FR04-B · A string can create its own screen

Electric Strings and Screening

How can a dynamical particle pair weaken an electric string? A particle pair can screen the fixed charges and lower the electric-field energy. Creating the pair costs mass energy, so light matter responds more strongly. On a small isolated chain, screening and electric flux can oscillate and return.

Release an electric string between fixed charges and follow its quantum evolution. Compare light and heavy matter, move the right-hand source, and watch the electric field and particle density together.

Mean string field $0.455$

Bare-pair expectation $0.992$

Total energy $2$

Gauss violation $0$

Release an electric string

Matter mass

m

3

Electric coupling

g^2

4.4

Matter hopping

w

1

Right source site

n_{\rm R}

5

Observation time

t

29.2

Drag the gold right-hand source to change the string length.

Chain size and electric cutoff

Chain sitesElectric cutoffFixed external charges

The no-source reference begins in the same staggered product state. Its pair excitation helps distinguish a general vacuum quench from the response to an external string.

Reading the scene and units

Cyan links have positive electric flux; violet links have negative flux. Gold and coral matter spheres show positive and negative charge; stronger color and glow mean a larger charge expectation. Their sizes mark sites. White arcs show occupation. Double gold rings mark fixed external sources, with signs identifying the charges. Site numbers begin at zero.

$a=\hbar=1$ sets lattice units. Spheres and links show basis values in the builder and quantum expectations in the evolving Labs.

$t=t_{\rm phys}w_{\rm ref}/\hbar,\quad w_{\rm ref}=1$ defines the dimensionless time axis using a fixed reference hopping energy. Moving Matter hopping changes the Hamiltonian, not the clock unit.

Playback traverses the displayed time interval in sixteen wall-clock seconds. The time slider and saved samples use physical lattice time.

Pair number counts occupied even sites relative to the bare staggered product state. It includes excitations that also occur without an external string. The finite chain evolves coherently and may return; increasing chain length or cutoff is a sensitivity check.

Finite-matrix diagonalization uses an open U(1) rotor chain, fixed half filling and a hard integer-flux cutoff. No continuum limit, transverse field or three-dimensional QCD is included.

Model assumptions

A finite, open, one-dimensional U(1) rotor Hamiltonian evolves staggered fermions in an exactly selected Gauss sector. Integer electric flux is truncated; the finite chain can revive. Colored links show electric-field expectations and spheres show matter-charge expectations, not individual particle tracks. The record is a deterministic model calculation, not QCD or experimental data.

Deterministic model calculation

Electric field and matter respond together

Numerical diagnostics and full record

\lVert\psi\rVert^2=1

$C_{1,N-2}=\langle Q_1Q_{N-2}\rangle-\langle Q_1\rangle\langle Q_{N-2}\rangle$ is the connected charge correlation between site one and the second-to-last chain site. It uses the same wavefunction as the scene. The chain length determines these sites; moving an external source leaves them fixed.

The record saves the frozen Hamiltonian parameters, initial state, sampled numerical trajectory and conservation diagnostics. Curves join exact finite-matrix samples; all data are simulated.

The edge probability measures weight on states that touch the electric cutoff. It does not bound the truncation error; compare both cutoffs and the observables directly.

The Four-ion model reference uses the source-free ideal four-site Hamiltonian of Martinez et al. (2016), arXiv:1605.04570v1, Figure 2a. This is a numerical model reference; experimental points and hardware errors are excluded.

Continue the course: Gauss law, strings and gauge protection

Start with the essentials

Focus question: How can a dynamical particle pair weaken an electric string?
One-sentence intuition: Screening couples a change of charge density to a change of electric flux through Gauss law.

Core mathematical model

The finite rotor Hamiltonian

H=-w\sum_n(c_n^\dagger U_nc_{n+1}+\mathrm{h.c.})+m\sum_n(-1)^nN_n+\frac{g^2}{2}\sum_nE_n^2

The three terms move matter with its link, set the staggered mass cost, and store electric energy. The link operator raises an integer flux by one; at the chosen electric cutoff its forbidden action is zero.

Counting bare particle pairs

N_{\rm pair}=\sum_{n\;\mathrm{even}}\langle N_n\rangle

At fixed half filling, each occupied even site has a matching hole on the odd sublattice. This counts excitations relative to the initial product state. A separate no-source product-state quench shows that some pairs also occur without external charges.

Evolution inside the physical sector

|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle,\qquad G_n|\psi(t)\rangle=0

The browser diagonalizes the small Hamiltonian and samples its exact finite-matrix evolution. Units set lattice spacing and the reduced Planck constant to one. A finite chain can return close to its starting state.

Common difficulties

An expectation is not a particle track

Typical misconceptionThe growing sphere shows one particle moving continuously through the chain.

Better mental modelEach sphere shows the expected local matter charge of a superposition. A single measurement has discrete occupations; the plotted expectation averages their probabilities.

Run the experiment

01
Follow field and matter together
Watch Light matter screens, then pause and scrub Observation time through the first oscillation.
What to observe: As matter charges develop near the external sources, central links dim. The field can grow again as the finite system evolves coherently.
02
Increase the mass
Compare Heavy matter resists with the light-matter preset. Keep the electric coupling fixed when making your own comparison.
What to observe: A larger mass increases the cost of particle-pair excitation. The string retains more of its original field for the selected settings.
03
Check the finite calculation
Open Chain size and electric cutoff. Compare both cutoffs and save each trajectory; then try A longer string.
What to observe: Differences measure sensitivity to the finite model. A late return on one short chain is a finite-volume feature; the exported norm, energy and Gauss diagnostics check numerical evolution.

Electric Strings and Screening

Electric field and matter respond together