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Register a new userIdentifying topological order through unsupervised machine learning
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Added by
Joaquin Rodriguez-Nieva
Publication date
2018
fields
Physics
and research's language is
English
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The Landau description of phase transitions relies on the identification of a local order parameter that indicates the onset of a symmetry-breaking phase. In contrast, topological phase transitions evade this paradigm and, as a result, are harder to identify. Recently, machine learning techniques have been shown to be capable of characterizing topological order in the presence of human supervision. Here, we propose an unsupervised approach based on diffusion maps that learns topological phase transitions from raw data without the need of manual feature engineering. Using bare spin configurations as input, the approach is shown to be capable of classifying samples of the two-dimensional XY model by winding number and capture the Berezinskii-Kosterlitz-Thouless transition. We also demonstrate the success of the approach on the Ising gauge theory, another paradigmatic model with topological order. In addition, a connection between the output of diffusion maps and the eigenstates of a quantum-well Hamiltonian is derived. Topological classification via diffusion maps can therefore enable fully unsupervised studies of exotic phases of matter.
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Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have advantages of theoretical understandings of the expressive power using entanglement properties, and as a bridge connecting the classical computation and the quantum computation. Despite the great potential, however, existing tensor-network-based unsupervised models only work as a proof of principle, as their performances are much worse than the standard models such as the restricted Boltzmann machines and neural networks. In this work, we present the Autoregressive Matrix Product States (AMPS), a tensor-network-based model combining the matrix product states from quantum many-body physics and the autoregressive models from machine learning. The model enjoys exact calculation of normalized probability and unbiased sampling, as well as a clear theoretical understanding of expressive power. We demonstrate the performance of our model using two applications, the generative modeling on synthetic and real-world data, and the reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor-network-based models and the restricted Boltzmann machines, and is competitive with the state-of-the-art neural network models.
The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
The topological order is equivalent to the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice boson models. We identify multiple independent minimal entangled states (MESs) in the groundstate manifold on a torus. The extracted modular $mathcal{S}$ matrix from MESs faithfully demonstrates the Majorana quasiparticle or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings support that MESs manifest the eigenstates of quasiparticles for the non-Abelian topological states and encode the full information of the topological order.
In this paper we develop a new unsupervised machine learning technique comprised of a feature extractor, a convolutional autoencoder (CAE), and a clustering algorithm consisting of a Bayesian Gaussian mixture model (BGM). We apply this technique to visual band space-based simulated imaging data from the Euclid Space Telescope using data from the Strong Gravitational Lenses Finding Challenge. Our technique promisingly captures a variety of lensing features such as Einstein rings with different radii, distorted arc structures, etc, without using predefined labels. After the clustering process, we obtain several classification clusters separated by different visual features which are seen in the images. Our method successfully picks up $sim$63 percent of lensing images from all lenses in the training set. With the assumed probability proposed in this study, this technique reaches an accuracy of $77.25pm 0.48$% in binary classification using the training set. Additionally, our unsupervised clustering process can be used as the preliminary classification for future surveys of lenses to efficiently select targets and to speed up the labelling process. As the starting point of the astronomical application using this technique, we not only explore the application to gravitationally lensed systems, but also discuss the limitations and potential future uses of this technique.
We propose an experiment to identify the topological order of the $ u=frac{5}{2}$ state through a measurement of the electric conductance of a mesoscopic device. Our setup is based on interfacing $ u=2, frac{5}{2}$ and $3$ in the same device. Its conductance can unambiguously establish or rule out the particle-hole symmetric Pfaffian topological order, which is supported by recent thermal measurements. Additionally, it distinguishes between the Moore-Read and Anti-Pfaffian topological orders, which are favored by numerical calculations.
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