finite-chain evolution in the energy basis, derive a two-state resonance, and read electric screening together with matter and conservation tests.
L3 · 5
What electric cutoffs and short chains change
Separate numerical precision, electric truncation and finite size using edge weights, matched local observables and boundary-return comparisons.
L3 · 6
Projecting out violations or penalizing them
Compare an exact physical basis with Gauss-energy protection under a defined gauge-breaking error, using leakage and physical-observable differences.
L4 · 7
Reading an electric string from atomic occupations
Follow an optical-lattice paper from atomic occupations to gauge observables, then calculate the strong-mass configuration-energy crossing.
L4 · 8
Independently recalculating a four-site pair-production curve
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
- Release a string and follow its electric field
- Compare physical projection with energy protection
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
- Sandbox Physics FR04 finite Hamiltonian, conventions and calculation scope
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.
- David Tong, Gauge Theory (2018 lecture notes)
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.
- Ying Liu et al., String breaking mechanism in a lattice Schwinger model simulator, arXiv:2411.15443v1 (23 November 2024)
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.
- 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)
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.
- Sandbox Physics FR04 original numerical references and reproduction scripts
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.
- ITensorMPS.jl, DMRG and MPS Time Evolution official tutorials
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.