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Physical review. B./Physical review. B

Deep learning for disordered topological insulators through their entanglement spectrum

Alejandro José Uría-Álvarez, Daniel Molpeceres-Mingo, Juan Jose Palacios

Published Apr 15, 2022• 8 citations
Publisher

Abstract

Calculation of topological invariants for crystalline systems is well understood in reciprocal space, allowing for the topological classification of a wide spectrum of materials. In this work, we present a technique based on the entanglement spectrum, which can be used to identify the hidden topology of systems without translational invariance. By training a neural network to distinguish between trivial and topological phases using the entanglement spectrum obtained from crystalline or weakly disordered phases, we can predict the topological phase diagram for generic disordered systems. This approach becomes particularly useful for gapless systems, while providing a computational speed-up compared to the commonly used Wilson loop technique for gapful situations. Our methodology is illustrated in two-dimensional models based on the Wilson-Dirac lattice Hamiltonian.

DOI
10.1103/physrevb.105.155128

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Calculation of topological invariants for crystalline systems is well understood in reciprocal space, allowing for the topological classification of a wide spectrum of materials. In this work, we present a technique based on the entanglement spectrum, which can be used to identify the hidden topology of systems without translational invariance. By training a neural network to distinguish between trivial and topological phases using the entanglement spectrum obtained from crystalline or weakly disordered phases, we can predict the topological phase diagram for generic disordered systems. This approach becomes particularly useful for gapless systems, while providing a computational speed-up compared to the commonly used Wilson loop technique for gapful situations. Our methodology is illustrated in two-dimensional models based on the Wilson-Dirac lattice Hamiltonian.
Calculation of topological invariants for crystalline systems is well understood in reciprocal space, allowing for the topological classification of a wide spectrum of materials. In this work, we present a new technique based on the entanglement spectrum, which can be used to identify the hidden topology of systems without translational invariance. By training a neural network to distinguish between trivial and topological phases using the entanglement spectrum obtained from crystalline or weakly disordered phases, we can predict the topological phase diagram for generic disordered systems. This approach becomes particularly useful for gapless systems, while providing a computational speed-up compared to the commonly used Wilson loop technique for gapful situations. Our methodology is illustrated in two-dimensional models based on the Wilson-Dirac lattice Hamiltonian.
Topological invariants allow one to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wave functions under twisted boundary conditions. However, those procedures do not allow one to calculate a topological invariant by evaluating the system locally, and thus require information about the wave functions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a one-dimensional topological superconductor and a two-dimensional quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the kernel polynomial method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.
Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge.The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network.When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of nonlocal properties.Here, we introduce quantum loop topography (QLT): a procedure of constructing a multidimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps.The loop configuration is guided by the characteristic response for defining the phase, which is Hall conductivity for the cases at hand.Feeding QLT to a fully connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish the Chern insulator and the fractional Chern insulator from trivial insulators with high fidelity.In addition to establishing the first case of obtaining a phase diagram with a topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.
The identification of topological phases in disordered systems is a significant field in condensed matter physics, where disorder breaks translational symmetry and invalidates conventional topological invariants defined in momentum space. Currently, machine learning offers promising alternatives in the identification of topological phases. In this work, by regarding the population dynamics as input data, we used feedforward neural networks (FNNs), vanilla recurrent neural networks (RNNs), and long short-term memory (LSTM) networks to identify the topology in the Su–Schrieffer–Heeger (SSH) model and the disordered SSH model, respectively. We also compared the identification capabilities of different neural networks using different input data. Our results show that FNN has the lowest training cost and a relatively high prediction accuracy. However, when increasing the time length and reducing the number of time points, vanilla RNN has higher prediction accuracy. Furthermore, we develop an interactive web-based tool, enabling real-time topological phase prediction based on user-specified parameters. This study not only lays the foundation for researchers to identify topology by using population dynamics as the input data of neural networks but also provides an accessible platform to support data-driven exploration of complex quantum phases.
In this Letter 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. We also make a couple of remarks regarding the role of the symmetry and the opposite effect of regularization techniques when applying machine learning to physical systems.
We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.