  In this study, the problem of strong Coulomb interactions in topological
superconducting wire is analyzed by means of the
density-matrix-renormalization-group (DMRG) approach. To analyze properties of
edge states in the BDI-class structure a quantity called Majorana polarization
is used. From its dependence on wire length and an entanglement-spectrum
degeneracy topological phase diagrams are obtained. The DMRG calculations for
the Shubin-Vonsovsky-type model of the wire show the transformation of phases
with Majorana single and double modes (MSMs and MDMs, respectively) under the
increase of on- and inter-site correlations. In particular, we demonstrate
different scenarios including the possibilities of both induction and
suppression of the MSMs and MDMs. It is shown that in the strongly correlated
regime the contributions of single-particle excitations to the Majorana-type
states significantly decrease at low magnetic fields. Moreover, the
$t-J^{*}-V$-model is derived allowing to study the effective interactions and
improve the DMRG numerics. It is found out that in the limiting case of the
effective Hamiltonian with infinitely strong on-site repulsion, $t$-model, the
topological phases are destroyed. Finally, the ways to probe the MSMs and MDMs
via the features of caloric functions are discussed.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons;Superconductivity