We solve a random two-matrix model with two real asymmetric matrices whose
primary purpose is to describe certain aspects of quantum chromodynamics with
two colours and dynamical fermions at nonzero quark chemical potential mu. In
this symmetry class the determinant of the Dirac operator is real but not
necessarily positive. Despite this sign problem the unquenched matrix model
remains completely solvable and provides detailed predictions for the Dirac
operator spectrum in two different physical scenarios/limits: (i) the
epsilon-regime of chiral perturbation theory at small mu, where mu^2 multiplied
by the volume remains fixed in the infinite-volume limit and (ii) the
high-density regime where a BCS gap is formed and mu is unscaled. We give
explicit examples for the complex, real, and imaginary eigenvalue densities
including Nf=2 non-degenerate flavours. Whilst the limit of two degenerate
masses has no sign problem and can be tested with standard lattice techniques,
we analyse the severity of the sign problem for non-degenerate masses as a
function of the mass split and of mu.
On the mathematical side our new results include an analytical formula for
the spectral density of real Wishart eigenvalues in the limit (i) of weak
non-Hermiticity, thus completing the previous solution of the corresponding
quenched model of two real asymmetric Wishart matrices.Comment: 45 pages, 31 figures; references added, as published in JHE
