Search for localized Wannier functions of topological band structures via compressed sensing

We investigate the interplay of band structure topology and localization properties of Wannier functions. To this end, we extend a recently proposed compressed sensing based paradigm for the search for maximally localized Wannier functions [Ozolins et al., Proc. Natl. Acad. Sci. USA 110, 18368 (2013)]. We develop a practical toolbox that enables the search for maximally localized Wannier functions which exactly obey the underlying physical symmetries of a translationally invariant quantum lattice system under investigation. Most saliently, this allows us to systematically identify the most localized representative of a topological equivalence class of band structures, i.e., the most localized set of Wannier functions that is adiabatically connected to a generic initial representative. We also elaborate on the compressed sensing scheme and find a particularly simple and efficient implementation in which each step of the iteration is an $O(NlogN)$ algorithm in the number of lattice sites $N$. We present benchmark results on one-dimensional topological superconductors demonstrating the power of these tools. Furthermore, we employ our method to address the open question of whether compact Wannier functions can exist for symmetry-protected topological states such as topological insulators in two dimensions. The existence of such functions would imply exact flat-band models with finite range hopping. Here, we find numerical evidence for the absence of such functions. We briefly discuss applications in dissipative-state preparation and in devising variational sets of states for tensor network methods.

Figure 1

Evolution of the extent of Wannier functions under the adiabatic continuity algorithm for a trivial 1D superconductor (upper panel) and a nontrivial 1D topological superconductor (lower panel). In both cases, the most localized, compactly supported representatives of the respective phases are found, i.e., a WF localized on a single site (upper panel) and on two sites (lower panel), respectively. Plotted is the real-space probability density ${\rho }_x$ [cf. Eq. (5)] on the horizontal $x$ axis with logarithmic color code from ${10}^0$ (yellow) to ${10}^{-30}$ (blue), and run time increases on the vertical $t$ axis in units of ten iterative steps. Initial Wannier functions obtained from the gauge constructed in Eq. (20) in Sec. 3b. Parameters are $\mu =1.5,2t=2\Delta =1$ and $\mu =0.3,2t=2\Delta =1$ for upper and lower panel, respectively. The home cell of both WFs is $x=101$, with total length $L=200$ for both plots.

Figure 2

Logarithmic plots of the probability density $\rho$ of WFs with home cell $x=101$ for a nontrivial 1D TSC with $\mu =0.3,2t=2\Delta =1$ (see Sec. 2c for definitions). Upper panel: Result of algorithm without additional symmetries and coupling constants $\xi =10,r=50,\lambda =20$. Central panel: Result of algorithm with $\mathcal{S}=\left\{\text{PHS}\right\}$ and coupling constants $\xi =10,r=50,\lambda =20$. Lower panel: WF from the gauge constructed in Eq. (20). $L=200$ has been chosen for all plots.

Figure 3

Logarithmic plot of the probability density $\rho$ of sets of Wannier functions from the gauge constructed in Eq. (20) for a trivial 1D superconductor with $\mu =1.5,2t=2\Delta =1$ (lower panel) and a nontrivial 1D TSC with $\mu =0.3,2t=2\Delta =1$ (upper panel). The home cell of both WFs is $x=101$. The linear tails demonstrate the asymptotic exponential decay. $L=200$ is chosen for both plots.

Figure 4

Logarithmic plot of the probability density $\rho$ of an initial Wannier function for the model Hamiltonian (21) for the topologically trivial mass parameter $M=2.5$ (leftmost panel). The home cell of the WFs is $(x,y)=(51,51)$; the size of the lattice is $101\times 101$. Adiabatically deformed WF after 100, 500, and 727 (rightmost panel) iterations iterations respectively with $\xi =\kappa =\lambda =50$.