Classification of Strongly Disordered Topological Wires Using Machine Learning
CClassification of Strongly Disordered Topological Wires Using Machine Learning
Ye Zhuang, Luiz H. Santos, and Taylor L. Hughes Department of Physics and Institute of Condensed Matter Theory,University of Illinois, Urbana, Illinois 61801, USA Department of Physics, Emory University, Atlanta, Georgia 30322, USA (Dated: January 30, 2020)In this article we apply the random forest machine learning model to classify 1D topologicalphases when strong disorder is present. We show that using the entanglement spectrum as trainingfeatures the model gives high classification accuracy. The trained model can be extended to otherregions in phase space, and even to other symmetry classes on which it was not trained and stillprovides accurate results. After performing a detailed analysis of the trained model we find that itsdominant classification criteria captures degeneracy in the entanglement spectrum.
INTRODUCTION
Since the discovery of topological insulator phases, theproblem of their classification has been an important sub-ject. A classification table, predicated on the stabilityof a strong topological phase in the presence of disor-der, was proposed for free fermion systems in the tenAltland-Zirnbauer symmetry classes [1, 2], though thetable does not determine the explicit phases and phasediagrams of model systems. Many successful determina-tions of the phase diagrams of low-dimensional disorderedtopological phases have been made in symmetry classesA, AIII, and BDI using techniques based on, e.g., en-tanglement properties, level-spacing statistical analysis,and real-space topological indices [3–7]. In this articlewe propose the use of a new technique based on the ran-dom forest (RF) machine learning model to determinethe phase diagrams of models of disordered topologicalphases.Recent work has shown that machine learning can pro-vide a new framework for solving problems in physics. Inour context, promising developments have been achievedin applying machine learning techniques to classifyingphases of matter in condensed matter systems. Super-vised learning has been used directly in characterizingphases in both classical spin systems [8–10] and quan-tum many-body systems [11–15]. Neural networks arethe most widely used model to identify phases, espe-cially topological phases of matter, and they are pow-erful models that have universal approximation capabili-ties [16, 17]. For example, Chern insulators and fractionalChern insulators can be classified by feeding quantumloop topography into a neural network [14]. Unfortu-nately, the black-box nature of neural networks makes ithard to interpret the trained models, and it is not easyto extract insightful physical intuition about the systemunder study. Additionally, the large number of hyper-parameters in a neural network can make it difficult totrain.In this paper, we use the RF method as our machinelearning model to detect topological phases with strong disorder. RF is an ensemble method that is capable ofrepresenting complicated functions with much fewer pa-rameters as compared with neural networks, and havingmore easily interpretable classification criteria once themodel is trained [18, 19]. RF is a collection of deci-sion trees, which can be understood as piecewise con-stant functions in feature space. An individual decisiontree cannot make good predictions because, in general,predictions of decision trees have large variance. Aver-aging over decision trees reduces variance, making RFa popular method [20]. Some major advantages of RFare that it has few hyper-parameters, and is immune toproblems such as over-fitting, collinearity, etc.To train our RF model we will use the entanglementspectrum (ES)[21] of our physical system as our inputdata. We will focus on 1D systems where the ES hasbeen widely used to characterize topological phases, anda robust degeneracy in the entanglement spectrum canserve as a general indicator for a topological phase [22].To benchmark our machine learning model we will con-sider 1D free-fermion wires having chiral symmetry inthe AIII class. We choose this system because the dis-ordered phase diagram of this model has been carefullystudied[6]. Here we find that the RF model, trained bythe ES data generated from a small fraction of the phasediagram, can be generalized to the full phase space withhigh prediction accuracy. Furthermore, the RF modeltrained from the AIII class data shows high predictionability for wires in symmetry class BDI as well. A de-tailed analysis reveals that the RF model is primarilycapturing the degeneracy in the ES to make its classifi-cation, and may provide new routes to identify disorderedtopological phases from their entanglement properties.
MODEL
We start from the disordered chiral Hamiltonian inRef. [6] defined on a one-dimensional chain with two sites a r X i v : . [ c ond - m a t . d i s - nn ] J a n A and B in one unit cell: H = (cid:88) n (cid:20) t n c † n ( σ x + iσ y ) c n +1 + h.c (cid:21) , + (cid:88) n m n c † n σ y c n . (1)where c † n = ( c † n,A , c † n,B ) are fermion creation operatorsin unit cell n . We have included disorder in both thehopping and mass terms, i.e., t n = 1 + W ω , and m n = m + W ω , where ω and ω are random variablesgenerated from a uniform distribution on [ − . , . , and W , W represent the strengths of the disorder. Themodel preserves chiral symmetry C H C − = − H with C = (cid:80) n c † n σ z c n .In the clean limit, the system has translational sym-metry and the Bloch Hamiltonian is H ( k ) = t cos kσ x + ( t sin k + m ) σ y . (2)The chiral symmetry operator χ = σ z anti-commuteswith H ( k ) , and one can use the topological winding num-ber ν to identify the Z classification [23]. If we writethe Bloch Hamiltonian in the form H ( k ) = d x ( k ) σ x + d y ( k ) σ y , then ν is the number of times the unit vector( ˆ d x , ˆ d y ) travels around the origin as k traverses the wholeBrillouin Zone. For example, when | m | < | t | , the systemis in symmetry protected topological (SPT) phase withwinding number ν = 1, and when | m | > | t | the systemhas ν = 0 . When disorder is turned on, and in the limit that W (cid:29) t , the system is completely dimerized within in-dividual unit cells, i.e., it is in the topologically trivialatomic limit. This result is independent of the value of m, and hence there must be a phase transition when W is gradually increased from zero when | m | < | t | . One sig-nature of the topological phase transition point is the di-vergence of the localization length of states at the Fermi-level. Using this criterion one can determine an analyticrelation that is satisfied at a topological critical point [6]. | W | /W +1 / | m − W | m/W − / | − W | /W − / | m + W | m/W +1 / = 1 . (3)Indeed, using this relation one can determine the phasediagram of this model even in the presence of disorder. RESULTS
To generate training and testing data we calculate thesingle-particle entanglement spectrum [24] using a cen-tral spatial cut of the lattice model. We use periodicboundary conditions on a chain of length L = 400 with t = 1 . To be explicit, let us first focus on a line in the3D phase space { m, W , W } with m = 0 . W = 1.For illustration we plot the ES of one disorder configura-tion for each value of W in Fig. 1(a). The black verticalline indicates the theoretical transition point calculated (a) (b) FIG. 1. (a) Single particle entanglement spectrum plottedvs disorder strength W . The vertical black dashed line isthe analytical transition point. Double degeneracy at 0.5 onthe left hand side indicates the non-trivial SPT phase. Theremay be accidental degeneracies in the trivial phase due todisorder.(b) Feature importance in our trained random forestmodel. The vertical axis is the average of the entanglementspectrum for each band in panel (a). We plot the feature im-portance of the bands in the horizontal direction. The highimportance values for the middle two bands, i.e., the two thatinclude the degenerate modes at 0.5, indicate their high influ-ence on model predictions. The bands are colored-coded thesame way in panels (a) and (b). from Eq. 3. We can clearly see double degeneracy at 0.5in the region of weak disorder, which is a signature forSPT phases [22]. In the region of strong disorder, onthe other hand, there are no such degeneracies in gen-eral, even though there may be accidental degeneraciesinduced by disorder.In order to test the predictive power of the RF model,we first train the model in regions deep in the two phasesusing a set of test data based on the entanglement spec-trum. We will further evaluate the RF model by test-ing if it can provide accurate predictions for other valuesof parameters different than those used to generate thetraining data. In particular, we test whether the modelis capable of detecting the behavior of the system nearthe topological phase transition despite training it deepin the phases. We will indeed verify that the RF modelcan accurately detect disorder-induced phase transitions.In order to implement the data training, 5000 train-ing samples were generated with W ranging from 0 to4 and from 7 to 10, which correspond, respectively, tothe topologically non-trivial and trivial phases. We in-tentionally skipped the region near the phase transitionpoint W ≈
5, in hopes that the RF model can locate itonly with knowledge deep in the phases.
Test data wasgenerated separately over the whole range of W from 0to 10, and includes the transition region. In order to testthe performance of the RF model with respect to otherwidely utilized algorithms, we trained three models us-ing the same training and testing data: the linear model(LM), neural network (NN), and random forest, and wecompare the predictions of the first two in relation tothe latter. In our numerical analysis, the python packagesklearn [25] was used for training and predicting.We show the prediction results of the three models inFig. 2. From left to right, the dots in the subfigures rep-resent predicted probabilities of being in the topologicalphase from LM, NN, and RF models respectively. We ob-serve that, while most ground states are correctly clas-sified for all the three models, two features stand out.First, the LM has a number of misclassified states inthe region of strong disorder due to the simple linear as-sumption. The trained LM gives a linear relationshipbetween the logarithm of the probability and the gap be-tween the middle two entanglement bands. When thegap is small the model predicts the state is topologicallynon-trivial with high probability even when it is a trivialstate. Second, while the NN and RF models give reli-able predictions for strong and weak values of disorder,they classify with much higher variance near the phaseboundary, which makes it hard to predict the resultingphase from a single disorder configuration. This behav-ior may be expected for, near the transition, the (entan-glement) spectral gap approaches zero, and the disordercauses strong fluctuations that make it difficult to cor-rectly distinguish the intrinsic degeneracies of the ES onthe topological side versus the frequently encountered ac-cidental degeneracies in the trivial phase near the phaseboundary.The predicted critical point can be obtained for eachmodel by fitting the predicted probability with the func-tion f ( x ) = 11 + e b + wx . (4)The fitted curves are shown in each of the subfigures inFig. 2. To help identify the phase boundary we choosea cutoff value of 0.5, i.e., when the predicted probabilityis larger than 0.5, we say the state is in the topologicalphase; otherwise it is in the trivial phase. Therefore,the transition happens at the crossing point of the fittedcurve and the horizontal dashed line at probability 0.5.For comparison, the black vertical dashed lines in eachsubfigure represent the true transition point. We see thatboth NN and RF models can predict the phase transitionpoint with high accuracy, while the LM is not as accurateat predicting the critical point.Quantitative assessments of the predictive propertiesof these three models can be obtained by evaluating accu-racy and error. Accuracy is defined as the percentage ofcorrectly predicted samples; higher accuracy means bet-ter prediction ability. We measure the error of the fittingby the cross entropy [26], which measures the closeness oftwo probability distributions p and q . The cross entropyis defined as (for discrete distributions) H ( p, q ) = − (cid:88) x p ( x ) log q ( x ) , (5) which is the expectation value of − log q ( x ) for the ran-dom variable x following distribution p . Here p is thetrue probability distribution and q is the predicted prob-ability distribution. For our problem, the distribution isdiscrete and has only two cases, topological and trivial,so the cross entropy reduces to just the log loss functionfor a binary classification problem L ( y, ˆ y ) = 1 n n (cid:88) i =1 [ − y i log ˆ y i − (1 − y i ) log(1 − ˆ y i )] , (6)where y i is the true probability of being in the topo-logical phase, and ˆ y i is the predicted probability. Smallerrors indicate a better model, and indeed the LM andNN models are trained on training data to reduce theerror. However, the the RF model is trained based moreon accuracy.The accuracies of the LM, NN, and RF models on thetest data are 0.923, 0.971, and 0.978, respectively. Thecorresponding errors are 0.202, 0.277, and 0.106. TheLM has the lowest accuracy among the three, due toits simple linear assumption, while NN and RF modelsperform similarly. Nevertheless, we emphasize the use ofthe RF method has some advantages including the factthat it captures a high level of accuracy while requiringmuch fewer parameters than the NN model.Another benefit of the RF model is the ability to inter-pret how it makes classification decisions. To illustratethis we can plot the feature importance of the model(Fig. 1(b)). Feature importance measures the number ofsplits in a tree that includes the feature [27], and higherfeature importance means that the feature is more influ-ential on prediction results. We use the ES as features inour model. For each band in the ES the feature impor-tance is calculated (these bands are the ES states near0, 1, and the two in the mid-gap region. To better un-derstand the roles played by each band, we put Fig. 1(b)and Fig. 1(a) side-by-side. The vertical axis of Fig. 1(b)is the same as Fig. 1(a) with the bands represented bytheir averaged position from Fig. 1(a). The horizontalaxis shows the importance of each feature. As shownin the figure, the middle two bands of the ES have thehighest influence on predictions, which strongly indicatesthat the RF model focuses on the degeneracy of the ES toperform its classification decisions. Similar feature prop-erties were also observed for our trained LM model, i.e.,the coefficients of most features are nearly zero exceptfor the features in the middle of the ES. We note that itis difficult to interpret coefficients of the NN model, sowe have little that we can interpret about its behavior.Let us move on to see how these methods can be ex-tended. We trained our models with data deep in thetopological and trivial phases, but a prior knowledge ofthe approximate phase transition point was needed todetermine a reasonable parameter region to produce thetraining data. It would be more ideal if the RF model (a) (b) (c) FIG. 2. (a) Predicted probability of being in the topological phase for the three models: (a) linear model, (b) neural network,and (c) random forest. The smooth curves are fits using Eq. 4.FIG. 3. Accuracy and log loss error of predictions for the RFmodel with different test choices for transition points W ( c )2 .The point with the highest accuracy (lowest log loss) agreeswell with the true transition point as anticipated from theconfusion scheme. could also be used to find an unknown critical point aswell. Indeed, this is possible if we use a scheme similarto the confusion scheme [15]. This method is a trial-and-test scheme that finds a point where two regions can bebest distinguished by the model. This point is then thephase transition point. To use this scheme we choose apossible transition point at W ( c )2 and calculate the pre-diction accuracy and error. If W ( c )2 is the true transitionpoint, we will produce high accuracy and low error. Oth-erwise the accuracy will be low and error will be high. Weplot the accuracy and error at different values of W ( c )2 inFig. 3 for the RF model. The ∧ shape of the accuracyand the ∨ shape of the error are consistent with eachother. These results suggest that the transition point is W ≈
5, which is consistent with the analytic result in-dicated by the vertical dashed line. This shows that inprinciple we could have employed the confusion schemeto find the disorder driven critical point in order to beginour model training, instead of knowing the exact critical (a) (b)
FIG. 4. The predicted phase diagram of (a) W = 1 . m = 0 .
5. The black solid lines are theoretical phase bound-aries. P is the predicted probability of being in the topologicalphase. point from an analytic calculation.So far we have found that the trained model can locatethe transition point with relatively high accuracy, eventhough the model is not given training data informationnear the phase boundary. Now we want to see if we canexpand the region of applicability of our model to a widerange of phase space. For example, we can take two othercross sections of the three-dimensional phase diagram:(i) fixed W = 1 with varying ( W , m ) or (ii) fixed m =0 . W , W ) . The phase diagrams for thesecross-sections are plotted in Fig 4a,b respectively. Thecolormap indicates the predicted probability of being inthe topological phase, and the exact phase boundariesare plotted as solid black lines. As can be seen, the RFmodel makes predictions with high confidence deep inthe phases. When disorder is small, the predicted phaseboundaries match very well with the exact ones, whilethere are some deviations near the phase boundaries atlarge disorder. This gives us confidence that our modelgeneralizes to a broader range of parameters than thoseon which it has been trained.Since the robust properties of the ES are character-istic features in topological phases, especially in 1D, weexpect that the RF algorithm that we trained for themodel in Eq. (1) can be applied to a much broader set
FIG. 5. The predicted probability of being in the topologicalphase of Kitaev model in symmetry class BDI. of symmetry-protected topological phases. Furthermore,since the stable features of the ES - used to train our ma-chine learning algorithm - rely on global symmetries ofthe system, we expect to be able to observe and quantifythe breakdown of the method once symmetry-breakingeffects are present.As an example of the former, let us test the applicabil-ity of the RF model to another system. As an example,we apply our class AIII trained RF model to a disorderedfermionic Kitaev chain in class BDI, whose Hamiltonianis given by [28] H = (cid:88) n [ t n ib n a n +1 + m n ia n b n ] , (7)where a n and b n are Majorana fermions. We add disorderto the parameters t n = 1 + W ω and m n = m + W ω where t n and m n here can be interpreted as inter-cell andintra-cell Majorana coupilng terms. In the clean limitcorresponding to W = W = 0, this model is a one-dimensional topological superconductor with one (zero)isolated Majorana end state at each end for | t | > | m | ( | t | < | m | ). For the test region we chose the line W = 2 W = W and m = 0 .
5. The prediction results forthis model are shown in Fig. 5, where, similar to Fig. 2,the dots represent predicted probabilities of being in thetopological phase. The orange curve is fitted using Eq. 4,and we find a predicted phase transition near W = 4.The black dots are winding numbers calculated in realspace for the same system [6]. The transition point de-termined by the two methods are consistent with eachother.Now let us try to characterize some effects of symmetrybreaking for the AIII model. To illustrate this we add asmall σ z term that breaks the chiral symmetry that pro-tects the phase. We fix the other parameters as m = 0 . W = W = 1, so that the system is in topologicalphase when the symmetry is not broken, and add a termproportional to (cid:80) n c † n σ z c n to Eq. 1. Since the topo-logical phase is protected by chiral symmetry this term (a) (b) FIG. 6. (a) Entanglement spectrum with chiral symmetrybreaking term. (b) The probability of being in the topologicalphase with symmetry breaking term added to the system.The model predicts that all configurations are in the trivialphase except when symmetry is preserved. (a) (b)
FIG. 7. (a) Predictions using the simple classifier that checksfor degeneracy in the entanglement spectrum (orange dots)compared with random forest classifier (blue squares). Thegreen line is average of predictions for the degeneracy method.The analytically calculated phase transition point is indicatedby a vertical dashed line. (b) Distribution of predictions ofthe random forest model for two different testing data sets.When the test entanglement spectrum data has (no) doubledegeneracy, most predictions give probability one (zero) asbeing in the topological phase. immediately breaks down the topological phase and thedegeneracy in the entanglement spectrum is lifted. Asthe symmetry breaking term becomes stronger, the hith-erto degenerate states in the middle of the entanglementspectrum are split away from each other, as indicatedin Fig. 6(a). The resulting predictions made by the RFmodel are shown in Fig. 6(b), where the blue dots arethe raw prediction probabilities. In the absence of sym-metry breaking terms, the model confidently predicts atopological phase for these model parameters. On theother hand, as soon as the symmetry breaking strengthis non-zero, the predicted probability immediately dropsbelow 0.5, showing the sensitivity of the RF model tothe removal of the degeneracy in the ES. Furthermore,this probability goes down gradually as the symmetrybreaking strength increase.Finally, we can compare the predictions of the trainedRF model with the very simple classifier of just checkingfor the degeneracy of the mid-gap entanglement modes.To use the entanglement degeneracy to make predictionsof the phase we classify our data using a threshold andassociate gaps in the ES smaller than 0.001 to the topo-logical phase with probability 1, and larger gaps to bein the trivial phase with probability 1. We find that thepredicted results using this simple method give accuracy0.977, which is close to the accuracy of our random for-est classifier. We plot the predictions from degeneracyin Fig. 7(a) with orange dots. Since we can only predictone or zero, we take the average of the predictions as theprobability (green line). The predictions of the randomforest model are shown with blue squares for comparisonand the two predictions match extremely well.We can dig a bit deeper into understanding how the RFmodel is classifying based on the ES degeneracy. We plotthe distribution of random forest predictions in Fig. 7(b)for two sets of testing data: one has degeneracies in theES while the other does not. Note that the y-axis is cutin the middle to reveal details for smaller y values. Fromthe figure we can see that if the ES has degeneracy theRF model almost always predicts the state as topologi-cal (probability > . < .
5) inmost cases. Among the 3% incorrect predictions made byeither the RF model or the simple degeneracy classifier,84% are wrong by both models; 9% are wrong by the RFmodel but correct by degeneracy; and 7% are correct bythe RF model but wrong by degeneracy. Therefore, theRF predictions are consistent with the predictions by de-generacies. So, after careful investigation we find that theRF model is making predictions based on the mid-gap de-generacy of the ES. We expect that the training processis essentially finding the best threshold for the gap size.The threshold ends up being about 0.0015, which is closeto the value we set for our simple degeneracy classifier.
SUMMARY
In summary, we applied the random forest model toclassify disordered topological phases. Compared withthe linear model, random forest gives better predictions.On the other hand, it preserves the easy interpretabilityof the linear model as compared to neural networks. Be-cause of the generality of the entanglement spectrum, themodel trained on a small training dataset can be gener-alized to test data in a larger phase space, and even toother models in different symmetry classes. A closer lookat the RF model indicates that the model is capturingthe degeneracy of the mid-gap entanglement spectrummodes and is very sensitive to any symmetry breakingwhich splits the modes.
Acknowledgements:
This article is presented as amemorial tribute for Shou-Cheng Zhang and combinestwo of his recent interests of topological insulators andmachine learning. TLH thanks him for the advice and support given during their long period of collaboration.YZ and TLH acknowledge support from the US Na-tional Science Foundation under grant DMR 1351895-CAR. LHS is supported by a faculty startup at EmoryUniversity. [1] Andreas P Schnyder, Shinsei Ryu, Akira Furusaki, andAndreas WW Ludwig. Classification of topological insu-lators and superconductors in three spatial dimensions.
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