Disorder-induced topological phase transitions on Lieb lattices

Motivated by the very recent experimental realization of electronic Lieb lattices and research interest on topological states of matter, we study the topological phase transitions driven by Anderson-type disorder on spin-orbit coupled Lieb lattices in the presence of spin-independent and -dependent staggered potentials. By combining the recursive Green's-function and self-consistent Born approximation methods, we found that both time-reversal-invariant and time-reversal-symmetry-broken spin-orbit coupled Lieb lattice systems can host the disorder-induced gapful topological phases, including the quantum spin Hall insulator (QSHI) and quantum anomalous Hall insulator (QAHI) phases. For the time-reversal-invariant case, the disorder induces a topological phase transition directly from a normal insulator (NI) to the QSHI, while for the time-reversal-symmetry-broken case, the disorder can induce either a QAHI-QSHI phase transition or a NI-QAHI-QSHI phase transition, depending on the initial state of the system. Remarkably, the time-reversal-symmetry-broken QSHI phase can be induced by Anderson-type disorder on the spin-orbit coupled Lieb lattices without time-reversal symmetry.

Figure 1

(a) Schematic illustration of the Lieb lattice ribbon used in our transport simulations. The unit cell of Lieb lattice consists of three sublattices: $A$ (the square dot), $B$ (the circle dot), and $C$ (the triangle dot). The two metal leads (blue site regions) are modeled by the tight-binding model Hamiltonian with only NN hopping on Lieb lattices. Different colors of sublattice represent distinct staggered potentials. (b)–(e) The bulk spectrum of system for $k_y=\pi$. The parameters are given by (b) $S_{\pm }<2\lambda$, ${\mathrm{\Delta }}_s=0$, (c) $S_{\pm }>2\lambda$, ${\mathrm{\Delta }}_s=0$, (d) $S_{+}>2\lambda$, $S_{-}<2\lambda$, ${\mathrm{\Delta }}_s\ne 0$, and (e) $S_{\pm }>2\lambda$, ${\mathrm{\Delta }}_s\ne 0$. The blue and red lines correspond to the spin-up and -down components of the bulk band, respectively. The energy gaps labeled with red, yellow, and green correspond to the topologically trivial, quantum spin Hall, and quantum anomalous Hall gaps, respectively. The spin Chern number of each band is labeled in the form of $(C_{+}^s,C_{-}^s)$ with $C_{\pm }^s=0,\pm 1$. We plot only the spin-up component of the bulk band in (b) and (c) since the two spin components are degenerate.

Figure 2

(a) Energy spectrum of the clean system with the open boundary condition along the $y$ direction and $S_{\pm }=0.55t$. The blue curves are the bulk subbands and the red lines correspond to the edge states. The up and down arrows denote the spin orientation. (c) The conductance as a function of the strength of disorder $U$ at different chemical potentials $\mu =0.70t$, $0.60t$, and $0.53t$, which correspond to the green, red, and blue dashed lines shown in (a). The error bars show the standard deviation of the conductance for 1000 samples. (e) Phase diagram for the conductance as functions of the disorder strength $U$ and the chemical potential $\mu$. The green solid and black dashed lines obtained from the self-consistent Born approximation denote the phase boundary defined as ${\mathrm{\Delta }}_{\tilde{E}}=0$ and ${\tilde{E}}_2<\tilde{\mu }<{\tilde{E}}_1$. The other panels, (b), (d), and (f), are the same as (a), (c), and (e), except that $S_{\pm }=0.62t$. In our numerical simulations, the width and length of the sample are chosen to be $L_y=200a$ and $L_x=300a$.

Figure 3

(a) Energy spectrum of the clean system with the open boundary condition along the $y$ direction for $S_{+}=0.62t$ and $S_{-}=0.55t$. The blue curves are the bulk subbands and the red lines correspond to the edge states. The up and down arrows denote the spin orientation. (c) The conductance as a function of the strength of disorder $U$ at the chemical potentials $\mu =0.70t$, $0.60t$, and $0.53t$, which are labeled with the green, red, and blue dashed lines shown in (a). The error bars show the standard deviation of the conductance for 1000 samples. (e) Phase diagram for the conductance as functions of the disorder strength $U$ and the chemical potential $\mu$. The green solid and black dashed lines obtained from the self-consistent Born approximation correspond to the phase boundary defined as ${\mathrm{\Delta }}_{\tilde{E}}=0$ and ${\tilde{E}}_2<{\tilde{E}}_f<{\tilde{E}}_1$. The other panels (b), (d), and (f) are the same as (a), (c), and (e), except that $S_{+}=0.62t$ and $S_{-}=0.70t$. In our numerical simulations, the width and length of the strip Lieb model are chosen to be $L_y=200a$ and $L_x=300a$.

Figure 4

The averaged nonequilibrium local current distribution for the disorder-induced topological phases of the Lieb lattice model with $S_{+}=0.62t$, $S_{-}=0.70t$, and $\mu =0.60t$. We take disorder strength $U=1.5t$ and $U=3t$ in (a) and (b) respectively. The red and blue arrows correspond to the spin-up and spin-down components of the local current, respectively. The arrow size means the strength of the local current.