Matryoshka approach to sine-cosine topological models
R. G. Dias, A. M. Marques
Abstract
We address a particular set of extended Su-Schrieffer-Heeger models with $2n$ sites in the unit cell $[\mathrm{SSH}(2n)]$, that we designate by sine-cosine models $[\mathrm{SC}(n)]$, with hopping terms defined as a sequence of $n$ sine-cosine pairs of the form ${sin({\ensuremath{\theta}}_{j}),cos({\ensuremath{\theta}}_{j})}$, $j=1,\ensuremath{\cdots},n$. These models, when squared, generate a block-diagonal matrix representation with one of the blocks corresponding to a chain with uniform local potentials. We further focus our study on the subset of $\mathrm{SC}({2}^{n\ensuremath{-}1})$ chains that, when squared an arbitrary number of times (up to $n$), always generate a block which is again a sine-cosine model, if an energy shift is applied and if the energy unit is renormalized. We show that these $n$-times squarable models $[\mathrm{SSC}(n)]$ and their band structure are uniquely determined by the sequence of energy unit renormalizations and by the energy shifts associated with each step of the squaring process. Chiral symmetry is present in all sine-cosine chains and edge-state levels at the respective central gaps are protected by it. Zero-energy edge states in a $\mathrm{SSC}(j)$ chain (with $j<n$) of the Matryoshka sequence obtained squaring the $\mathrm{SSC}(n)$ chain with open boundary conditions become finite-energy edge states in noncentral band gaps of the $\mathrm{SSC}(n)$ chain. The extension to higher dimensions is also discussed.