Thirty years ago Sidney Coleman discussed the peculiar case of the half-asymptotic particles in the Schwinger model (quantum electrodynamics in a one-dimensional world). Normally, an electron and a positron in one dimension are joined by a string of constant electric flux,

which means the electric field energy is proportional to their distance apart. Thus the charges can never separate to infinity: they are confined, and a single, isolated charge can never exist.

If you now impose a background electric field of half a unit (e.g. using condenser plates at either end of the universe), the situation changes. Inserting a single charge, thus,

produces a change of sign of the electric flux, but does not change the electric field energy. Therefore at this one special value, the charge is unconfined, and can move freely along the line.

We have been performing numerical calculations on the lattice version of the Schwinger model, using a new technique called the Density Matrix Renormalization Group (DMRG). We have been able to get results 50 times more accurate than any previously obtained. An example is shown below, where we have shown that at a certain critical value of the fermion mass, the half-asymptotic particles form a massless condensate.

Energy gaps for various states in the background field, as functions of the fermion mass m. The squares correspond to the single half-asymptotic particle state.

Tim Byrnes, Pradeep Sriganesh,
Rob Bursill and Chris Hamer