† Corresponding author. E-mail:
Project supported by the National Basic Research Program of China (Grant No. 2019YFA0308403), the National Natural Science Foundation of China (Grant No. 11822407), Undergraduate Training Program for Innovation and Entrepreneurship, Soochow University (Grant No. 201810285022Z), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.
We study the disorder-induced phase transition in two-dimensional non-Hermitian systems. First, the applicability of the noncommutative geometric method (NGM) in non-Hermitian systems is examined. By calculating the Chern number of two different systems (a square sample and a cylindrical one), the numerical results calculated by NGM are compared with the analytical one, and the phase boundary obtained by NGM is found to be in good agreement with the theoretical prediction. Then, we use NGM to investigate the evolution of the Chern number in non-Hermitian samples with the disorder effect. For the square sample, the stability of the non-Hermitian Chern insulator under disorder is confirmed. Significantly, we obtain a nontrivial topological phase induced by disorder. This phase is understood as the topological Anderson insulator in non-Hermitian systems. Finally, the disordered phase transition in the cylindrical sample is also investigated. The clean non-Hermitian cylindrical sample has three phases, and such samples show more phase transitions by varying the disorder strength: (1) the normal insulator phase to the gapless phase, (2) the normal insulator phase to the topological Anderson insulator phase, and (3) the gapless phase to the topological Anderson insulator phase.
Since the spin Hall model proposed by Kane and Mele in 2005,[1–3] topological states based on Hermitian systems have been greatly developed.[4–13] Among them, ten classes of topological phases guaranteed by time-reversal symmetry, inversion symmetry, and particle–hole symmetry are the vital research objects.[3,14–16] With the help of the space group and the point group, topological crystalline insulators were also proposed by Fu.[17] These topological states receive great interest over the past few years.[18–25] Very recently, three groups[26–28] extended the study of topological states in Hermitian systems to the complete inorganic crystal structure database.
With the comprehensive understanding of the topological states in Hermitian systems, physicists begin to suspect whether topological states exist in non-Hermitian systems as well.[29] For Hermitian systems, the topological order is mainly determined by the bulk states.[3,5,30] To be specific, the bulk–edge correspondence[3,5] is achieved. This method was also widely used in early studies of non-Hermitian systems.[31–40] Nevertheless, several studies illustrated that non-Hermitian systems could be completely different from the Hermitian ones[33–36] in their unique property — the non-Hermitian skin effect,[41–43] which means that all the eigenstates concentrate on one side or even one point of the samples’ open boundary. This characteristic makes the topological properties of the open boundary conditions in the non-Hermitian system notably different from the periodic boundary case.[34–36] For example, the emergence of zero energy state is regarded as the benchmark to define the topological state in the one-dimensional model for both Hermitian[44–47] and non-Hermitian[32] systems. It is believed that the winding number,[46,47] which is calculated by the bulk states of a sample with the periodic boundary condition, will determine the existence of zero-energy modes and give a reasonable phase diagram. However, the distribution of zero energy states obtained with the open boundary samples is quite different from that with the periodic boundary condition in non-Hermitian systems.[33–35] To determine the topological phase diagram of non-Hermitian systems, Yao et al. proposed a special transformation to the non-Hermitian Hamiltonian.[35,36] After such a kind of transformation, the effective Hamiltonian becomes Hermitian at high symmetric points. When a phase transition happens, the bandgap at a high symmetric point is closed and then reopened. Therefore, the correct phase boundaries can be obtained. These works significantly advanced the study of topological states in non-Hermitian systems.
Notably, the effect of disorder is inevitable in experiments because there always exist impurities or non-uniformity in the samples. After the pioneering work by Anderson et al.,[48] the study of disorder effect has received extensive advancement.[49–51] For a long period, disorder is considered to make things worse, e.g., the Anderson localization.[49–56] However, an unexpected phase transition induced by disorder was reported in 2009,[57,58] in which disorder can turn a normal insulator into a topological insulator. Such a nontrivial phase induced by disorder is called the topological Anderson insulator, and has been widely studied in the Hermitian system.[57–60] Since all systems could have impurities, it is also essential to study the disorder effect for non-Hermitian systems. Previously, the disorder in non-Hermitian systems was investigated in terms of the distribution of eigenvalues.[61–63] These works mainly focus on the stability of the topological states, but the investigation of the interplay between TAI and the non-Hermitian effect is still insufficient.
In this paper, the disorder-induced topological phase transition of the non-Hermitian insulator is numerically studied with the help of noncommutative geometric method (NGM).[46,47,54,55,64,65] We first investigate the applicability of NGM in non-Hermitian systems for two different cases (the effective model in periodic boundary condition and the original model in open boundary condition). These two cases show the coincident phase transition points, which agrees quite well with the analytical one. We also find that the Chern number calculated by NGM slightly deviates from the quantized value (the maximum deviation is about 0.1) when the non-Hermitian effect appears, and the deviation could be eliminated by enlarging the sample size. Next, after checking the stability of the numerical method, the disorder effect in non-Hermitian systems is considered. We confirm that non-Hermitian Chern insulators are robust against disorder. Meanwhile, the disorder makes non-Hermitian systems transform from the trivial phase to the nontrivial phase, which is similar to the topological Anderson transition in Hermitian systems. Finally, cylindrical samples are investigated, and they show more phase transitions.
The first selected non-Hermitian Hamiltonian is[36]
![](cpb_29_5_050502/cpb_29_5_050502_ieqn1.gif)
![](cpb_29_5_050502/cpb_29_5_050502_ieqn2.gif)
![](cpb_29_5_050502/cpb_29_5_050502_ieqn3.gif)
As shown in Figs.
To better understand the difference between H under the condition of the open boundary and Heff under the condition of the periodic boundary, we compare the Chern numbers for these two cases. The Chern number for the finite-sized sample is calculated by combining the generalized definition of non-Bloch Chern number[36] and the theory of Chern number in real space.[64,65] Due to the non-Hermitian properties of the Hamiltonian, the eigenvalues of H and H† are separately defined as[36]
The above eigenvectors can be used to define the non-Bloch Chern number[36,67]
![](cpb_29_5_050502/cpb_29_5_050502_ieqn4.gif)
![](cpb_29_5_050502/cpb_29_5_050502_ieqn5.gif)
![](cpb_29_5_050502/cpb_29_5_050502_ieqn6.gif)
![](cpb_29_5_050502/cpb_29_5_050502_ieqn7.gif)
Here, based on the evolution of the Chern number (calculated by Eq. (
In order to make the phase transition point more accurate, we compact the grid near the phase transition point (see Fig.
In the above section, we have shown that the phase transition point obtained by combining Heff under the condition of periodic boundary with NGM is consistent with the analytical prediction. However, more data is needed to ensure whether the feature will hold for the whole parameter regions or not. Sequentially, we study the reliability of NGM for non-Hermitian systems with different γ.
In Fig.
To explain this phenomenon, we plot the evolution of the Chern number with the change of sample size N for γ = 0.6 (Fig.
Conclusively, the numerical results obtained by combining NGM and non-Bloch Chern number can characterize the topological phase transition. However, when the non-Hermitian features are strong, especially when the parameters are close to the phase transition point, a sample with a large size is needed to achieve reliable results.
The established NGM does not require the translation symmetry of the system; thus it has an advantage when dealing with the disorder systems. In this section, we use NGM to investigate the disorder-induced phase transition in two-dimensional non-Hermitian systems. Notably, we examine whether topological Anderson insulators can exist in non-Hermitian systems. In our calculations, the disorder effect is based on the standard Anderson disorder.[48–51] The random on-site energy for each site satisfies the unitary distribution [−W/2,W/2], where W represents the disorder strength.
Since NGM is not suitable to calculate the Chern number in systems with strong non-Hermitian features, we only study the case with small γ (γ = 0.4). The main results are shown in Fig.
Interestingly, when m > 2.16, the clean samples are normal insulators initially. However, the nontrivial region occurs in a wide range of disorder strength (see Fig.
The Chern number gradually decreases to zero when W > 4.2, which means the system transmits from the topological Anderson insulator to normal Anderson insulator due to Anderson localization. In order to further study the numerical results obtained by Heff under the periodical boundary conditions, we also investigate the disorder effect of H under the open boundary conditions. As shown in Figs.
In this section, we investigate the disorder-induced phase transition in cylindrical samples to further illustrate that NGM is also applicable to other non-Hermitian Hamiltonians. Following the study,[36] we use another Hamiltonian
![](cpb_29_5_050502/cpb_29_5_050502_ieqn11.gif)
Then, we investigate the topological phase transition with the disorder. By setting γ = 0.4, the evolution of the Chern number is calculated. For a clean sample, the system has three different phases: when m < 1 + e−0.4, the system is topological nontrivial; when m > 1 + e0.4, the system is topological trivial; and when 1 + e−0.4 < m < 1 + e0.4, the system is gapless. With the increase of the disorder strength, the topological region with a quantized Chern number gradually extends to the gapless area. This specific Chern number evolution corresponds to a transition from the gapless phase to the topological Anderson insulator. Similarly, the gapless region gradually expands towards the normal insulator region with increasing disorder strength, which leads to the transition from the trivial phase to the gapless one. Besides, the area of the gapless phase is so vast that the topological nontrivial insulator state caused by disorder cannot extend to the insulator region. Thus, there exists no disorder-induced phase transition from a trivial state to a topological state in this case.
To better understand the influence of the non-Hermitian intensity on the topological phase transition, we set W = 3 and calculate the evolution of the Chern number by changing γ and m. The main results are shown in Fig.
A recent paper of Song et al.[67] also showed that the Chern number obtained in real space could be greater than one or less than zero, which is very similar to our results (see Fig.
In summary, we investigate the applicability of NGM in the two-dimensional non-Hermitian system. By applying NGM, we study the disorder-induced phase transition in the two-dimensional non-Hermitian system, including both the situations of the square sample and the cylindrical sample. In the square sample, we find a disorder-induced direct phase transition from normal insulator to topological Anderson insulator. For the cylindrical sample, the system mainly has the following three phase transitions: (1) the normal insulator phase to the gapless phase, (2) the normal insulator phase to the topological Anderson insulator phase, and (3) the gapless phase to the topological Anderson insulator phase. We notice that a very recent paper proposed a method that can be used to investigate the topological index under the condition of open boundary[67] (a comparison of the methods in this paper and Ref. [67] is presented in appendix
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