Topological Anderson insulators induced by random binary disorders

In the past decades, topological states of matter have been widely investigated in condensed-matter physics Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016); Armitage et al. (2018); Chiu et al. (2016); Gao et al. (2018); Zhang and Cao (2016); Qin and Guo (2016) and artificial systems such as ultracold atoms Zhang et al. (2018); Cooper et al. (2019); Goldman et al. (2016), superconducting circuits Schroer et al. (2014); Roushan et al. (2014); Tan et al. (2018, 2019a, 2019b), and photonic lattices Lu et al. (2014); Ozawa et al. (2019). Under strong disorders, the topological states usually become trivial Anderson insulators due to the fact that the eigenstates become localized and the band gap closes Anderson (1958); Klitzing et al. (1980); Evers and Mirlin (2008). Surprisingly, it was found that topological insulators can be induced by random disorders from trivial phases Li et al. (2009), leading to the novel interplay effect of topology and disorder. The disorder-induced topological phases are dubbed topological Anderson insulators (TAIs) and have been widely investigated Li et al. (2009); Groth et al. (2009); Jiang et al. (2009); Sanchez-Palencia and Lewenstein (2010); Meier et al. (2018); Li et al. (2020); Mondragon-Shem et al. (2014); Hsu and Chen (2020); Jiang et al. (2019); Zhang et al. (2020); Tang et al. (2020); Titum et al. (2015); Yang et al. (2021); Zhang et al. (2021a); Stützer et al. (2018); Peng et al. (2021); Zhang and Shen (2013); Shi et al. (2021); Liu et al. (2020a); Hsu and Chen (2020); Liu et al. (2020b, 2021); Zhang et al. (2012); Song et al. (2012); Xu et al. (2012); Zhang et al. (2019); Yamakage et al. (2013); Orth et al. (2016); Xing et al. (2011); Shen (2017); Prodan (2011); Wu et al. (2019); Altland et al. (2014); Longhi (2020). Experimental observations of TAIs were reported in several artificial systems, such as 1D cold atomic wires Meier et al. (2018), 2D photonic waveguide arrays Stützer et al. (2018); Liu et al. (2020a), electric circuits Zhang et al. (2021a, 2019), coupled waveguides Stützer et al. (2015), and photonic quantum walks Lin et al. (2021). Notably, TAIs are induced by the uncorrelated disorders in these studies. For instance, both uncorrelated random Meier et al. (2018) and quasiperiodic disorders Zhang et al. (2021b); Tang et al. (2022); Lu et al. (2022) can give rise to the TAI phase in 1D Su-Schrieffer-Heeger (SSH) chains. It was shown that the spatial correlations in the disorder can entirely suppress the emergence of the TAI phase on two-dimensional quantum spin Hall systems with Gaussian correlated on-site potentials Girschik et al. (2013). However, the effects of correlated disorders on other topological phases are largely unexplored. In particular, it remains unclear whether the TAI phase exhibits in 1D SSH chiral chains with correlated disorders.

As a typical kind of spatially short-range correlated disorder, the random binary disorder has been studied in localization phenomena. It was found that the random binary disorder can induce unconventional localization-delocalization transition Flores (1989); Hilke (1997); Dunlap et al. (1990); Sedrakyan (2004); Schaff et al. (2010), and the phase transitions between Mott insulators and Bose glasses  Krutitsky et al. (2008) or metal phases Semmler et al. (2010). Moreover, the random site energies with a discrete binary probability distribution can significantly influence the electronic structure and transport properties Evangelou and Wang (1993); Alvermann and Fehske (2005). The random binary disorder in random dimer models can be used to explain the insulator-metal transition with new metallic states in conducting polymers Phillips and Wu (1991) and a new transport mechanism Wu and Phillips (1991). Realizations of the random-dimer model have been achieved in GaAs-AlGaAs superlattices Bellani et al. (1999), 1D waveguide array fabricated infused silica Naether et al. (2013), and a ultracold atomic mixture in 1D optical lattices Schaff et al. (2010). Except for these localization effects, it is interesting to study the topological phases under the random binary disorder. A nature question is whether the TAI phase can be induced by this kind of correlated disorder.

In this paper, we study the topological and localization properties of the 1D SSH model  Su et al. (1979); Heeger et al. (1988) with random binary disordered hoppings. We obtain the topological phase diagrams by numerically calculating the real-space winding number, the energy gap, and the edge states. It is found that the random binary disorder can induce the TAI phase from the trivial insulator in a wide range of parameter regions. We also study the influence of the bimodal probability on the topological phase. By generalizing the existing self-consistent Born approximation (SCBA) approach to our model, we reveal the trivial-topological phase transition driven by the disorder-renormalization of the intracell hopping strength. An analytical formula for the localization length of the zero-energy state is derived, whose divergence indicates the topological phase transition. We further study the localization properties of the system by computing the inverse participation ratio (IPR) in both real and momentum space. We numerically show that the mean chiral displacement of the bulk dynamics can be used to reveal the topological phases, which can already be implemented in ultracold atomic and photonic systems Cardano et al. (2017); Maffei et al. (2018); Meier et al. (2018); Xiao et al. (2021); Wang et al. (2018); D’Errico et al. (2020); Xie et al. (2020, 2019); Wang et al. (2021); Meng et al. (2020). Finally, we discuss the TAI phase in a more general case with both binary disordered inter- and intra-cell hoppings.

The rest of this paper is organized as follows. We first introduce the 1D SSH model with random binary disordered hoppings in Sec. II. Section III is denoted to present the results of the topological phase diagrams with disorder-induced TAIs, the localization properties, and the influence of the bimodal probability. In Sec. IV, we propose to detect the topological phases by measuring the mean chiral displacement. A short discussion and a brief summary are finally given in Sec. V.

We begin by considering the 1D SSH model with random binary disordered hopping strengths. The tight-binding Hamiltonian consists of two sublattices (labeled by A and B, respectively) per unit cell, which can be expressed as

where $c_{A(B),n}^{\dagger}$ and $c_{A(B),n}$ are the creation and annihilation operators of electron on $A(B)$ sublattice in the $n$-th unit cell, respectively. We consider a system has $N$ unit cells, and the total number of sublattice sites is $L=2N$ with each sublattice site indexed by $x\in[1,L]$.

For simplicity, we first consider the dilute case where the intercell hopping strengths to be site-independent $t_{n}=t$, and the intracell hopping strengths $m_{n}$ are binary disorders. In the last section, we will generalize our results by considering $t_{n}$ also binary disordered. To do this, $m_{n}$ is drawn from a bimodal probability distribution function Saha et al. (1996); Mondal et al. (2019)

which means $m_{n}$ takes bimodal value $V_{0}+w_{a}$ or $V_{0}+w_{b}$ with probability $P$ or $1-P$ on $n$-th unit cell. $V_{0}=w_{a}=1$ is chosen as the energy unit, and the disorder strength is defined as the difference between $w_{a}$ and $w_{b}$: $W\equiv w_{a}-w_{b}$. For the special case that the probability $P=0$ or $P=1$, it reduces to the clean SSH model Su et al. (1979); Heeger et al. (1988) of free fermions.

It is well-known that the SSH model has two topologically distinct phases and exists two-fold degenerate zero-energy edge states  Asbóth et al. (2016) in the topological insulating phase under open boundary conditions (OBCs). In experiments, the SSH model has been realized in different systems, such as cold atoms in 1D optical superlattices Atala et al. (2013). In this clean case, the 1D winding number of Bloch vectors in momentum space can be used to characterize topologically trivial and nontrivial phases. In the presence of disorders, the translational symmetry breaking and the wave vectors in momentum space are no longer good quantum numbers, so the conventional winding number defined in terms of Bloch vectors should be extended to the real space Mondragon-Shem et al. (2014). Here we use the real-space winding number to characterize the topological property of our model. For a given disorder configuration denoted by the label $s$, the corresponding real-space winding number is given by  Mondragon-Shem et al. (2014):

where $X=\mathrm{diag}(1,1,2,2,\cdots,N,N)$ denotes the unit-cell coordinate operator, $Q_{s}=\sum_{j}(\ket{\psi_{j}}_{ss}\bra{\psi_{j}}-\ket{\tilde{\psi}_{j}}_{ss}\bra{\tilde{\psi}_{j}})$ is the flat-band Hamiltonian with $\ket{\psi_{j}}_{s}$ being the $j$-th eigenstate of $H$ and $\ket{\tilde{\psi}_{j}}_{s}=\Gamma^{-1}\ket{\psi_{j}}_{s}$, $\Gamma=\mathrm{diag}(1,-1,1,-1,\cdots,1,-1)$ is the chiral operator with diagonal elements $1$ and $-1$ for A and B sublattices, respectively. Here $L^{\prime}=L-2l$ is slightly less than the chain length $L$ to get rid of the $l$ sites boundary effect at both ends, and ${\mathrm{Tr}}^{\prime}$ stands for the trace within the central region of the length $L^{\prime}$ in the chain. In numerical simulations, we choose $l=L/4$ to avoid boundary effects and get quantized winding numbers. For the disordered SSH model, we use the disorder-averaged (over $N_{s}$ configurations) winding number

as the topological invariant, where a modest configuration number $N_{s}$ is usually sufficient in practice. Thus, $\nu=0$ and $1$ correspond to topologically trivial and nontrivial in the limit of $L\rightarrow\infty$, respectively.

In this section, we systematically investigate the topological and localization properties of the 1D SSH model with random binary disordered hoppings. First, we study the winding number, energy spectrum, and edged state to consistently characterize the topological phase transition induced by random binary hoppings. This phase transition can also be revealed by using the SCBA method which accounts for the disorder effects into the renormalization of the hopping strengths. We then analytically derive the localization length of the zero-energy edge mode and numerically calculate the real-space and momentum-space IPR. Finally, we study the influence of bimodal probability on the topological and localization properties.

Finally, we propose that the TAI phase in this binary disordered model can be detected from the mean chiral displacement, as recently demonstrated in atomic and photonic systems Cardano et al. (2017); Maffei et al. (2018); Meier et al. (2018); Xiao et al. (2021); Wang et al. (2018); D’Errico et al. (2020); Xie et al. (2020, 2019); Wang et al. (2021); Meng et al. (2020); Xu et al. (2018); St-Jean et al. (2021). The time evolution of the mean chiral displacement $C_{s}(\tau)$ for a single disorder configuration $s$ is given by

where $\tau$ denotes time, $\Gamma$ and $X$ are chiral and unit-cell coordinate operators previously used in the real-space winding number. The disorder average mean chiral displacement is then written as

After a long time evolution $\tau=T$, the mean chiral displacement averaged over disorders and time is given by

We choose the initial state at the center of the lattice and simulate the quantum dynamics for many disorder configurations, with typical results of $C(\tau)$ and $\bar{C}$ shown in Fig. 4. As shown in Figs. 4(a) and 4(b), the dynamics of $C(\tau)$ shows a transient and oscillatory behavior and its time- and disorder-averaged $\bar{C}$ converges to the winding number $\nu$ in both clean and disorder cases in the long time limit Cardano et al. (2017); Maffei et al. (2018).

We further plot $\bar{C}$ as a function of $W$ for $P=1/2$ and $t=1.5$ in Fig. 4(c) for different lattice sizes. We can see that $\bar{C}$ converges to an integer value which approaches to $0$ in the trivial phase and $1$ in the topological phase by increasing the lattice size $L$, which agrees with the result of $\nu$ shown in Fig. 1(b). We also numerically calculate $\bar{C}$ as a function of $P$ for $W=1$ and $t=1.5$ in Fig. 4(d), which shows that $\bar{C}\approx 1$ for small $P$ and drops to zero for $P\gtrsim 0.5$. The decrement of $\bar{C}$ by increasing $P$ is in good agreement with the winding number $\nu$ obtained in Fig. 3(c).

Before concluding, we discuss the 1D SSH model with overall random binary disordered hoppings and reveal the TAI phase in this general case. The intercell hoppings $t_{n}$ are drawn from the following bimodal probability distribution function

where $W^{\prime}$ is the disorder strength of the intercell hopping. When $W^{\prime}=0$, it returns to the previous case discussed in Sec. II. The trivial-topological and topological-trivial phase transition points ($W\simeq 0.88$ and $W\simeq 3.12$) shown in Fig. 5(a) consist with results in Fig. 1(b) when $t=1.5$. When increasing the disorder strength $W^{\prime}$ of intercell hoppings, we find the TAI region shrinking and finally disappearing when $W^{\prime}=1.5$. Further increasing $W^{\prime}$ can revive and broaden the topological region. When $0<W^{\prime}<1.5$, the effective intercell hopping $\bar{t}<t=1.5$, and from the SCBA renormalized relation Eq. (10) one can see that the critical disorder $W_{c}$ from trivial to topology is larger than the corresponding point for $W^{\prime}=0$ to satisfy $\bar{m}=\bar{t}$, and the TAI region induced by $W$ is shrinking. When $W^{\prime}=1.5$, the probability of $t_{n}=0$ leads to a breaking inter-cell hopping and vanishes the TAI phase. Further increasing $W^{\prime}$ effectively enlarges the hopping strength of $|t_{n}|$ and thus the topological phase revives. In this general case, we obtain the inverse of the localization length as

When $W^{\prime}=0$, Eq. (V) returns to Eq. (16). In Fig. 5(a), we plot the solid red lines where $\Lambda^{-1}=0$ on the $W$-$W^{\prime}$ plane for $t=1.5$ and $P^{\prime}=P=0.5$, which are consistent with the real-space winding number. We also calculate the disorder-averaged real-space IPR $\bar{I}_{r}$ in Fig. 5(b), which indicates the bulk states are localized in the topological phase region.

In summary, we have explored the topological and localization properties of the SSH model with the random binary disorder. We have shown that the random binary disorder can induce TAIs in a wide range of parameter regions of topological phase diagrams. We have analyzed trivial-topological phase transitions by calculating the renormalization of hopping strengths with the SCBA method and topological phase boundaries by calculating the localization length of the zero-energy mode. The localization and the effect of the bimodal probability have been studied. We have also numerically demonstrated that the revealed topological phases can be detected from the mean chiral displacement, which could be measured from the bulk dynamics in atomic or photonic systems. This work provides an extension of the TAIs to the case of spatially correlated disorders.