Proper treatment of 1D finite systems: an study on strongly tilted lattices and Majorana-like excitations

In quantum mechanics, the momentum space and its conjugate, the position, are related via Fourier transforms and thus the properties are interwoven with their structure. In particular, for lattice systems possessing an underlying discrete position space the momentum becomes finite. Moreover, if the lattice length is finite, L infinite, the momentum space also becomes both finite and discrete breaking altogether the continuity of the dispersion relation. By imposing the proper hard wall boundary conditions exhibited by finite chains, we are able to reproduce analytically the rich physical phenomenology of tilted one-dimensional lattice systems in a scenario of many interacting quantum particles, the so-called many-body Wannier-Stark system which we can map to a non-interacting diagonal system in momentum representation. The aspect of properly treating boundary conditions is also relevant in novel systems such as topological materials, which is also reviewed for the Kitaev chain.