FR04-C · An energy cost can suppress a wrong hop

Protecting the Gauge Subspace

Does a large Gauss penalty keep an imperfect simulator physical? A large energy penalty makes gauge-violating transitions costly and can reduce leakage caused by an imperfect hopping term. Only selecting the physical subspace enforces Gauss law exactly; finite penalties retain small violations and can alter the motion.

Compare exact selection of physical states with an imperfect full-space evolution. Turn up the energy penalty for violating Gauss law and see how much unwanted motion survives.

Physical-basis leakage $0$

Unprotected leakage $0.596$

Penalty leakage $0.00705$

Mean squared Gauss residual $0.00353$

Make the wrong hop costly

Gauss penalty

\lambda

8

Wrong-hop strength

\epsilon

0.65

Observation time

t

1.2

Drag the gold shield handle to change the penalty.

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.

All three branches begin in the same physical string. Only the full-space branches include a matter hop without its link update. The penalty branch adds an energy cost for the sum of local squared Gauss residuals. Leakage is the probability outside the physical sector.

The full basis has four sites at half filling and three electric links with flux values minus one, zero and one. A finite penalty suppresses violations and can also shift physical dynamics.

Model assumptions

The comparison uses a four-site open chain with finite electric cutoff. The physical calculation projects onto the Gauss sector. The full-space calculations add an explicit matter-hop error that leaves its electric link unchanged; the protected case also adds the sum of squared Gauss generators as an energy penalty. A finite penalty suppresses leakage but does not impose an exact constraint. The plotted records are model simulations.

Deterministic model calculation

Probability outside the physical sector

Numerical diagnostics and full record

\lVert\psi\rVert^2=1

$\Delta\langle E_1\rangle=\langle E_1\rangle_\lambda-\langle E_1\rangle_{\rm phys}$ compares the central link in the penalty branch with exact physical-subspace evolution at the same time. A small leakage probability can coexist with a visible field shift.

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.

Continue the course: Gauss law, strings and gauge protection

Start with the essentials

Focus question: Does a large Gauss penalty keep an imperfect simulator physical?
One-sentence intuition: A finite energy penalty can suppress a gauge-breaking error, but exact selection of the physical sector is a different operation.

Core mathematical model

A controlled gauge-breaking error

H_{\rm err}=-\epsilon\sum_n(c_n^\dagger c_{n+1}+\mathrm{h.c.})

This term deliberately omits the link operator. It preserves total fermion number but changes two neighboring Gauss generators.

The energy penalty

H_{\lambda}=H+H_{\rm err}+\lambda\sum_nG_n^2

Every violating configuration gains a positive energy cost. The strength of suppression depends on the error and on the other energy scales; finite penalties need not improve every observable monotonically.

Probability outside the physical sector

\mathcal L(t)=1-\langle P_{\rm phys}\rangle_t

The data plot measures probability in any Gauss-violating basis state. A vanishing mean Gauss generator is insufficient because positive and negative violations can cancel.

Common difficulties

Suppression is not projection

Typical misconceptionA sufficiently large finite penalty makes the computation identical to deleting every forbidden state.

Better mental modelForbidden states remain in the full basis and can contribute through virtual transitions. Compare leakage and physical observables as the penalty grows.

Run the experiment

01
Let the error act
Choose Errors spread and watch the three chains at the same observation time.
What to observe: The physical-basis calculation stays in its selected sector. The other two agree when the penalty is zero, and the wrong-hop term spreads probability into forbidden states.
02
Turn up the penalty
Move Gauss penalty continuously, then compare Weak protection and Strong protection.
What to observe: The gold trace usually drops as the energy separation grows. It stays finite at the shown settings, and its physical observables can differ from the exact physical-basis trace.
03
Remove the cause
Choose No hopping error. Vary the penalty again.
What to observe: Without the explicit gauge-breaking term, all three evolutions preserve the initial Gauss sector. A gauge-invariant Hamiltonian does not need a penalty to enforce its own symmetry.

Protecting the Gauge Subspace

Probability outside the physical sector