  We study a critical behavior for the eigenvalue statistics in the two-matrix
model in the quartic/quadratic case. For certain parameters, the eigenvalue
distribution for one of the matrices has a limit that vanishes with an exponent
1/2 in the interior of the support. The main result of the paper is a new
kernel that describes the local eigenvalue correlations near that critical
point. The kernel is expressed in terms of a 4 x 4 Riemann-Hilbert problem
related to the Hastings-McLeod solution of the Painlev\'e II equation. We then
compare the new kernel with two other critical phenomena that appeared in the
literature before. First, we show that the critical kernel that appears in case
of quadratic vanishing of the limiting eigenvalue distribution can be retrieved
from the new kernel by means of a double scaling limit. Second, we briefly
discuss the relation with the tacnode singularity in non-colliding Brownian
motions that was recently analyzed. Although the limiting density in that model
also vanishes with an exponent 1/2 at a certain interior point, the process at
the local scale is different from the process that we obtain in the two-matrix
model.
