The work is supported by the Foundation for Polish Science through the International Research Agendas program co-financed by the European Union within the Smart Growth Operational Programme. This work was financially supported by the National Science Center in the framework of the "PRELUDIUM" (Decision No.: DEC- 2020/37/N/ST3/02338). G. C. acknowledges financial support from "Fondazione Angelo Della Riccia". A. L. acknowledges support from a Marie Skłodowska-Curie Individual Fellowship under grant MagTopCSL (ID 101029345). W. B. acknowledges the support by Narodowe Centrum Nauk (NCN, National Science Centre, Poland) Project No. 2019/34/E/ST3/00404. The work at Northeastern University was supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0322 and benefited from the computational resources of Northeastern University's Advanced Scientific Computation Center (ASCC) and the Discovery Cluster. The work at TIFR Mumbai is supported by the Department of Atomic Energy of the Government of India under project number 12-R&D-TFR-5.10-0100. We acknowledge the access to the computing facilities of the Interdisciplinary Center of Modeling at the University of Warsaw, Grant No. GB84-1 and No. GB84-7. We acknowledge the CINECA award under the ISCRA initiative IsC81 "DISTANCE" Grant, for the availability of high-performance computing resources and support.

In search of materials with three-dimensional flat band dispersions, using ab-initio computations we investigate how topological phases evolve as a function of hydrostatic pressure and uniaxial strain in two types of superlattices: HgTe/CdTe and HgTe/HgSe. In short-period HgTe/CdTe superlattices, our analysis unveils the presence of isoenergetic nodal lines, which could host strain-induced three-dimensional flat bands at the Fermi level without requiring doping, when fabricated, for instance, as core-shell nanowires. In contrast, HgTe/HgSe short-period superlattices are found to harbor a rich phase diagram with a plethora of topological phases. Notably, the unstrained superlattice realizes an ideal Weyl semimetal with Weyl points situated at the Fermi level. A small-gap topological insulator with multiple band inversions can be obtained by tuning the volume: under compressive uniaxial strain, the material transitions sequentially into a Dirac semimetal to a nodal-line semimetal, and finally into a topological insulator with a single band inversion. The provided repository contains data to reproduce the figures of the corresponding article.