Thirty years
ago Sidney Coleman discussed the peculiar case of the halfasymptotic
particles in the Schwinger model (quantum electrodynamics in a onedimensional
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 halfasymptotic 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
halfasymptotic particle state. 
Tim
Byrnes, Pradeep Sriganesh,
Rob Bursill and Chris
Hamer
