Machine learning the dynamics of quantum kicked rotor
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Machine learning the dynamics of quantum kicked rotor
Tomohiro Mano, Tomi Ohtsuki
Physics Division, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan
Abstract
Using the multilayer convolutional neural network (CNN), we can detect thequantum phases in random electron systems, and phase diagrams of two andhigher dimensional Anderson transitions and quantum percolations as well asdisordered topological systems have been obtained. Here, instead of using CNNto analyze the wave functions, we analyze the dynamics of wave packets vialong short-term memory network (LSTM). We adopt the quasi-periodic quan-tum kicked rotors, which simulate the three and four dimensional Andersontransitions. By supervised training, we let LSTM extract the features of thetime series of wave packet displacements in localized and delocalized phases.We then simulate the wave packets in unknown phases and let LSTM classifythe time series to localized and delocalized phases. We compare the phase dia-grams obtained by LSTM and those obtained by CNN.
Keywords:
Anderson transition, quantum phase transition, quantum kickedrotor, machine learning, convolutional neural network, long short-termmemory network
1. Introduction
Critical behaviors of the Anderson transition[1] have been attracting con-siderable attraction for more than half a century. The problem is related toquantum percolation[2, 3, 4, 5, 6], where the wave functions on the randomlyconnected lattice sites begin to be extended [7]. Electron states on random lat-tice systems are difficult to study, because the conventional methods of usingthe transfer matrix[8] are not applicable. The scaling analyses of the energylevel statistics[9] are also difficult, if not impossible[10, 11], owing to the spikydensity of states [6].To overcome these difficulties, we used neural networks[12, 13, 14, 15] to classify the states [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. That is, instead ofclassifying images of photos, we input the wave functions (actually the squaredmodulus of them) at the Fermi energy and classify them to metal, insulator,topological insulator etc. First we train the convolutional neural network in Email address: [email protected] (Tomi Ohtsuki)
Preprint submitted to Elsevier January 26, 2021 nderson model of localization, whose phase diagram is well known. We thenapply the CNN to classify the eigenfunctions of quantum percolation to metalor insulator relying on the generalization capability of CNN. We have shown inrefs. [27, 28] that this method is free from the above difficulties and works wellin determining the phase diagrams of quantum percolation.The above method, however, requires many eigenfunctions, which are dif- ficult to obtain in higher dimensions. One of the way to study the Andersontransition without diagonalizing the Hamiltonian is to use quantum kicked rotor(QKR) [29, 30, 31], where we analyze the wave packet dynamics in one dimen-sion. The simple quantum kicked rotor can be mapped to one-dimensional (1D)Anderson model[32], whereas with incommensurate modulation of the strengthof kick, the model is mapped to tight binding models in higher dimensions[33].In this paper, we draw phase diagrams of the three dimensional (3D) andfour dimensional (4D) tight binding models that correspond QKR using theCNN trained for standard Anderson models of localization. We also analyzethe time series of QKR via long short-term memory (LSTM) network[34, 35], let LSTM classify the time series to localized/delocalized phases, draw the phasediagrams, and compare them with those drawn by the CNN analyses of tightbinding models. We demonstrate that the phase boundaries of localized anddelocalized phases are less noisy in the case of LSTM.
2. Model and Method
We consider QKR with incommensurate modulation of the kick as follows; H ( t ) = p K cos x × X n δ ( t − n ) × F ( t ) , (1)with F ( t ) = (cid:26) ǫ cos( ω t + θ ) × cos( ω t + θ ) 3D1 + ǫ cos( ω t + θ ) × cos( ω t + θ ) × cos( ω t + θ ) 4D (2)where ω i are irrational numbers that are incommensurate with each other, K the strength of the kick, ǫ the modulation strength, and θ i the initial phases.We took ω = 2 π √ , ω = 2 π √
13 and ω = 2 π √
23 [29, 30, 31]. We analyze the QKR in two ways. One way is to map this model to tightbinding models, H tb = X m ε m | m ih m | + X m,r W r | m ih m − r | , (3)diagonalize H tb numerically to obtain the eigenfunctions, and let CNN deter-mine whether they are localized or delocalized. Details of the CNN method isreviewed in ref. [14]. Note that H tb is defined on 3D cubic lattice for the caseof two incommensurate frequencies ( ω and ω ), whereas it is on 4D hypercubiclattice in the case of three incommensurate frequencies ( ω , ω , and ω ). Note2lso that we use CNN that has been trained for Anderson models of localization[27, 14].The other way is to solve the time dependent Schr¨odinger equation i ¯ h ddt ψ ( t ) = H ( t ) ψ ( t ) , (4)with ¯ h set to 2.89[29, 30, 31], calculate the time dependence of “displacement”in momentum space, p ( t ) = h ψ ( t ) | p | ψ ( t ) i , (5)and analyze the time series of p ( t ) via LSTM. Figure 1: p ( t ) vs. t for various K and ǫ with two incommensurate frequencies (3D case).(a) the plot before normalization. p ( t ) is proportional to t for delocalized states, whereas itsaturates to finite values for localized states. (b) after normalizing p ( t ) to x ( t ) , the mean andstandard deviation of which are 0 and 1, respectively. We have calculated the wave packetdynamics up to T = 10 time steps, and recorded p ( t ) at every 50 time steps.
3. Results
We first apply the CNN trained for the Anderson model to the wave functionsobtained by diagonalizing Eq. (3). For 3D systems, the system size is 32 × ×
32, whereas for 4D it is 10 × × ×
10. We diagonalize the systems withperiodic boundary conditions, obtain the eigenfunctions at the center of theenergy spectrum, input squared modulus of the eigenfunctions to the CNN, andlet CNN calculate the probabilities for the inputs being delocalized. The results are shown in Fig. 2 (a) (3D) and (c) (4D) as a heat map.We next analyze the time series of p ( t ), Fig. 1. We first note that p ( t ) ∼ Dt delocalized ,ξ localized ,t /d critical , (6)3ith D the diffusion constant, ξ the localization length, and d the dimension. Atthe critical point, p ( t ) ∝ t /d , so for 3D p ( t ) ∝ t / and for 4D p ( t ) ∝ t / [36].Note that we discuss here the localization/delocalization in momentum space.(a) 3D CNN (b) 3D LSTM(c) 4D CNN (d) 4D LSTM Figure 2: Phase diagrams of QKR in ǫ - K plane. Those obtained by CNN[(a) and (c)] andthose by LSTM[(b) and (d)]. 3D cases [(a) and (b)] and 4D cases[(c) and (d)]. The CNN istrained by Anderson model of localization. The training regions of LSTM are indicated asgreen arrows. White crosses are obtained by the critical behaviors of p ( t ). We find the critical strength (
K, ǫ ) by detecting the behaviors p ( t ) ∝ t /d ,and use this information for supervised training of LSTM. The values of p ( t ),however, strongly depend on K and ǫ , and the neural network tends to learn onlymaxima and minima of p ( t ). We therefore preprocessed the data by normalizingthem, i.e., normalize p ( t ) to x ( t ) , whose mean and standard deviation are 0 and
1, respectively.We first determine the critical point along a straight line in ǫ - K plane byfinding a point that shows p ( t ) ∝ t /d (see white crosses in Fig. 2). Oncethe critical point is determined, we prepare time series p ( t ) for localized anddelocalized phases by varying ( K, ǫ ) along a straight line indicated by greenarrows in Fig. 2(b),(d). We then normalize p ( t ) to x ( t ) and use them fortraining bidirectional LSTM. Once the LSTM is trained, we vary parameters4n ǫ - K plane and calculate x ( t ) , and feed x ( t ) to LSTM, which outputs theprobability that the input time series x ( t ) belongs to the delocalized phase.The results are shown in Fig. 2(b) for 3D and Fig. 2(d) for 4D, which nicely distinguish localized and delocalized phases.Now we compare the phase diagrams (heat maps) for 3D and 4D systems. Inthe case of 3D, both CNN and LSTM give reasonably sharp phase boundaries[see Fig. 2(a), (b)]. On the other hand, in the case of 4D, the phase boundarybecomes noisy if we use 4D CNN [Fig. 2(c)]. This is because in the case of4D, only small system can be diagonalized, and CNN fail to learn the detailedfeatures of localized and delocalized states. In the case of LSTM [Fig. 2(d)], thephase boundary is less noisy, since we do not need to diagonalize the system,and we can follow as long time series as in 3D case.
4. Summary To summarize, we have analyzed the quantum kicked rotor with time mod-ulated kick strength. The systems are analyzed in two ways. One is to mapthe Hamiltonian to static higher dimensional tight binding models and studythe eigenfunctions via the deep convolutional neural network. The other is toanalyze the time series of the original time dependent one dimensional systemsvia bidirectional long short-term memory network. We have demonstrated thatthe latter approach gives less noisy phase boundary between the localized anddelocalized phases. The latter approach works especially well for analyzing theAnderson transitions in higher dimensions.
Acknowledgement
This work was supported by JSPS KAKENHI Grant Nos.
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