  We study the symmetry behavior of the Gross-Neveu model in three and two
dimensions with random chemical potential. This is equivalent to a four-fermion
model with charge conjugation symmetry as well as Z_2 chiral symmetry. At high
temperature the Z_2 chiral symmetry is always restored. In three dimensions the
initially broken charge conjugation symmetry is not restored at high
temperature, irrespective of the value of the disorder strength. In two
dimensions and at zero temperature the charge conjugation symmetry undergoes a
quantum phase transition from a symmetric state (for weak disorder) to a broken
state (for strong disorder) as the disorder strength is varied. For any given
value of disorder strength, the high-temperature behavior of the charge
conjugation symmetry is the same as its zero-temperature behavior. Therefore,
in two dimensions and for strong disorder strength the charge conjugation
symmetry is not restored at high temperature.

High Energy Physics - Theory;High Energy Physics - Lattice