Topological gaps in quasiperiodic spin chains: A numerical and K-theoretic analysis

Abstract

Topological phases supported by quasi-periodic spin-chain models and their bulk-boundary principles are investigated by numerical and K-theoretic methods. We show that, for both the un-correlated and correlated phases, the operator algebras that generate the Hamiltonians are non-commutative tori, hence the quasi-periodic chains display physics akin to the quantum Hall effect in two and higher dimensions. The robust topological edge modes are found to be strongly shaped by the interaction and, generically, they have hybrid edge-localized and chain-delocalized structures. Our findings lay the foundations for topological spin pumping using the phason of a quasi-periodic pattern as an adiabatic parameter, where selectively chosen quantized bits of magnetization can be transferred from one edge of the chain to the other.

A Groupoid Approach to Interacting Fermions

“…Thus, in these special cases, the stably-homotopic classification of the self-binding states of N fermions is no more complicated than that of single fermion states. We point out that this has been already observed in numerical experiments (see [33]).…”

“…For example, for the many-body quantum system simulated in Fig. 11 of [33], the energy spectrum separates into several spectral islands and [33] found that the states associated with top spectral island display all the trades of selfbinding dynamics. On the other hand, the states associated with the other spectral islands display all the trades of a scattered dynamics (see Fig.…”

“…On the other hand, the states associated with the other spectral islands display all the trades of a scattered dynamics (see Fig. 23 in [33] for direct evidence of the self-binding/scattered characters). Furthermore, the spectral projection onto the top spectral island belongs to the groupoid C * -algebra, while the spectral projections onto the remaining spectral islands belong to the corona algebra.…”

“…The finite N -sectors are equally interesting, from both a theoretical and a practical point of view. For example, [33] supplied numerical evidence that the K -theories of the finite N -sectors can become extremely complex even for N = 3. As explained in [33], this can be exploited to generate new topological dynamics in meta-materials and new manifestations of the bulk-boundary principle.…”

Mechanical metamaterials

“…Further work can be found in [298,398,399,409,[414][415][416][417][418][419][420][421][422][423]. It should be also noted that noncommutative differential geometry developed in the late 70s and 80s by Connes [424] (building upon the earlier work of Grothendieck, Atiyah and Hirzebruch on topological Ktheory [425]), has been applied to the quantum Hall effect [426] and it offers an interesting avenue to study topological insulators in disordered and quasiperiodic systems [427][428][429]. From an engineering standpoint, group theoretic and topological concepts have been employed to geometrically design Reproduced from [401].…”