Higher-order topological Anderson insulators

We study disorder effects in a two-dimensional system with chiral symmetry and find that disorder can induce a quadrupole topological insulating phase (a higher-order topological phase with quadrupole moments) from a topologically trivial phase. Their topological properties manifest in a topological invariant defined based on effective boundary Hamiltonians, the quadrupole moment, and zero-energy corner modes. We find gapped and gapless topological phases and a Griffiths regime. In the gapless topological phase, all the states are localized, while in the Griffiths regime, the states at zero energy become multifractal. We further apply the self-consistent Born approximation to show that the induced topological phase arises from disorder renormalized masses. We finally introduce a practical experimental scheme with topoelectrical circuits where the predicted topological phenomena can be observed by impedance measurements. Our work opens the door to studying higher-order topological Anderson insulators and their localization properties.

Figure 1

(a) Schematics of our model (1). (b) The phase diagram with respect to the disorder strength $W$ mapped out based on the topological invariant $P$, the quadrupole moments $q_{xy}$, and the bulk energy gap shown in Fig. 2. We observe distinct phases including gapped and gapless HOTAI, Griffiths phase, and trivial I and II phases, separated by vertical dashed lines. Here $m_x=m_y=1.1$.

Figure 2

(a) The bulk energy gap versus $W$. (b) The quadrupole moment (red circles) versus the system size $L$ when $W=2.6$, which is fitted by a power-law function plotted as the black line. The zero-energy local density of states (LDOS) obtained under open boundary conditions for (c) $W=2.6$ and (d) $W=3.4$. The zoomed-in view of the LDOS around one corner is shown in the insets. Here $m_x=m_y=1.1$.

Figure 3

(a) The normalized localization length ${\mathrm{\Lambda }}_x$ at $E=0$ versus $W$ for several distinct $L_y$. The scaling of ${\mathrm{\Lambda }}_x$ at different energies for (b) $W=3.2$ and (c) $W=4.6$. (d) The level-spacing ratio (LSR) and inverse participation ratio (IPR) versus $W$ for the eigenstates around zero energy in a system with size $L=500$. (e) The fractal dimension $D_2$ with respect to $W$ for the eigenstates around zero energy. The vertical dashed lines separate different phases. (f) The topological invariant $P$ in the $(W,m_0)$ plane with $m_0=m_x=m_y$. The red dotted line indicates the topological phase boundary determined by the SCBA. In [(a)–(e)], $m_x=m_y=1.1$.

Figure 4

The LSR versus energy $E$ for (a) $W=1$, (b) $W=2.8$, (c) $W=3.2$, and (d) $W=4.6$ in the trivial-I, gapped HOTAI, gapless HOTAI, and Griffiths phases, respectively. The blue, red, and yellow lines describe results for the system size $L_x=L_y=L=300$, 400, and 500, respectively. The inset in (a) displays the LSR as a function of $E$ for $W=1.5$. The inset in (b) shows the normalized localization length ${\mathrm{\Lambda }}_x$ with respect to $L_y$ at different energies. Here $m_x=m_y=1.1$.

Figure 5

(a) The DOS at zero energy $\rho \left(0\right)$ versus the disorder strength $W$ calculated by the kernel polynomial method (KPM) for the system size $L=100$ and the expansion order $N_c=2^{13}$. (b) The DOS $\rho \left(E\right)$ versus $E$ for different $W$ with the zoomed-in view around zero energy in the inset. The vertical dashed lines separate different phases. Here $m_x=m_y=1.1$.

Figure 6

(a) The phase diagram with respect to $W$, where $P$ and the quadrupole moments are displayed. Here the gapped and gapless HOTI, the Griffiths phase, and the trivial-II phase are observed. (b) ${\mathrm{\Lambda }}_x$ at $E=0$ versus $W$ for different $L_y$. (c) The LSR and IPR versus $W$ for the eigenstates around zero energy of a system with $L=200$. The vertical dashed lines separate different phases. Here $m_x=m_y=0.5$.

Figure 7

The LSR versus energy $E$ for (a) $W=1$, (b) $W=2$, (c) $W=3.2$, and (d) $W=5$ in gapped HOTI, gapless HOTI, and Griffiths phase, respectively. The top insets in (b), (c), and (d) describe the normalized localization length ${\mathrm{\Lambda }}_x$ with respect to $L_y$ at different energies. The bottom inset in (c) plots the LSR as a function of system size $L$ at $E=-1$. Here $m_x=m_y=0.5$.

Figure 8

(a) Schematics of an electric network for realizing our Hamiltonian (1). Here each node in the circuit represents one lattice site in the Hamiltonian, and four nodes form a unit cell as shown in the blue dashed box. The hopping between two neighboring sites $H_{ab}$ is simulated by the admittance $R_{ab}$ of the electric element connecting them. The different values of $H_{ab}$ correspond to different electric elements, including capacitors, inductors, or negative impedance converters with current inversion (INICs) whose structure is shown in the red box. Each node $a$ should be grounded through an electric element with an appropriate admittance $R_a$. [(b),(c)] The averaged magnitude of the impedance $\left|Z\right(x,y\left)\right|$ of each unit cell under open boundary conditions for $W=2.6$ and $W=3.4$, respectively. The insets show the zoomed-in view of $\left|Z\right(x,y\left)\right|$ around one corner. Here $m_x=m_y=1.1$.

Figure 9

The quadrupole moment $q_{xy}$ for different disorder configurations when $W=2.6$, showing that $q_{xy}=0.5$ for most disorder samples and $q_{xy}=0$ for others. Here $L_x=L_y=80$ and $m_x=m_y=1.1$.

Figure 10

The polarizations [(a),(c)] $p_x$ and [(b),(d)] $p_y$ for the 801st to 1000th iterations corresponding to different disorder configurations. The region between two vertical lines correspond to the Griffiths regime. [(a),(b)] $m_x=m_y=1.1$ and [(c),(d)] $m_x=m_y=0.5$. Here $L_x=L_y=500$.

Figure 11

(a) The LSR with respect to the system size $L$ for $W=1$ and (b) the IPR with respect to the system size $L$ for $W=3.2$ for different energy levels around zero energy. The inset shows the fractal dimension $D_2$ averaged over these energy levels as a function of $L$. Here $m_x=m_y=1.1$.