One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges

Self-consistent theory of mobility edges in quasiperiodic chains

“…which recovers precisely the exact result [11]. Note that the MEs never exit the spectrum, some states always being extended (such that V + → ∞), while the intersection of ω ME,± and ω (l) ± gives V − = √ 2.…”

“…The simplest and arguably most famous member of the family, the Aubry-André-Harper (AAH) model [2,3], hosts a localization transition [2] already in one dimension. Quasiperiodic chains also commonly show other interesting phenomena such as mobility edges, multifractal eigenstates both at and away from criticality, and "mixed phases" with both extended and localized eigenstates [4][5][6][7][8][9][10][11][12][13][14][15][16], and are readily implemented in experimental quantum emulators with ultracold atoms [17,18].…”

“…The second family is the so-called mosaic models [11] parametrized by an integer l. These non-self-dual models have an AAH potential on every lth site, while all remaining sites have l = 0. Formally,…”

“…where k ∈ Z. While MEs are known analytically for arbitrary l, for brevity we here consider explicitly the l = 2 model, where the spectrum hosts two MEs given by ω ME = ±2/V [11].…”

“…The local propagator of the end site can be computed as G 0 (ω) = . Black lines denote the known mobility edges [10,11]. In panels (a), (b) a finite typ indicates the presence of extended states and a vanishingly small value indicates localised states.…”