  The spin-1/2 transverse field two-leg Ising ladder with nearest-neighbor
exchange and plaquette four-spin interaction $J_{4}$ is studied analytically
and numerically with the density matrix renormalization group approach. The
quantum phase diagram in the transverse field $B$ versus $J_{4}$ plane has been
obtained. There are three different phases: a paramagnetic (PM) phase for high
values of the transverse field and, for low values of $B$, a ferromagnetic (FM)
ordered phase for small $J_{4}$ and a dimerized-rung (DR) phase for large
negative values of $J_{4}$. All phases are separated by quantum phase
transition lines meeting at a multicritical point. The critical lines have been
obtained by exploring the entanglement entropy. The results show that along the
critical lines the central charge is $c=1/2$, while at the multicritical point
one has $c=1$. The scaling dimension of the energy operator is
$X_{\epsilon}=1$, in agreement with the universality class of the critical
behavior of the quantum Ising chain. An effective field theory for the
multicritical point is also discussed. The FM and the DR order parameters have
also been computed and we found a region where the FM and the DR phases
coexist.
