Topologically Localized Insulators

We show that fully localized, three-dimensional, time-reversal-symmetry-broken insulators do not belong to a single phase of matter but can realize topologically distinct phases that are labeled by integers. The phase transition occurs only when the system becomes conducting at some filling. We find that these novel topological phases are fundamentally distinct from insulators without disorder: they are guaranteed to host delocalized boundary states giving rise to the quantized boundary Hall conductance, whose value is equal to the bulk topological invariant.

Figure 1

(a) TLI with $N_{\mathrm{TLI}}=1$ for open boundary conditions along $z$ direction only: all the bulk states are localized by disorder, while the two boundary surfaces (orange and blue) host delocalized states and have opposite quantized Hall conductance. (b) The energies $E^{\mathrm{c}}(\pm {\widehat{e}}_z)$ of delocalized states are not constrained by the bulk. (c) Comparison between TLIs and strong topological insulators (TIs). (A TI is called strong if its existence does not rely on translational symmetry.).

Figure 2

(a) TLI construction from stack of two-dimensional layers in $z$ direction. The Hilbert space of each layer is divided into two subspaces spanned by blue and orange orbitals: occupying only blue (orange) orbitals with electrons results in quantized Hall conductance of the layer ${\sigma }_{12}=e^2/h$ (${\sigma }_{12}=-e^2/h$). The blue (orange) orbitals cannot all have exponential localization; exponential localization is obtained by hybridizing differently colored orbitals from neighboring layers. (b) Distance dependence of the hopping elements $\ln |t_{\overrightarrow{0}1}^{\overrightarrow{R}1}|$ of the TLI with $\overrightarrow{R}=(x,y,0)$ for the system size $N_x=N_y=31$, where $t_{\overrightarrow{0}1}^{\overrightarrow{R}1}$ is defined below Eq. (1). We observe an exponential decay of the hopping elements $t_{\overrightarrow{0}1}^{\overrightarrow{R}1}$ in all directions.

Figure 3

(a) Estimate of ${\lambda }_c$ from mean level spacing ratio $r$ as function of $\lambda .r$ is computed over 500 eigenstates in the middle of the spectrum, considering ${10}^3$ disorder realizations. (b) The winding number $N_{\mathrm{TLI}}$ is computed for different system sizes with increasing hopping strength $\lambda$, where the quantized plateau increases with system size for $\lambda <{\lambda }_c$. (c) Surface Chern number for the hard-wall open boundary conditions (OBC) in $z$ direction as function of $\lambda$. (d) Same as (c), for OBC in $y$ direction. No disorder averaging was performed for the winding and Chern numbers, apart from the blue curves where average was performed over five disorder realizations, as well as the red curves in (b) and (c) for $\lambda <{\lambda }_c$. The dashed curves in (c) and (d) are for the finite-range hopping version of the model (1) with $N_t=4$, see the main text.

Figure 4

Sketch of the layer construction for the three-dimensional TLI. A stack along $z$ direction of two-band disordered Chern insulators, with the bandwidth ${\mathrm{\Delta }}_B$ and the gap $\mathrm{\Delta }$ or $2\mathrm{\Delta }$, becomes fully localized due to interlayer coupling $t$ such that ${\mathrm{\Delta }}_B\ll t\ll \mathrm{\Delta }$. For each layer, density of states (DOS) is shown, and the delocalized states depicted in orange (blue) carry a positive (negative) quantized Hall conductance.