One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges

The mobility edges (MEs) in energy that separate extended and localized states are a central concept in understanding the localization physics. In one-dimensional (1D) quasiperiodic systems, while MEs may exist for certain cases, the analytic results that allow for an exact understanding are rare. Here we uncover a class of exactly solvable 1D models with MEs in the spectra, where quasiperiodic on-site potentials are inlaid in the lattice with equally spaced sites. The analytical solutions provide the exact results not only for the MEs, but also for the localization and extended features of all states in the spectra, as derived through computing the Lyapunov exponents from Avila’s global theory and also numerically verified by calculating the fractal dimension. We further propose a novel scheme with experimental feasibility to realize our model based on an optical Raman lattice, which paves the way for experimental exploration of the predicted exact ME physics.

Figure 1

The 1D quasiperiodic mosaic model with $\kappa =2$ and $\kappa =3$. The red and black spheres denote the lattice sites whose potentials are quasiperiodic and zero, respectively, as shown by the corresponding red and black dashed lines. The blue sphere denotes a particle, and the nearest-neighbor hopping strength is $t$.

Figure 2

Fractal dimension $\mathrm{\Gamma }$ of different eigenstates as a function of the corresponding eigenvalues and quasiperiodic potential strength $\lambda$ for (a) $\kappa =2$ with size $L=F_{14}=610$ and (b) $\kappa =3$ with size $L=F_{15}=987$. The red dashed lines represent the MEs given in Eq. (3).

Figure 3

Spatial distributions of two eigenstates correspond to (a) $E=-2.205(9)$ and (b) $E=-1.741(7)$, which, respectively, correspond to the nearest-neighbor eigenvalue below and above the ME of the system. Eigenenergies versus the corresponding index (c) from 67 to 72 and (d) from 73 to 79, which are, respectively, below and above the ME ($E_c=-2$), here the eigenenergies in ascending order. Here we fix $\kappa =2$, $\lambda =0.5$, and $L=610$.

Figure 4

Realization of the quasiperiodic model with $\kappa =2$ in cold atoms. (a) Realization scheme. The spin-dependent primary lattice ${\mathcal{V}}_p(x)$ [$-{\mathcal{V}}_p(x)$] locates spin-up (-down) atoms at odd (even) sites, with an incommensurate potential ${\mathcal{V}}_s(x)$ being applied only to the spin-down atoms. The primary lattice is deep enough such that the spin-conserved hopping can be ignored. A Raman coupling $M_0$ is then used to induce the spin-flipped hopping, which plays the role of nearest-neighbor tunneling. (b) Fractal dimension $\mathrm{\Gamma }$ of the lowest-band eigenstates of the lattice model as a function of the lattice depth $V_s$. The eigenvalues $E$ have been shifted a constant value such that the center of the band is zero for $V_s=0$. Here we set $V_p=10E_r$, $M_0=1.5E_r$, and $k_s/k_p=[(\sqrt{5}-1)/2]$, with $E_r\equiv k_p^2/(2m)$. The red dashed curves represent the analytical MEs $E_c=\pm t^2/\lambda$, with $t\simeq 0.353E_r$ and $\lambda \simeq 0.215V_s$.