Chiral symmetry breaking and the Banks--Casher relation in lattice QCD with Wilson quarks

Understand

The Banks--Casher relation links the spontaneous breaking of chiral symmetry in QCD to the presence of a non-zero density of quark modes at the low end of the spectrum of the Dirac operator.

• Spectral observables like the number of modes in a given energy interval are renormalizable and can therefore be computed using the Wilson formulation of lattice QCD even though the latter violates chiral symmetry at energies on the order of the inverse lattice spacing.
• Using numerical simulations, we find (in two-flavour QCD) that the low quark modes do condense in the expected way.
• In particular, the chiral condensate can be accurately calculated simply by counting the low modes on large lattices.