  We explore the quantum phase transition between Peierls and
charge-density-wave insulating states in the one-dimensional, half-filled,
extended Hubbard model with explicit bond dimerization. We show that the
critical line of the continuous Ising transition terminates at a tricritical
point, belonging to the universality class of the tricritical Ising model with
central charge $c=7/10$. Above this point, the quantum phase transition becomes
first order. Employing a numerical matrix-product-state based (infinite)
density-matrix renormalization group method we determine the ground-state phase
diagram, the spin and two-particle charge excitations gaps, and the
entanglement properties of the model with high precision. Performing a
bosonization analysis we can derive a field description of the transition
region in terms of a triple sine-Gordon model. This allows us to derive field
theory predictions for the power-law (exponential) decay of the density-density
(spin-spin) and bond-order-wave correlation functions, which are found to be in
excellent agreement with our numerical results.
