⇦Chinese Physics Letters, 2021, Vol. 38, No. 5, Article code 051101 Finding Short-Range Parity-Time Phase-Transition Points with a Neural Network Songju Lei (雷松炬)1, Dong Bai (柏栋)2*, Zhongzhou Ren (任中州)2,3*, and Mengjiao Lyu (吕梦蛟)4,5 Affiliations 1School of Physics, Nanjing University, Nanjing 210093, China 2School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 3Key Laboratory of Advanced Micro-Structure Materials (Ministry of Education), Shanghai 200092, China 4College of Science, Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing 210016, China 5Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT, Nanjing 211106, China Received 7 January 2021; accepted 11 March 2021; published online 2 May 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11535004, 11975167, 11761161001, 11375086, 11565010, 11881240623 and 11961141003), the National Key R&D Program of China (Grant Nos. 2018YFA0404403 and 2016YFE0129300), the Science and Technology Development Fund of Macau (Grant No. 008/2017/AFJ), and the Fundamental Research Funds for the Central Universities (Grant Nos. 22120210138 and 22120200101).
*Corresponding author. Email: dbai@tongji.edu.cn; zren@tongji.edu.cn Citation Text: Lei S J, Bai D, Ren Z Z, and Lyu M J 2021 Chin. Phys. Lett. 38 051101    Abstract The non-Hermitian $PT$-symmetric system can live in either unbroken or broken $PT$-symmetric phase. The separation point of the unbroken and broken $PT$-symmetric phases is called the $PT$-phase-transition point. Conventionally, given an arbitrary non-Hermitian $PT$-symmetric Hamiltonian, one has to solve the corresponding Schrödinger equation explicitly in order to determine which phase it is actually in. Here, we propose to use artificial neural network (ANN) to determine the $PT$-phase-transition points for non-Hermitian $PT$-symmetric systems with short-range potentials. The numerical results given by ANN agree well with the literature, which shows the reliability of our new method. DOI:10.1088/0256-307X/38/5/051101 © 2021 Chinese Physics Society Article Text Parity-time ($PT$) symmetric non-Hermitian Hamiltonians have nontrivial physical properties.[1] They could have either real spectra or complex spectra coming in conjugate pairs. The non-Hermitian systems are said to live in the unbroken $PT$-symmetric phase in the case of real spectra, while they are said to live in the broken $PT$-symmetric phase in the case of complex conjugate spectra. Conventionally, one has to solve the Schrödinger equation to determine which phase the non-Hermitian Hamiltonian lives in. It is interesting to develop alternative methods to identify different phases. In recent years, artificial intelligence has been extensively applied to physics research.[2–17] In this Letter, we design an artificial neural network (ANN) to predict whether a non-Hermitian $PT$-symmetric Hamiltonian is in unbroken or broken $PT$-symmetric phase without solving the corresponding Schrödinger equations explicitly. We assume the potential to be short-range. In other words, $V(x)$ vanishes as $|x|\rightarrow \infty$. For the parameterized non-Hermitian $PT$-symmetric Hamiltonians, we also use the trained ANN to determine the $PT$-phase transition point. This separates the unbroken and broken $PT$-symmetric phases and is an important physical quantity to characterize the parameterized non-Hermitian $PT$-symmetric Hamiltonians. The rest of this article is organized as follows. First, the $PT$-symmetric non-Hermitian quantum mechanics is introduced. Then, the ANN and its training process are discussed briefly. Furthermore, we verify the reliability of ANN. Finally, a summary is given, as well as some short remarks on some future directions. $PT$-Symmetric Quantum Mechanics. For a one-dimensional single-particle non-Hermitian Hamiltonian with a complex potential $V(x)=u(x)+iw(x)$, it is said to have the $PT$-symmetry if it satisfies[1] $$ u(x)=u(-x),~~~~w(-x)=-w(x).~~ \tag {1} $$ An early example of the non-Hermitian $PT$-symmetric Hamiltonians is $$ H=p^{2}+x^{2}(i x)^{\varepsilon}.~~ \tag {2} $$ Bender and Boettcher showed that the eigenvalues of this class of Hamiltonians are real, discrete, and positive for all $\varepsilon \geq 0$. When $\varepsilon < 0$, the eigenvalues of this class of Hamiltonians are, however, complex and come in conjugate pairs. Therefore, these Hamiltonians are said to live in the unbroken $PT$-symmetric phase for $\epsilon>0$ and in the broken $PT$-symmetric phase for $\epsilon < 0$; $\epsilon=0$ is referred to as the $PT$-phase-transition point. This picture is also applicable to other non-Hermitian $PT$-symmetric Hamiltonians. The $PT$-symmetry and the $PT$-phase-transition point have been studied from the experimental side. There are many optical systems that are suitable for realizing $PT$-symmetry in experiments, such as microcavities,[18] optically induced atomic lattices,[19] metamaterials,[20] and exciton-polariton condensates.[21–23] In particular, the $PT$-phase-transition point is observed in some optical waveguide and laser systems.[24–29] From the theoretical side, the $PT$-phase-transition points are conventionally determined by solving the Schrödinger equations directly. In this Letter, we propose an alternative approach based on ANN, which is a helpful complement to the conventional method.
cpl-38-5-051101-fig1.png
Fig. 1. Schematic description of how to generate the potential $u_{_{\scriptstyle \rm R}}(x)$ by the random walk process. Here $\delta x_{i}$ and $\delta u_{i}$ obey Gaussian distributions, and the purple dots represent the values of the random potential.
Artificial Neural Network. In this study, we attempt to determine the $PT$-phase-transition points using an ANN. We first discuss how the training data are prepared in our study. We generate a number of random short-range $PT$-symmetric potentials in the following forms: $$ u(x)=\begin{cases} u_{_{\scriptstyle \rm R}}(x), & -L < x < 0, \\ u_{_{\scriptstyle \rm R}}(x), & 0 < x < L, \\ \end{cases}~~ \tag {3} $$ and $$ w(x)=\begin{cases} -w_{_{\scriptstyle \rm R}}(x), & -L < x < 0, \\ w_{_{\scriptstyle \rm R}}(x), & 0 < x < L, \\ \end{cases}~~ \tag {4} $$ where $u(x)$ and $w(x)$ are nonzero only within the interval $[-L,L]$. The functions $u_{_{\scriptstyle \rm R}}(x)$ and $w_{_{\scriptstyle \rm R}}(x)$ are generated by using the random walk process. For example, $u_{_{\scriptstyle \rm R}}(x)$ is chosen to be $$ u_{_{\scriptstyle \rm R}}(x)=u_{i}, ~~{\rm for}~ x_{i} < x < {\min}\{x_{i+1}, L\},~~ \tag {5} $$ with $i=1,\ldots,n$. Here $u_{j}$ and $x_i$ are generated by the recurrence relations $u_{i}=u_{i-1}+\delta u_{i}$ and $x_{i}=x_{i-1}+\delta x_{i}$. The initial value $u_0$ at $x_0=0$ is chosen from some interval $[a,b]$ following the uniform distribution. $\delta x_{i}$ and $\delta u_{i}$ are all random variables satisfying the distributions $\theta(\delta x_{i}) N(\delta x_{i}, 4\,L / n, \sqrt{2} L /n)$ and $N(\delta u_{i}, 0, u_{0} / 10)$. $N(x, \mu, \sigma)$ is the normal distribution with the expectation $\mu$ and the standard deviation $\sigma$, and $\theta(x)$ is the Heaviside function. In Fig. 1, a schematic description is given on how to generate $u_{_{\scriptstyle \rm R}}(x)$ from the random walk process. In Fig. 2, we give a concrete example of the generated $u_{_{\scriptstyle \rm R}}(x)$. Similarly, we can also generate the imaginary part $w(x)$ of the complex potential $V(x)$. Using the algorithm given above, we generate $10^5$ random $PT$-symmetric potentials for the training set and $10^4$ $PT$-symmetric potentials for the test set. Their $PT$-symmetric phases are determined by solving the corresponding Schrödinger equations directly. The training and test data are labelled explicitly as ``unbroken $PT$ symmetry'' and ``broken $PT$ symmetry''. The structure of the ANN used in this study is shown in Fig. 3. We use the “scaled exponential linear units” (SELU) function as the activation function:[30] $$ \operatorname{selu}(x)=\begin{cases} \lambda x, & {\rm if }~ x>0, \\ \lambda\alpha e^{x} - \alpha, & {\rm if }~ x \le 0, \end{cases}~~ \tag {6} $$ with $\alpha=1.0507009873554804934193349852946$ and $\lambda=1.6732632423543772848170429916717$. The number of neurons in the input layer is $N_{\rm in}=258$. The number of neurons in each middle layer is simply set to be 50. An advantage of such an ANN is that it can give satisfying results within five epochs. When epochs are chosen to be greater than 30, the accuracy of the ANN could even be more than $99\%$. This suggests that ANN can be helpful in distinguishing whether a non-Hermitian $PT$-symmetric Hamiltonian is in the unbroken or broken $PT$-symmetric phase.
cpl-38-5-051101-fig2.png
Fig. 2. An example of random potentials obtained by random walk. Here, we choose $L=1$, $n=64$ for $u_{_{\scriptstyle \rm R}}(x)$ at $x > 0$. After interpolation, the random potential becomes smooth.
cpl-38-5-051101-fig3.png
Fig. 3. A demonstration of the ANN used in this study. There are four middle layers in our ANN.
Results. We test the reliability of our ANN in two examples. In the first example, we choose the following square well potential[31] $$\begin{align} &\operatorname{Re} V(x)=0,~~ x \in(-1,1), \\ &\operatorname{Im} V(x)=-g,~~ x \in(-l,0), \\ &\operatorname{Im} V(x)=g,~~ x \in(0,l).~~ \tag {7} \end{align} $$ This is closely related to the potential in Eq. (2) as $\varepsilon\to\infty$.[32–34] This potential is solvable. Its $PT$-phase-transition point is known to be $g_{\rm c} \approx 4.475$[35] when $l=1$. We start with $g=0$ and increase $g$ gradually. For each $g$, we determine the $PT$-symmetric phase using the ANN trained above. Initially, the Hamiltonian is in the unbroken $PT$-symmetric phase. The $PT$-phase-transition point is identified as the $g$ value where the ANN starts to output ``broken $PT$ symmetry''. The $PT$-phase-transition point is found to be $g_{\rm c}=4.48$ by our ANN, which is in good agreement with $g_{\rm c} \approx 4.475$ in literature. To verify the robustness of our ANN, we change the value of $l$ and calculate the corresponding $g_{\rm c}$. As shown in Table 1, the results of ANN are in good agreement with the results of Ref. [35]. The relationship between $l$ and $g_{\rm c}$ is also shown in Fig. 4. It is found that $g_{\rm c}$ increases rapidly as $l \rightarrow 0$.
Table 1. The $PT$-phase-transition point $g_{\rm c}$ for square-well potentials at different $l$ values. The second row gives the results of Ref. [35]. The third row gives the results of ANN.
$l$ 1 0.7 0.5 0.4 0.3 0.2
$g_{\rm c}$ (numerical results) 4.475 4.813 6.436 8.601 13.43 27.27
$g_{\rm c}$ (neural network) 4.48 4.81 6.35 8.81 13.39 26.51
Table 2. The critical value $V_{\rm 2c}$ for the Scarf-II potential with different $V_1$ values given by ANN. They are in good agreement with the theoretical results given by $V_{\rm 2c}=V_1+1/4$.[36]
$V_{1}$ 1 2 3 5 8 10
$V_{\rm 2c}$(neural network) 1.26 2.26 3.25 5.26 8.25 10.25
cpl-38-5-051101-fig4.png
Fig. 4. Numerical results for the $PT$-phase-transition points $g_{\rm c}$ given by ANN and Ref. [35]. The red dots are the results of Ref. [35]. The blue dots are the results of ANN.
In the second example, we choose the complex $PT$-symmetric Scarf-II potential $$\begin{alignat}{1} V(x)={}&-V_{1} \operatorname{sech}^{2} x+i V_{2} \operatorname{sech} x \tanh x, \\ & V_{1},V_{2}>0.~~ \tag {8} \end{alignat} $$ When $V_2$ exceeds the critical value $V_{\rm 2c}=V_1+1/4$ from below, there is a $PT$-phase-transition from the unbroken phase to the broken phase.[36] We fix $V_{1}=1,2,3,5,8,10$ and use our ANN to find the corresponding critical value $V_{\rm 2c}$. The results are shown in Table 2. The results of our ANN agree well with the theoretical results. In summary, it is straightforward for human eyes to determine whether a complex potential is $PT$-symmetric or not from its shape. However, it is less straightforward for human eyes to determine whether a $PT$-symmetric non-Hermitian potential lives in the unbroken or broken $PT$-symmetric phase. In this Letter, we show that this task could be handled easily with an ANN, which is an artificial counterpart of neural network in the human brain. We construct an ANN with four middle layers and prepare the training and test sets with help of the random walk process. We test the reliability of ANN with two examples: the square-well potential and the Scarf-II potential. It is found that the $PT$-phase-transition points given by ANN agree well with theoretical results in literature. Our method could also be applied to some other complex problems, such as the $PT$-symmetric non-Hermitian Bose–Hubbard system,[37] which is an interacting quantum system. It may be interesting to explore this direction in the future. References Real Spectra in Non-Hermitian Hamiltonians Having P T SymmetryMachine learning phases of matterRor2 signaling regulates Golgi structure and transport through IFT20 for tumor invasivenessPredictions of nuclear charge radii and physical interpretations based on the naive Bayesian probability classifierMachine Learning Phases of Strongly Correlated FermionsFinding Quantum Many-Body Ground States with Artificial Neural NetworkQuantum Loop Topography for Machine LearningMachine learning Z 2 quantum spin liquids with quasiparticle statisticsDeep Learning the Quantum Phase Transitions in Random Two-Dimensional Electron SystemsProbing many-body localization with neural networksKernel methods for interpretable machine learning of order parametersDiscovering phase transitions with unsupervised learningDetection of Phase Transition via Convolutional Neural NetworksLearning phase transitions by confusionActive learning algorithm for computational physicsA quantum Hopfield neural network model and image recognitionDrawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave FunctionsParity–time-symmetric whispering-gallery microcavitiesObservation of Parity-Time Symmetry in Optically Induced Atomic Lattices P T Metamaterials via Complex-Coordinate Transformation OpticsMultistability and condensation of exciton-polaritons below thresholdObservation of non-Hermitian degeneracies in a chaotic exciton-polariton billiardPermanent Rabi oscillations in coupled exciton-photon systems with PT -symmetryObservation of P T -Symmetry Breaking in Complex Optical PotentialsObservation of parity–time symmetry in opticsParity–time synthetic photonic latticesObservation of optical solitons in PT-symmetric latticesParity-time-symmetric microring lasers PT -symmetric laser absorberSelf-Normalizing Neural Networks-symmetric square wellSchrödinger operators with complex potential but real spectrumA family of complex potentials with real spectrumComplex square well - a new exactly solvable quantum mechanical model${\mathcal{PT}}$-SYMMETRIC SUPERSYMMETRY IN A SOLVABLE SHORT-RANGE MODELReal and complex discrete eigenvalues in an exactly solvable one-dimensional complex -invariant potentialMean-Field Dynamics of a Non-Hermitian Bose-Hubbard Dimer
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