Projecting out violations or penalizing them

Introduce one definite error process

A gauge-preserving Hamiltonian cannot take a physical initial state outside its Gauss sector. Raising forbidden-state energies therefore changes nothing in that ideal evolution. Testing energy protection requires a term that can actually reach those states.

We add bare matter hopping of strength $\epsilon$ : a particle moves without updating its link flux. One such hop creates opposite residuals on neighboring sites. This defines the error being tested and makes its strength an explicit parameter.

\hat H_{\mathrm{err}}=-\epsilon\sum_n(\hat f_n^\dagger\hat f_{n+1}+\mathrm{h.c.})

This term omits the link raising or lowering operator. Its deliberate violation of Gauss’s law tests the protection mechanism.

Reading: David Tong · Sandbox Physics FR04 finite Hamiltonian · Sandbox Physics FR04 original numerical references and reproduction scripts

Assign an energy to each local violation

Keep the larger product space of half-filled matter and truncated links. Add the error and $\lambda\sum_n\hat G_n^2$ to the gauge-preserving Hamiltonian. A pair of unit residuals contributes two to the sum and therefore costs $2\lambda$ in penalty energy.

A large energy separation usually suppresses error transitions. Their amplitude still depends on the original configuration energies, couplings and observation time. Different contributions can also compensate and approach resonance. Plotting leakage as a function of time and penalty tests the actual finite system.

\hat H_\lambda=\hat H+\hat H_{\mathrm{err}}+\lambda\sum_n\hat G_n^2

The penalty coefficient has energy units. A laboratory implementation would need to map this term to available interactions.

Specify the projected reference Hamiltonian

Let $P$ project onto configurations with every Gauss residual zero. Hard selection evolves with $P(\hat H+\hat H_{\mathrm{err}})P$ . A single bare hop ends outside the physical space, so its first-order matrix element is removed in this example.

With a finite penalty, a state can briefly visit a high-energy violating configuration and return. These virtual processes can alter evolution inside the physical subspace. Small leakage therefore need not make a field curve identical to the hard-projection result.

Use the same initial physical configuration, electric range and couplings in both calculations. Hard projection is the ideal reference; the penalty calculation includes the specified error. Their basis dimensions can differ, but each observable must have the same definition.

Two diagnostics answer different questions

Leakage $1-\langle P\rangle$ is the total probability outside the physical subspace. The mean squared Gauss residual additionally weights each violating configuration by the size of its residuals. They measure related but different properties.

Now compare a physical observable, such as the middle-link field, with its hard-projection reference. Its difference measures the error in the prediction you wanted. Retain norm and total-energy checks for the full Hamiltonian while scanning the penalty, so a numerical problem cannot masquerade as physical leakage.

\mathcal L(t)=1-\langle\psi(t)|P|\psi(t)\rangle,\qquad \delta E_n(t)=\langle E_n\rangle_\lambda-\langle E_n\rangle_{P}

Leakage asks where the state is. The field difference asks how much the chosen prediction changed. Both references use the same electric truncation.

Estimate one isolated leakage channel

Suppose one physical state couples to one violating state with strength $\epsilon$ and total detuning approximately $2\lambda$ . The two-state result gives maximum leakage $\epsilon^2/(\lambda^2+\epsilon^2)$ . For a large penalty, it decreases as the squared ratio of error strength to penalty.

This isolated-channel estimate explains a trend. The finite chain contains many channels and detunings, which the workbench calculates together. Comparing the estimate with those curves tests when the simple picture is useful.

Try the prediction

Expose the same error to different protection strengths

Hold the gauge-breaking error fixed and compare leakage with zero and large penalty.
Open Numerical diagnostics and full record to read the central-link field difference from the physical-basis reference. Vary the penalty and compare field error with leakage at the same time. Do they always decrease together?
Set the gauge-breaking error to zero and test whether changing the penalty still changes the physical trajectory. Explain your result.

Exercises and explanations

A configuration has Gauss residuals of plus one and minus one at neighboring sites and zero elsewhere. What penalty does it pay?

Read the explanation

Each nonzero residual contributes one to its square. The sum is two, so the added energy is $2\lambda$ . Opposite signs do not cancel in the squared sum.

Why compare a field curve even after leakage becomes small?

Read the explanation

Virtual excursions into the violating space can shift effective couplings and phases inside the physical space. A state that remains mostly physical can still accumulate an appreciable error in an observable.

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.

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.