Skip to main content
Sandbox Physics
Lattice gauge fields and string breaking

L1 · Chapter 1

Building an electric-flux chain with Gauss’s law

Sandbox Physics ·

Electric flux increases when you cross a positive charge and decreases when you cross a negative charge. A valid chain must satisfy this rule at every site. Checking only its total charge can miss a pair of local mistakes.

Walk along the chain

A chain has no sideways direction for the field to spread into. Choose rightward flux as positive. Start with zero flux at the left boundary. Passing a positive charge adds one unit. Neutral sites leave it unchanged. An equal negative charge at the far end brings the flux back to zero.

Label the sites n=0,1,…,N−1n=0,1,\ldots,N-1. Write EnE_n for the flux on the link from site nn to its right neighbor. At each site, outgoing rightward flux minus incoming rightward flux equals the total charge there.

En−En−1=Qn+qnextE_n-E_{n-1}=Q_n+q_n^{\mathrm{ext}}
QnQ_n is the mobile matter charge and qnextq_n^{\mathrm{ext}} is a fixed external charge. The fixed left and right boundary fluxes enter the first and last site equations.

Fixed sources at the endpoints · zero boundary flux

q0ext=+1q_0^{\mathrm{ext}}=+1q3ext=−1q_3^{\mathrm{ext}}=-1

Intact string

00
11
11
11
00
11
11

One link screened

00
11
00
00
11
11
11
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.

Reading: David Tong · Sandbox Physics FR04 finite Hamiltonian

An occupied site need not carry positive charge

Staggered fermions use alternating reference occupations: empty on even sites, occupied on odd sites. Charge is the actual occupation minus this reference. The alternating product configuration is therefore neutral. It is a convenient counting reference; finding the interacting ground state is a separate calculation.

Each actual occupation fnf_n is zero or one. Relative to the background bnb_n, filling an even site produces positive charge. Emptying an odd site produces negative charge. The missing particle is called a hole. Moving one particle between those sites can create a positive-negative pair without changing total particle number.

bn=1−(−1)n2,Qn=fn−bnb_n=\frac{1-(-1)^n}{2},\qquad Q_n=f_n-b_n
The four-site reference reads empty, occupied, empty, occupied. Charge counts departures from that pattern.
A neutral reference and one neighboring hop, listed from left to right.
ConfigurationOccupationsMatter charges
Neutral reference(0,1,0,1)(0,1,0,1)(0,0,0,0)(0,0,0,0)
Middle particle hops right(0,0,1,1)(0,0,1,1)(0,−1,+1,0)(0,-1,+1,0)

Work through all four sites

Set both boundary fluxes to zero. Put a fixed positive source on site zero and a fixed negative source on site three. Start the matter in its neutral reference configuration. The first source raises the flux to one; the two middle sites leave it unchanged; the final source lowers it to zero. The internal links carry (1,1,1)(1,1,1).

Let the middle particle hop right, giving occupations (0,0,1,1)(0,0,1,1). A negative hole appears on site one and a positive particle on site two. Adding the charges again gives (1,0,1)(1,0,1). The new matter charges remove the field from the middle link. This is local screening.

If you changed the occupations but left all three fluxes at one, total charge would still vanish. Two sites would nevertheless fail Gauss’s law. A global test cannot detect where the error occurred.

Gn=En−En−1−Qn−qnextG_n=E_n-E_{n-1}-Q_n-q_n^{\mathrm{ext}}
The residual GnG_n measures the mismatch at one site. Every residual must vanish in an allowed configuration.

The boundary restricts the allowed configurations

Sum Gauss’s law over the chain. Every internal flux appears once with each sign and cancels. Right-boundary flux minus left-boundary flux equals total charge. With both boundaries fixed, adding one unbalanced charge must fail the last boundary test.

The same observation gives a useful construction. Start at the left boundary and add each site’s matter charge and external source. This determines every internal flux. Check the right boundary at the end, then check whether each flux lies inside the retained electric range. The next chapter uses this construction to build a smaller physical state space.

En=E−1+∑j=0n(Qj+qjext)E_n=E_{-1}+\sum_{j=0}^{n}(Q_j+q_j^{\mathrm{ext}})
On an open one-dimensional chain, accumulated charge fixes the links. A periodic chain also needs a winding-flux choice, so its construction is different.

Try the prediction

Make a local violation and repair it

  1. Choose “An allowed string” and predict each link flux between the two external sources.
  2. Change one occupation or one link and identify which sites acquire a residual.
  3. Rebuild the electric field from left to right, then check whether the fixed right boundary permits your configuration.

Exercises and explanations

  1. Why does increasing one internal link flux by one change the Gauss residual at two sites?

    Read the explanation

    That link is the right-hand flux of one site and the left-hand flux of its neighbor. Its contributions have opposite signs. One residual increases by one and the other decreases by one, so their sum can remain unchanged.

  2. In the four-site example, change the occupations to (1,0,0,1)(1,0,0,1). Which fluxes follow? Can an electric cutoff of one retain them?

    Read the explanation

    The matter charges become (+1,−1,0,0)(+1,-1,0,0). Including the fixed endpoint sources gives internal fluxes (2,1,1)(2,1,1) and returns the right boundary to zero. The configuration satisfies Gauss’s law but exceeds the cutoff on the first link.

Sources and calculation scope

  1. 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.

  2. 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.

FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy