Machine learning Z2 quantum spin liquids with quasiparticle statistics

Abstract

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 (QLT), a notion we have recently introduced. Specifically, we propose a construction of QLT that is sensitive to quasi-particle statistics. We then use mutual statistics between the spinons and visons to detect a Z 2 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.

From DFT to machine learning: recent approaches to materials science–a review

“…Therefore, artificial-neuralnetwork quantum states are not necessarily needed to use ML for topological states. In the same spirit, Zhang et al showed that the phase boundary between the topological and trivial phases for the Z 2 quantum spin liquid can be identified by feed-forward neural networks by defining quantum loop topography sensitive to quasiparticle statistics [503].…”

“…The success of ML techniques in topological phase models aroused interest in experimentally fabricated systems, which in turn gave rise to the study of topological band insulators models [503], e.g., the Su-Schrieffer-Heeger model. Thus, using the Hamiltonian in the momentum space as an input of convolutional neuralnetworks, Zhang et al found a model for the topological invariant of general one-dimensional models, i.e., the winding number [503]. Although the winding number for a one-dimensional Hamiltonian k h ih x y , in [503], the authors found an equivalent neural-network-based expression for more general Hamiltonians.…”

“…Thus, using the Hamiltonian in the momentum space as an input of convolutional neuralnetworks, Zhang et al found a model for the topological invariant of general one-dimensional models, i.e., the winding number [503]. Although the winding number for a one-dimensional Hamiltonian k h ih x y , in [503], the authors found an equivalent neural-network-based expression for more general Hamiltonians. In other work, the authors extend this methodology for four-band insulators in AIII class and two-dimensional two-band insulators in A class, arguing that the output of some intermediate hidden layers leads to either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, respectively [503].…”

“…Although the winding number for a one-dimensional Hamiltonian k h ih x y , in [503], the authors found an equivalent neural-network-based expression for more general Hamiltonians. In other work, the authors extend this methodology for four-band insulators in AIII class and two-dimensional two-band insulators in A class, arguing that the output of some intermediate hidden layers leads to either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, respectively [503]. This suggests that neural networks can capture the mathematical formula of topological invariants.…”

Learning epidemic threshold in complex networks by Convolutional Neural Network

“…Recently, this topic has attracted much attention from physicists because of the strong performance of detecting, predicting and uncovering various phases of matter in quantum many-body systems [2][3][4][5][6][7][8][9][10][11][12][13]. Neural network based models play an irreplacable role in this field because they're able to not only learn generate phases or states of matter that are previously known [4,5,7,12] or uncovering phase transitions [2,3,6], but predict out-ofequilibrium phases of matter that have not been known yet [13]. Both Feed-Forward Neural Network (FFNN) and Convolutional Neural Network (CNN) have been ex-ploited to learn and discover the hidden pattern of phase transition in Ising-type lattice structures [7].…”