**6. Conclusions**

An aperiodic solid could be thermodynamically stable by the growth of entropy, the appearance of electronic energy gap around of the Fermi energy as occurred in the Peierls instability [218], or mechanisms described by the Hume–Rothery rules [219]. Such structural asymmetry represents a singular opportunity to achieve many unique physical properties. For example, the union of positively and negatively doped semiconductors constitutes the base of current microelectronics and modern illumination. Nevertheless, the presence of these structural interfaces requires new approaches for its study and design.

From the theoretical point of view, the tight-binding or Hubbard Hamiltonian based on the Wannier functions provides an atomic scale modelling of measurable physical quantities, where the huge degrees of freedom should be e fficient and accurately addressed by taking the advantage of all visible and hidden symmetries. For instance, the exciton di ffusion in organic solar cells has been recently analyzed by means of an attractive Hubbard Hamiltonian and the real-space renormalization method (RSRM) [220]. For aperiodic lattices with hierarchically structured inhomogeneities, the RSRM seems to be an ideal candidate because the structural scaling rule of these aperiodic lattices can be used as the starting point of RSRM. However, this procedure is truly useful only in 1D systems because they have a constant number of boundary atoms, in contrast to multidimensional systems whose boundary-atom number grows with the system size. These boundary atoms are extremely important for the Green's function determination, that is, a precise counting of all possible paths between two arbitrary atoms. For separable Hamiltonians, such as nearest-neighbor tight-binding Hamiltonian in cubically structured aperiodic lattices, a combination of the convolution theorem and RSRM has demonstrated its e ffectiveness [17]. Beyond cubically structured systems, the labyrinth lattice has been the first non-hypercubic aperiodic network recently addressed by the renormalization plus convolution scheme, where a new convolution theorem for a product of Hamiltonians instead of summation in the traditional convolution theorem was developed [184]. This fact opens a new horizon for the applicability of RSRM to more complex multidimensional aperiodic structures. On the other hand, the design of electronic and optical devices based on quantum mechanical calculations has been one of the biggest dreams of physicists and engineers, and the recent advances in RSRM bring it closer because these electronic and optical devices usually contain multiple aperiodic located structural interfaces. For example, first-principle calculations have been used in the multiscale design of omnidirectional dielectric reflectors [221] and Fabry–Perot resonators [222], whose results were experimentally confirmed.

Finally, despite the proven e fficiency of RSRM in the study of systems with huge degrees of freedom, there are still many challenges in the development and application of new RSRM and they might be summarized as follows: (1) extend the applicability of RSRM to multidimensional lattices with complex structural symmetry; (2) combine the RSRM with the density functional theory to address multielectron systems; and (3) apply the RSRM to strong correlated phenomena, such as the superconductivity.

**Author Contributions:** V.S. and C.W. both participated in the conceptualization, reference analysis, figure preparation, manuscript writing, and final revision. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by the Consejo Nacional de Ciencia y Tecnología of Mexico through gran<sup>t</sup> 252943 and by the National Autonomous University of Mexico (UNAM) through projects PAPIIT-IN115519 and PAPIIT-IN110020. Computations were performed at Miztli of DGTIC, UNAM.

**Acknowledgments:** We would like to thank Antonio Galeote for his participation at early stages of this article and Fernando Sánchez for stimulating conversations and technical assistance.

**Conflicts of Interest:** The authors declare no conflict of interest.
