Session P34: Machine Learning in Condensed Matter Physics III

Machine learning quantum states and many-body entanglement

Recently, machine learning techniques have been introduced to many-body quantum condensed matter physics, raising considerable interest across different communities. In this talk, I will briefly introduce a neural-network representation of quantum many-body states and show that this representation can describe certain topological states in an exact and efficient fashion. I will talk about the entanglement properties, such as entanglement entropy and spectrum, of those quantum states that can be represented efficiently by neural networks. I will also show that neural networks can be used, through reinforcement learning, to solve a challenging problem of calculating the massively entangled ground state for a model Hamiltonian with long-range interactions.   

Restricted-Boltzmann-Machine Learning for Solving Hubbard and Heisenberg Models

We propose a versatile machine learning technique to construct accurate ground state wave functions of strongly-entangled spin/boson systems as well as fermionic lattice models [1]. We construct a variational wave function, which we call RBM+PP, by combining concepts from machine learning (restricted Boltzmann machine (RBM)) and physics (pair-product (PP) wave functions). The RBM is a type of artificial neural networks, allowing for a flexible and unbiased description of a wide variety of quantum states. The PP wave function or geminal wave function used in conventional wave-function methods such as the variational Monte Carlo (VMC) method, properly describes nonlocal entanglement, helping machine learning to learn many-body ground states more efficiently. Combined RBM+PP substantially improves accuracies of the RBM and VMC method applied separately in Heisenberg and Hubbard models. The high accuracy and flexible applicability of the RBP+PP wave function opens up a new route in the study of strongly-correlated systems. [1] Y. Nomura, A. Darmawan, Y. Yamaji, and M. Imada, arXiv:1709.06475.   

Applications of multilayer convolutional neural network to quantum phase transitions in disordered topological and non-topological systems

Quantum phase transitions in disordered topological and non-topological systems show rich phase diagrams. Examples of the phases are the band gap insulator, Anderson insulator, strong and weak topological insulators, Weyl semimetal, and diffusive metal. Here we obtain these phase diagrams using the machine learning method. We prepare thousands of wave functions in each phase, train the multi-layer convolutional neural network, and apply the neural network to determine the phase of the unknown systems from its wave function characteristics. Two- and three-dimensional Anderson transitions, as well as three dimensional topological insulators are discussed. We also compare this method with the conventional method.

[1] Tomoki Ohtsuki and Tomi Ohtsuki: J. Phys. Soc. Jpn. 85, 123706 (2016)
[2] Tomi Ohtsuki and Tomoki Ohtsuki: J. Phys. Soc. Jpn. 86, 044708 (2017)
[3] Tomohiro Mano and Tomi Ohtsuki: J. Phys. Soc. Jpn. 86, 113704 (2017)   

Machine Learning Entanglement Structure of Disordered Topological Phases and Competing Orders

The entanglement spectrum is expected to provide a characterization of topologically ordered systems beyond traditional order parameters. Nevertheless, so far attempts at accessing this information relied on the presence of translational symmetry. Here we introduce a framework for using a simple artificial neural network (ANN) to detect defining features of a fractional quantum Hall state, a charge density wave state and a localized state from entanglement spectra, even in the presence of disorder. We then successfully obtain a phase diagram for Coulomb-interacting electrons at fractional filling \nu = 1/3, perturbed by modified interactions and disorder. Our results bench-mark well against existing measures in parts of the phase space where such measures are available. Hence we explicitly establish a finite region of robust topological order. Moreover, we establish that the ANN can indeed access and learn defining traits of topological as well as broken symmetry phases using only the entanglement spectra of ground states as input.   

Machine Learning Disordered Topological Phases by Statistical Recovery of Symmetry

In this talk, we apply the artificial neural network (ANN) in a supervised manner to map out the quantum phase diagram of disordered topological superconductor in class DIII [1]. Given the disorder that keeps the discrete symmetries of the ensemble as a whole, translational symmetry which is broken in the quasiparticle distribution individually is recovered statistically by taking an ensemble average. This enables us to classify the phases by the ANN that learned the quasiparticle distribution in the clean limit. The consistency of the result with the calculation by the transfer matrix method is shown. If all three of the Z2, trivial, and the thermal metal (ThM) phases appear in the clean limit, the machine can classify them with high confidence over the entire phase diagram. If only the former two phases are present, we find that the machine remains confused in the certain region, leading us to conclude the detection of the unknown phase which is eventually identified as the ThM. In our method, only the first moment of the quasiparticle distribution is used for input, but application to a wider variety of systems is expected by including higher moments.

[1] N. Yoshioka, Y. Akagi, and H. Katsura, arXiv:1709.05790.
  

Machine Learning Topological Invariants with Neural Networks

We supervisedly train neural networks to distinguish different topological phases in the context of topological band insulators. After training with Hamiltonians of one-dimensional insulators with chiral symmetry, the neural network can predict their topological winding numbers with nearly 100% accuracy, even for Hamiltonians with larger winding numbers that are not included in the training data. These results show a remarkable success that the neural network can capture the global and nonlinear topological features of quantum phases from local inputs. By opening up the neural network, we confirm that the network does learn the discrete version of the winding number formula.   

Machine Learning Z2 Quantum Spin Liquids with Quasi-particle Statistics

After decades of progress and effort, obtaining a phase diagram for a strongly-correlated topological system still remains a challenge. Although in principle one could turn to Wilson loops and long-range entanglement, evaluating these non-local observables at many points in phase space can be prohibitively costly. With growing excitement over topological quantum computation comes the need for an efficient approach for obtaining topological phase diagrams. Here we turn to machine learning using quantum loop topography, a notion we have recently introduced. Specifically, we propose a construction of quantum loop topography that is sensitive to quasi-particle statistics. We then use mutual statistics between the spinons and visons to detect a Z₂ quantum spin liquid in a multi-parameter phase space. We successfully obtain the quantum phase boundary between the topological and trivial phases using a simple feed forward neural network. Furthermore we demonstrate advantages of our approach for the evaluation of phase diagrams relating to speed and storage. Such statistics-based machine learning of topological phases opens new efficient routes to studying topological phase diagrams in strongly correlated systems.   

Machine learning inverse problem for topological photonics.

The rapidly growing interest in the field of topological photonics is leading to the study of more and more complex structures to explore the properties of topological insulators.
This asks for a technique of properties computation speed and reliable. More challenging is to achieve an effective design methodology capable of solving the inverse problem in which desired optical properties result from topological characteristics.
Although various computational techniques are available for this process these tend to require specific implementations tailored to the task at hand.
Machine learning (ML) algorithms can give the right tools to do this and their recent growth in sophistication and application offers exciting perspectives on applying them to the field of topological photonics.
We apply ML techniques to solve the inverse problem of designing 1D photonic topological insulators, modelling the multidimensional nonlinear relationships among the structure parameters, whose custom tailoring can enable enhanced innovative applications to be realized.
We show that the architecture of ML with an appropriate label’s choice makes it possible to handle multivalued scenarios in different regimes of the modelling problem.   

Sparse Representation of Wannier functions from L1 regulariztion

Traditionally, Wannier functions are obtained by minimizing their spread functional with respect to the gauge of Bloch states, and therefore exponentially localized. We borrow the concept of sparsity from the LASSO method by adding an L₁ penalty term ∑_i ∫_v |W_i(r)| dr to the total energy minimization and achieve a sparse representation of Wannier functions, which means they are nonzero only within a finite spatial region. The exponentially localized representation will be the limit of this sparse one when the weight of L₁ term approaches zero. Our method is fully k-separable and works equally for both insulators and metals, as evidenced by our calculation on silicon, copper and SnSe. No disentanglement procedure is required for entangled bands. First order algorithms to tackle this optimization problem will be discussed.   

Free Energy–Based Reinforcement Learning Using Quantum Monte Carlo and Quantum Annealing

We investigate whether quantum or numerical sampling from select many-body systems can be used to improve upon classical methods in reinforcement learning. We introduce free energy–based reinforcement learning (FERL) as an application of such sampling methods. In our experiments, we focus on transverse field Ising spin Hamiltonians with layouts of qubits similar to that of deep Boltzmann machines (DBM), and use simulated quantum annealing (SQA) as a subroutine of a reinforcement learning framework. In the absence of a transverse field, the DBMs train more effectively than restricted Boltzmann machines with the same number of weights. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field. This further improves the DBM-based reinforcement learning method. To perform physical experiments, we propose a method for processing a quantum annealer’s measured qubit spin configurations in approximating the free energy of a QBM. We then apply this method to perform reinforcement learning on the grid-world problem using the D-Wave 2000Q quantum annealer. The experimental results show that our technique is a promising method for harnessing the power of quantum sampling in reinforcement learning tasks.   

Entanglement Entropy From Tensor Network States for Stabilizer Codes

We present the construction of tensor network states (TNS) for some of the degenerate ground states of 3D stabilizer codes, and use the TNS formalism to obtain the entanglement spectrum and entropy of these ground-states for some special cuts. In particular, we work out the examples of the 3D toric code, the X-cube model and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: for these, the constructed TNS is the singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground state degeneracy. We show that the transfer matrices are projectors. The number of nonzero eigenvalues is tightly related to the ground state degeneracy.   

Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter

We study the entanglement entropy of gapped phases of matter in three spatial dimensions. We focus in particular on size-independent contributions to the entropy across entanglement surfaces of arbitrary topologies. We show that for low energy fixed-point theories, the constant part of the entanglement entropy across any surface can be reduced to a linear combination of the entropies across a sphere and a torus. We first derive the constant part of the entanglement entropy of the fixed-point models across arbitrary entanglement surfaces, and identify the topological contribution by considering the renormalization group flow; in this way we give an explicit definition of topological entanglement entropy in (3+1)D, which sharpens previous results. We illustrate our results using several concrete examples and independent calculations, and show adding ``twist'' terms to the Lagrangian can change $S_{\mathrm{topo}}$ in (3+1)D. For the generalized Walker-Wang models, we find that the ground state degeneracy on a 3-torus is given by $\exp(-3S_{\mathrm{topo}}[T^2])$ in terms of the topological entanglement entropy across a 2-torus.