  We study Majorana chain with the shortest possible interaction term and in
the presence of hoping alternation. When formulated in terms of spins the model
corresponds to the transverse field Ising model with nearest-neighbor
transverse and next-nearest-neighbor longitudinal repulsion. The phase diagram
obtained with extensive DMRG simulations is very rich and contains six phases.
Four gapped phases include paramagnetic, period-2 with broken translation
symmetry, $\mathbb{Z}_2$ with broken parity symmetry and the $\mathbb{Z}_2$
period-2 phase with both symmetries broken. In addition there are two floating
phases: gapless and critical Luttinger liquid with incommensurate correlations,
and with an additional spontaneously broken $\mathbb{Z}_2$ symmetry in one of
them. By analyzing an extended phase diagram we demonstrate that, in contrast
with a common belief, the Luttinger liquid phase along the self-dual critical
line terminates at a weaker interaction strength than the end point of the
Ising critical line that we find to be in the tri-critical Ising universality
class. We also show that none of these two points is a Lifshitz point
terminating the incommensurability. In addition, we analyzed topological
properties through Majorana zero modes emergent in the two topological phases,
with and without incommensurability. In the weak interaction regime, a
self-consistent mean-field treatment provides a remarkable accuracy for the
description of the spectral pairing and the parity switches induced by the
interaction.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics