Machine learning of mirror skin effects in the presence of disorder
MMachine learning of mirror skin effects in the presence of disorder
Hiromu Araki, ∗ Tsuneya Yoshida, and Yasuhiro Hatsugai Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Non-Hermitian systems with mirror symmetry may exhibit mirror skin effect which is the extremesensitivity of the spectrum and eigenstates on the boundary condition due to the non-Hermitiantopology protected by mirror symmetry. In this paper, we report that the mirror skin effect sur-vives even against disorder which breaks the mirror symmetry. Specifically, we demonstrate therobustness of the skin effect by employing the neural network which systematically predicts thepresence/absence of the skin modes, a large number of localized states around the edge. Thetrained neural network detects skin effects in high accuracy, which allows us to obtain the phasediagram. We also calculate the probability by the neural network for each of states. The aboveresults are also confirmed by calculating the inversed participation ratio.
Introduction.—
Topological properties of condensedmatter systems have been extensively studied in thesedecades [1–5]. Among them, non-Hermitian topologyattracts much attention in these years [6–47]. The plat-forms of non-Hermitian topological physics extends toa wide variety of systems such as, open quantum sys-tems [12–15, 30, 48], electric circuits [17, 21, 22, 49, 50],photonic crystals [6, 16, 18, 19, 51–55], equilibrium sys-tem of quasiparticles [46, 56–64], and so on.In non-Hermitian topological systems, various novelphenomena have been reported which do not have Her-mitian counterparts [11, 34, 65–73]. One of the typi-cal examples is the skin effect [7, 31–34, 74–78] whichcan be observed for Hatano-Nelson model [25, 26],a one-dimensional tight-binding model with the non-reciprocal hopping. This non-reciprocity results in thenon-Hermitian topological properties characterized bythe winding number [76–78]; the energy spectrum of theBloch Hamiltonian winds around the origin of the com-plex plane with increasing the momentum from − π to π .Corresponding to the non-trivial properties, the systemshows extreme sensitivity of the energy spectrum and theeigenstates. In particular, the non-reciprocity inducesskin modes which are a large number of localized statesaround the edge due to the finite winding number in thebulk. This remarkable topological phenomenon in non-Hermitian systems is further extended by taking into ac-count symmetry [49, 78–81]. In particular, it turned outthat mirror symmetry protects a type of non-Hermitiancrystalline symmetry inducing the mirror skin effect [82].Despite the above progress of non-Hermitian crys-talline topology, the robustness of the perturbationsbreaking the relevant symmetry remains unclear. Suchperturbations is inevitable in experiments because crys-talline symmetry is generically broken by impurities [83].In this paper, we show that the mirror skin effect sur-vives even in the presence of disorders, which demon-strates that the non-Hermitian crystalline topology is ro-bust against perturbation breaking the crystalline sym-metry. The robustness of the mirror skin effect is eluci-dated by a machine learning approach [84–104]. Namely,we train the neural network in the clean limit so that it predicts the presence/absence of the skin modes in thepresence of disorder, which is a signal of the mirror skineffect. The trained neural network systematically pre-dicts the presence/absence of the skin effects, which al-lows us to obtain the phase diagram. The obtained phasediagram elucidate the robustness of non-Hermitian topol-ogy with mirror symmetry against disorders. We alsoconfirm the robustness by computing the inverse partici-pation ratio (IPR). Finally, we apply the neural networkto each of states and calculate the probability of the lo-calized states.The rest of this paper is organized as follows. Firstly, atoy model with disorder is introduced. Secondly, effectsof disorder are discussed for specific strength of disorder.The distribution of complex energies and the edge statesfrom the skin effects are also shown. In next, the modelfor the machine learning and the method for training areexplained. Then the results by applying the machinelearning model to the non-Hermitian model are shown.Then, the machine learning model is applied to each ofstates and we calculate the probability of the localizedstates about them. Lastly, we conclude our study. Non-Hermitian model.—
In order to demonstrate therobustness of the mirror skin effect against perturbationsbreaking the relevant symmetry, we analyze a fermionicbilayer system with disorder. The Hamiltonian reads, H W = H + (cid:88) jα w jα c † jα c jα , (1)where W is the strength of the randomness and w jα is auniform random variable in [ − W/ , W/ c † jα ( c jα ) creates (annihilates) a spinless fermion at site j of layer α . In the clean limit W = 0 [49], the system isdescribed by H ; H = (cid:88) k αβ c † k α h αβ ( k ) c k α , (2) h ( k ) = [2 t (cos k x + cos k y ) − µ ] ρ + i ∆ sin k x ρ + i ∆ sin k y ρ , (3)where c † jα ( c jα ) denotes the Fourier transformed cre-ation (annihilation) operator. The vector k = ( k x , k y ) a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1. (a): The eigenenergies of the model with ( t , µ , ∆) =(1, 2, 1 .
8) and M = 18. The boundary conditions are periodicfor the x -axis and open for the y -axis. (b): The density plotof the eigen states corresponding to the eigenenergies in thered circles in panel (a). The particles are localized on edges.(c): The density plot of the eigen states corresponding to theeigenenergies outside the red circles in panel (a). The particlesare itinerant in the bulk. Panels (b) and (c) are obtained for M = 12. ( − π ≤ k x ( y ) < π ) describes the momentum of the two-dimensional system. The matrices ρ i ( i = 0 , , ,
3) arethe Pauli matrices acting on the pseudo-spin space of thetwo layers. The Hamiltonian in the clean limit H pre-serves the mirror symmetry; M x H ( k ) M − x = H ( M x k )with M x = ρ P x . Here, P x flips the momentum from k = ( k x , k y ) to M x k := ( − k x , k y ).The mirror symmetry of H protects the non-Hermitian point-gap topology [105] which is character-ized by the mirror winding number [49]; ν M ( k ∗ x ) = (cid:88) m = ± sgn( m ) (cid:90) dk y πi ∂ k y log det[ h m ( k ∗ x , k y ) − E pg ] , (4)with sgn( m ) taking 1 ( −
1) for m = + ( m = − ). Here, h +( − ) is the Bloch Hamiltonian for plus (minus) sectorof the mirror operator M x [106]. For k ∗ x = 0, the mirrorwinding number takes ν M (0) = − E pg = 2 t − µ ,while for k ∗ x = π , it takes ν M ( π ) = − E pg = − t − µ . Effect of disorder on the mirror skin effect.—
Prior tothe systematic analysis based on the machine learningapproach, we discuss the effect of disorder for specificvalues of disorder strength, W = 1 and W = 10.For comparison, we firstly analyze the spectrum andeigenstates in the clean limit.Figure 1 shows the spectrum and eigenstates for the FIG. 2. (a) and (b): The eigenenergies of the model with ( t , µ , ∆) = (1, 2, 1 .
8) and M = 18. (b) and (d): The density plotof the 4 M eigenstates around E +pg and E − pg for M = 12. Panels(a) and (c) [(b) and (d)] are obtained for W = 1 [ W = 10]. system of M × M sites. Here, the boundary conditions areperiodic for the x -axis and open for the y -axis. [Fig. 1(a)is obtained for M = 18 while Fig. 1(b) and 1(c) areobtained for M = 12].Figure 1(b) indicates that due to the topology char-acterized by the mirror winding number, the 4 M -statesenclosed in red circles in Fig. 1(a) are localized. Theseskin modes are signals of the skin effects. Here, the num-ber 4 M can be calculated as follows: at given k ∗ x taking 0or π , the mirror skin effect induces 2 M -localized states,where the prefactor 2 is arising from the number of layers.We note that the all of states outside the red circlesare extended to the bulk [see Fig. 1(c)].We analyze the system for specific values of disorderstrength W = 1 ,
10. The obtained results imply that thepoint-gap topology remains for weak W , although intro-ducing the disorder breaks the mirror symmetry. Furtherincreasing the interaction suppresses the skin effects asthe localized states are buried in the bulk states.Figure 2 shows the energies and a part of eigenstates ofthe model with randomness. Boundary conditions are thesame as the Fig. 1 – periodic for the x -axis and open forthe y -axis. Figures 2(a) and 2(b) show the eigenenergiesof the model with ( t , µ , ∆, W ) = (1, 2, 1 .
8, 1) and ( t , µ , ∆, W ) = (1, 2, 1 .
8, 10), respectively. The figuresshow that the edge-localized states from the skin effectare still in the gap of E ± pg for W = 1, however the gapis closed for W = 10. Next we consider the eigenstatesaround the point E ± pg . Figures 2(c) and 2(d) show thedensity plot of the 4 M states around E ± pg for W = 1 and W = 10, respectively. For W = 1, the in-gap states arestill localized at edges. It suggests that the skin effectstill remains. However, for W = 10, the gap is closed FIG. 3. Schematic picture of our CNN model. Input data are batched two dimensional images. The model consists of two CNNlayers and two linear layers. The numbers in this figure are the output layers for the CNN layers and the output parameters forthe linear layers. Calculating the softmax for the output of the second linear layer, the output data of the model are obtained. and there are no localized states around the E ± pg .The above results imply that the mirror skin effectsurvives even in the presence of the disorder. In next, weelucidate the robustness by systematic analysis based onthe machine learning technique. Machine learning of disordered non-Hermitiansystems.—
Hereafter, we try to draw the phase diagramof the non-Hermitian system with randomess. Toaccomplish it, we use a machine learning. By using theconvolutional neural network (CNN), we can detect theedge states and draw a phase diagram systematically foreach parameters [82, 107].As input data, we take the absolute square of the wavefunction. The boundary conditions of the system areperiodic for the x -axis and open for the y -axis and systemsize is 2 M . Here after, we set M = 6. Since the systemis in two dimensions and has two degrees of freedom ateach point, each of input data is ( M × M ×
2) matrix.In addition, we take average of the 4 M wave functionsaround the reference energies E ± pg .The output data are whether the edges states from theskin effect are in the point-gap of the E ± pg . We use thedata of the clean systems as the training data. Then,the trained model predicts the phase of the disorderedsystems.Here after, we show the architecture of the neural net-work. The schematic picture of our model is shown inFig. 3. It consists of two convolutional layers and twolinear layers. The number of filters comprising the con-volutional layers is 32 and 64. The kernel size of each con-volutional layer is 3 ×
3. The first linear layer transformsthe output of the previous layer to 128 dimensional vec-tor and the second one transforms it to two-dimensionalvector. The output data are obtained by applying soft-max functions y i = e x i / (cid:80) k =1 e x k , where i labels one ofthe phases. All the activation functions are the ReLUfunctions. For input x , the ReLU function returns x if x >
0, otherwise it returns 0. To prevent overfitting,the dropout layer is inserted after the second convolu-tional layer and the first linear layer. The probability of
FIG. 4. (a) Probabilities that are culculated by the machinelearning model for the input data specificated by W and ∆.(b) The IPR for the input data specificated by W and ∆. dropout is set to 0 . M edge states. Then, we take theaverage of the probability density of the 4 M states asthe data of localized states. On the other hand, there FIG. 5. The energies of the system with various ∆ and W . (a) ∆ is fixed to 1 .
8. (b) ∆ is fixed to 2 .
2. The color on the dotscorresponds with the probability of the localized states calculated by the machine learning model. are 2 M − M bulk states. In a similar way, we takeaverage of the probability density of the 2 M − M bulkstates as the data of bulk states. We take 2000 samplesas input data by randomly selecting ∆ ∈ (1 . , . w . For the disordered system,the input data are prepared as follows. Firstly, the 2 M states near the each of reference energies E ± pg are selected.Then the input data are the average of the probabilitydensity of the states. The output data of the trainedmodel tells us whether the states are localized or bulkstates. Figure 4 (a) shows the probability of the localizedstates for ∆ and w . The figure shows that the two phasesare clearly distinct. It implies that the skin effect stillexists for small w but the localized states are buried inthe bulk states for w lager than the critical value, which isdetermined by the competition between the size of point-gaps and the strength of disorder. In this model, the sizeof the point-gap is determined by t and ∆. Then, ifthe strength of disorder w exceeds the critical value, thephase change to the disordered phase (Fig. 4(a)).Next, we consider IPR as an indicator for the local- ization. The IPR p is calculated as p = (cid:80) k,j | φ k ( r j ) | for a state φ where r j is the j th lattice point and k isthe index of inner degree of freedom. Figure 4 (b) showsthe IPR for the averaged states. The figure shows thatthe large (small) p corresponds with the localized (bulk)state. Furthermore, the phase diagram is consistent withthe result by the trained machine learning model. Machine learning of disordered non-Hermitiansystems.—
Here, we apply the machine learning modelto the each of states of the non-Hermitian model withimpurities so that the probability of the localizedstate is calculated by the machine learning model. Toaccomplish it, we use both the averaged states and thenon-averaged states as the supervised data. This isbecause the non-averaged states are localized at oneof the edges, which are not appeared in the averagedstates.The supervised data contain 4000 averaged states and4320 non-averaged states. Half of the states are the bulkstates and the rest states are the localized states. Thesystem and the architecture of the neural network are thesame as before.Figure 5 shows the energies of the system with severalvalues of ∆ and W . The color on the dots correspondswith the probability of the localized states. In Fig. 5(a)and 5(b), ∆ is fixed to 1 . .
2, respectively. In bothFig. 5(a) and 5(b), the system shown in left, rightand center figures corresponds with the white, dark andboundary region in Fig. 3. The figures show that thereare more localized states in the in-gap states for weak W and there are more extended states for large W . At thephase transition point, both are included. Conclusion.—
Employing the neural network, we havedemonstrated that the mirror skin effect survives even inthe presence of the disorder, which elucidates the robust-ness of the non-Hermitian crystalline topology againstperturbations breaking the relevant symmetry. The neu-ral network, which is trained in the clean limit, system-atically predicts the presence/absence of the skin modesin high accuracy, which allows us to obtain the phasediagram. The obtained phase diagram indicates thateven in the presence of disorder, the system shows theskin modes due to the point-gap topology with mirrorsymmetry. We have also confirmed that the above re-sults are consistent with the analysis based on the IPR.Finally we have applied the machine learning model toeach of states and calculated the probability of the local-ized states about them. The robustness of non-Hermitiancrystalline topology against disorders is considered to beexperimentally verified by electric circuits.
Acknowledgments.—
We thank Tomonari Mizoguchi forcollaboration in Ref. 49 proposing mirror skin effect.This work is supported by JSPS Grant-in-Aid for Scien-tific Research on Innovative Areas “Discrete GeometricAnalysis for Materials Design”: Grants No. JP20H04627.This work is also supported by JSPS KAKENHI GrantsNo. JP17H06138, No. JP19J12315, and No. JP19K21032. ∗ [email protected][1] X.-L. Qi and S.-C. Zhang: Rev. Mod. Phys. (2011)1057.[2] C. L. Kane and E. J. Mele: Phys. Rev. Lett. (2005)146802.[3] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W.Ludwig: New Journal of Physics (2010) 065010.[4] Y. Hatsugai: Phys. Rev. Lett. (1993) 3697.[5] X.-G. Wen: Rev. Mod. Phys. (2017) 041004.[6] K. Takata and M. Notomi: Phys. Rev. Lett. (2018)213902.[7] T. E. Lee: Phys. Rev. Lett. (2016) 133903.[8] H. Shen, B. Zhen, and L. Fu: Phys. Rev. Lett. (2018) 146402.[9] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-gashikawa, and M. Ueda: Phys. Rev. X (2018) 031079.[10] S. Lieu: Phys. Rev. B (2018) 045106.[11] H. 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